Number theory Books

Book SynopsisGeodesic and Horocyclic Trajectories presents an introduction to the topological dynamics of two classical flows associated with surfaces of curvature −1, namely the geodesic and horocycle flows.Table of ContentsDynamics of Fuchsian groups.- Examples of Fuchsian Groups.- Topological dynamics of the geodesic flow.- Schottky groups.- Topological dynamics.- The Lorentzian point of view.- Trajectories and Diophantine approximations.

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

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

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

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

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

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

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

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

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

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

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

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

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Book SynopsisG. H. Hardy ranks among the great mathematicians of the twentieth century, doing essential research in number theory and analysis. This book is a feast of Hardy's writing, featuring articles ranging from the serious to the humorous. The G. H. Hardy Reader is a worthy introduction to an extraordinary individual.Trade Review'The editors are to be congratulated on putting together this beautiful 'reader' with material from so many different sources, which illustrates so well the life, character and work of one of the great mathematicians of the twentieth century, Godfrey Harold Hardy (1877-1947). Even if you are familiar with Hardy's masterpiece A Mathematician's Apology or his book on Ramanujan, Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work you will find a wealth of new and fascinating material in this 'reader' about Hardy.' Kenneth S. Williams, Canadian Mathematical Society NotesTable of ContentsPart I. Biography: 1. Hardy's life; 2. The letter from Ramanujan to Hardy, 16 January 1913; 3. A letter from Bertrand Russell to Lady Ottoline Morrell, 2 February 1913; 4. The Indian mathematician Ramanujan; 5. Epilogue from the man who knew infinity; 6. Posters of 'Hardy's years at Oxford'; 7. A glimpse of J. E. Littlewood; 8. A letter from Freeman Dyson to C. P. Snow, 22 May 1967, and two letters from Hardy to Dyson; 9. Miss Gertrude Hardy; Part II. Writings by and about G. H. Hardy: 10. Hardy on writing books; 11. Selections from Hardy's writings; 12. Selections from what others have said about Hardy; Part III. Mathematics: 13. An introduction to the theory of numbers; 14. Prime numbers; 15. The theory of numbers; 16. The Riemann zeta-function and lattice point problems; 17. Four Hardy gems; 18. What is geometry?; 19. The case against the mathematical tripos; 20. The mathematician on cricket; 21. Cricket for the rest of us; 22. A mathematical theorem about golf; 23. Mathematics in war-time; 24. Mathematics; 25. Asymptotic formulæ in combinatory analysis (excerpts) with S. Ramanujan; 26. A new solution of Waring's problem (excerpts), with J. E. Littlewood; 27. Some notes on certain theorems in higher trigonometry; 28. The Integral _∞0sin xx dx and further remarks on the integral _∞0sin xx dx; Part IV. Tributes: 29. Dr. Glaisher and the 'messenger of mathematics'; 30. David Hilbert; 31. Edmund Landau (with H. Heilbronn); 32. Gösta Mittag-Leffler; Part V. Book Reviews: 33. Osgood's calculus and Johnson's calculus; 34. Hadamard: the psychology of invention in the mathematical field; 35. Hulburt: differential and integral calculus; 36. Bôcher: an introduction to the study of integral equations.

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Book SynopsisAn Introduction to Cryptography.- Discrete Logarithms and Diffie Hellman.- Integer Factorization and RSA.- Combinatorics, Probability and Information Theory.- Elliptic Curves and Cryptography.- Lattices and Cryptography.- Digital Signatures.- Additional Topics in Cryptography.Trade ReviewFrom the reviews: "The book is devoted to public key cryptography, whose principal goal is to allow two or more people to exchange confidential information … . The material is very well organized, and it is self-contained: no prerequisites in higher mathematics are needed. In fact, everything is explained and carefully covered … . there is abundance of examples and proposed exercises at the end of each chapter. … This book is ideal as a textbook for a course aimed at undergraduate mathematics or computer science students." (Fabio Mainardi, The Mathematical Association of America, October, 2008) "This book focuses on public key cryptography … . Hoffstein, Pipher, and Silverman … provide a thorough treatment of the topics while keeping the material accessible. … The book uses examples throughout the text to illustrate the theorems, and provides a large number of exercises … . The volume includes a nice bibliography. … Summing Up: Highly recommended. Upper-division undergraduate through professional collections." (C. Bauer, Choice, Vol. 46 (7), March, 2009) "For most undergraduate students in mathematics or computer science (CS), mathematical cryptography is a challenging subject. … it is written in a way that makes you want to keep reading. … The authors officially targeted the book for advanced undergraduate or beginning graduate students. I believe that this audience is appropriate. … it could even be used with students who are just learning how to execute rigorous mathematical proofs. … I strongly believe that it finds the right tone for today’s students … ." (Burkhard Englert, ACM Computing Reviews, March, 2009) "The exercises and text would make an excellent course for undergraduate independent study. … This is an excellent book. Hoffstein, Pipher and Silverman have written as good a book as is possible to explain public key cryptography. … This book would probably be best suited for a graduate course that focused on public key cryptography, for undergraduate independent study, or for the mathematician who wants to see how mathematics is used in public key cryptography." (Jintai Ding and Chris Christensen, Mathematical Reviews, Issue 2009 m)Table of ContentsAn Introduction to Cryptography.- Discrete Logarithms and Diffie-Hellman.- Integer Factorization and RSA.- Probability Theory and Information Theory.- Elliptic Curves and Cryptography.- Lattices and Cryptography.- Digital Signatures.- Additional Topics in Cryptology.

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Book Synopsis1 The Romance of Numbers.- 2 Figures from Figures: Doing Arithmetic and Algebra by Geometry.- 3 What Comes Next?.- 4 Famous Families of Numbers.- 5 The Primacy of Primes.- 6 Further Fruitfulness of Fractions.- 7 Geometric Problems and Algebraic Numbers.- 8 Imagining Imaginary Numbers.- 9 Some Transcendental Numbers.- 10 Infinite and Infinitesimal Numbers.Trade ReviewFrom the reviews: "This is a really fascinating book either to read or to browse in, or for reference - there is a good index, and I can strongly recommend it - it should be in every school and college library!" The Mathematical Gazette "… A delightful look at numbers and their roles in everything from language to flowers to the imagination." Science News "… The great feature of the book is that anyone can read it without excessive head scratching … You'll find plenty here to keep you occupied, amused, and informed. Buy, dip in, wallow." New ScientistTable of Contents1. The Romance of Numbers 2. Figures from Figures Doing Arithmetic and Algebra by Geometry 3. What Comes Next? 4. Famous Families of Numbers 5. The Primacy of Primes 6. Further Fruitfulness of Fractions 7. Geometric Problems and Algebraic Numbers 8. Imagining Imaginary Numbers 9. Some Transcendental Numbers 10. Infinite and Infinitesimal Numbers

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Book SynopsisPreface to the Second Edition.- Preface to the First Edition.- 1. Logic.- 2. Sets.- 3. Functions.- 4. Finite and Infinite Sets.- 5. Combinatorics.- 6. Number Theory.- 7. Complex Numbers.- Hints and Partial Solutions to Selected Odd-Numbered Exercises.- Index.Table of Contents-Preface.- 1. Logic.- 2. Sets.- 3. Functions.- 4. Finite and Infinite Sets. - 5. Permutations and Combinations.- 6. Number Theory.- 7. Complex Numbers.- Hints and Partial Solutions to Selected Odd-Numbered Exercises.- Index

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Trade Review“This book is an outstanding book on counting integer points of polytopes … . The book contains lots of exercises with very helpful hints. Another essential feature of the book is a vast collection of open problems on different aspects of integer point counting and related areas. … The book is reader-friendly written, self-contained and contains numerous beautiful illustrations. The reader is always accompanied with deep research jokes by famous researchers and valuable historical notes.” (Oleg Karpenkov, zbMATH 1339.52002, 2016)Reviews of the first edition:“You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics.”— MAA Reviews“The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the material, exercises, open problems and an extensive bibliography.”— Zentralblatt MATH“This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron.”— Mathematical Reviews“Many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying way. Beck and Robins have written the perfect text for such a course.”— CHOICETable of ContentsPreface.- The Coin-Exchange Problem of Frobenius.- A Gallery of Discrete Volumes.- Counting Lattice Points in Polytopes: The Ehrhart Theory.- Reciprocity.- Face Numbers and the Dehn-Sommerville Relations in Ehrhartian Terms.- Magic Squares.- Finite Fourier Analysis.- Dedekind Sums.- The Decomposition of a Polytope into Its Cones.- Euler-MacLaurin Summation in Rd.- Solid Angles.- A Discrete Version of Green's Theorem Using Elliptic Functions.- Appendix A: Triangulations of Polytopes.- Appendix B: Hints for Selected Exercises.- References.- Index.- List of Symbols.-

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Book SynopsisThe Coin-Exchange Problem of Frobenius.- A Gallery of Discrete Volumes.- Counting Lattice Points in Polytopes: The Ehrhart Theory.- Reciprocity.- Face Numbers and the DehnSommerville Relations in Ehrhartian Terms.- Magic Squares.- Finite Fourier Analysis.- Dedekind Sums.- Zonotopes.- h-Polynomials and h*-Polynomials.- The Decomposition of a Polytope Into Its Cones.- EulerMaclaurin Summation in Rd.- Solid Angles.- A Discrete Version of Green's Theorem Using Elliptic Functions.Trade Review“This book is an outstanding book on counting integer points of polytopes … . The book contains lots of exercises with very helpful hints. Another essential feature of the book is a vast collection of open problems on different aspects of integer point counting and related areas. … The book is reader-friendly written, self-contained and contains numerous beautiful illustrations. The reader is always accompanied with deep research jokes by famous researchers and valuable historical notes.” (Oleg Karpenkov, zbMATH 1339.52002, 2016)Reviews of the first edition:“You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics.”— MAA Reviews“The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the material, exercises, open problems and an extensive bibliography.”— Zentralblatt MATH“This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron.”— Mathematical Reviews“Many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying way. Beck and Robins have written the perfect text for such a course.”— CHOICETable of ContentsPreface.- The Coin-Exchange Problem of Frobenius.- A Gallery of Discrete Volumes.- Counting Lattice Points in Polytopes: The Ehrhart Theory.- Reciprocity.- Face Numbers and the Dehn-Sommerville Relations in Ehrhartian Terms.- Magic Squares.- Finite Fourier Analysis.- Dedekind Sums.- The Decomposition of a Polytope into Its Cones.- Euler-MacLaurin Summation in Rd.- Solid Angles.- A Discrete Version of Green's Theorem Using Elliptic Functions.- Appendix A: Triangulations of Polytopes.- Appendix B: Hints for Selected Exercises.- References.- Index.- List of Symbols.-

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Book SynopsisPreface.- Introduction.- 1 An Introduction to Cryptography.- 2 Discrete Logarithms and Diffie-Hellman.- 3 Integer Factorization and RSA.- 4 Digital Signatures.- 5 Combinatorics, Probability, and Information Theory.- 6 Elliptic Curves and Cryptography.- 7 Lattices and Cryptography.- 8 Additional Topics in Cryptography.- List of Notation.- References.- Index.Trade Review“This book explains the mathematical foundations of public key cryptography in a mathematically correct and thorough way without omitting important practicalities. … I would like to emphasize that the book is very well written and quite clear. Topics are well motivated, and there are a good number of examples and nicely chosen exercises. To me, this book is still the first-choice introduction to public-key cryptography.” (Klaus Galensa, Computing Reviews, March, 2015)“This is a text for an upper undergraduate/lower graduate course in mathematical cryptography. … It is very well written and quite clear. Topics are well-motivated, and there are a good number of examples and nicely chosen exercises. … An instructor of a fairly sophisticated undergraduate course in cryptography who wants to emphasize public key cryptography should definitely take a look at this book.” (Mark Hunacek, MAA Reviews, October, 2014)Table of ContentsPreface.- Introduction.- 1 An Introduction to Cryptography.- 2 Discrete Logarithms and Diffie-Hellman.- 3 Integer Factorization and RSA.- 4 Digital Signatures.- 5 Combinatorics, Probability, and Information Theory.- 6 Elliptic Curves and Cryptography.- 7 Lattices and Cryptography.- 8 Additional Topics in Cryptography.- List of Notation.- References.- Index.

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Book SynopsisThis proceedings volume contains articles related to the research presented at the 2017 Simons Symposium on p-adic Hodge theory. This symposium was focused on recent developments in p-adic Hodge theory, especially those concerning integral questions and their connections to notions in algebraic topology. This volume features original research articles as well as articles that contain new research and survey some of these recent developments. It is the first of three volumes dedicated to p-adic Hodge theory.Table of ContentsNotes on the Ainf-cohomology of Integral p-adic Hodge theory (M. Morrow).- On the cohomology of the affine space (P. Colmez, W. Nizioł).- Arithmetic Chern-Simons Theory II (H.-J. Chung, D. Kim, M. Kim, J. Park, H. Yoo).- Some ring-theoretic properties of Ainf (K.S. Kedlaya).- Sure une q-déformation locale de la théorie de Hodge non-abélienne en caractéristique positive (M. Gros).- Crystalline Zp-representations and Ainf-representations with Frobenius (T. Tsuji).

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Book SynopsisThis graduate textbook offers an introduction to modern methods in number theory. It gives a complete account of the main results of class field theory as well as the Poitou-Tate duality theorems, considered crowning achievements of modern number theory.Assuming a first graduate course in algebra and number theory, the book begins with an introduction to group and Galois cohomology. Local fields and local class field theory, including Lubin-Tate formal group laws, are covered next, followed by global class field theory and the description of abelian extensions of global fields. The final part of the book gives an accessible yet complete exposition of the Poitou-Tate duality theorems. Two appendices cover the necessary background in homological algebra and the analytic theory of Dirichlet L-series, including the Čebotarev density theorem. Based on several advanced courses given by the author, this textbook has been written for graduate students. Including complete proofs and numerous exercises, the book will also appeal to more experienced mathematicians, either as a text to learn the subject or as a reference.Trade Review“This book is a very good textbook for studying important roles of Galois cohomology in algebraic number theory … . This book consists of four parts, and the chapters are well constructed. … Examples and exercises are also well selected and interesting. This English translation is also very useful not only for students but also for researchers who study algebraic number theory in modern mathematical language, such as that of category theory or homological algebra.” (Yasushi Mizusawa, Mathematical Reviews, April, 2022)“For students with no prior understanding of class field theory, the book is ideal. It is self-contained, and is based on a concatenation of master’s level courses given by the author. … his book seamlessly stitches together all the components in a neat and lucid manner. The whole process of learning this classical theory from Harari’s book makes it a painless and enjoyable experience. … The author takes a lot of care to make illuminating remarks in each chapter …” (Balasubramanian Sury, zbMATH 1466.11086, 2021)Table of ContentsPreface.- ​Part I Group cohomology and Galois cohomology: generalities.- 1 Cohomology of finite groups.- 2 Cohomology of cyclic groups.- 3 p-groups, the Tate-Nakayama theorem.- 4 Cohomology of profinite groups.- 5 Cohomological dimension.- 6 First notions of Galois cohomology.- Part II Local fields.- 7 Basic facts about local fields.- 8 Brauer group of a local field.- 9 Local class field theory: the reciprocity law.- 10 The Tate local duality theorem.- 11 Local class field theory: Lubin-Tate theory.- Part III Global fields.- 12 Basic facts about global fields.- 13 Cohomology of the idèles.- 14 Reciprocity law.- 15 The abelianized absolute Galois group of a global field.- Part IV Duality theorems.- 16 Class formations.- 17 Poitou-Tate duality.- 18 Some applications.- Appendix.- A Some results from homological algebra.- B A survey of analytic methods.- References.- Index.

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Book SynopsisBridging the gap between novice and expert, the aim of this book is to present in a self-contained way a number of striking examples of current diophantine problems to which Arakelov geometry has been or may be applied. Arakelov geometry can be seen as a link between algebraic geometry and diophantine geometry. Based on lectures from a summer school for graduate students, this volume consists of 12 different chapters, each written by a different author. The first chapters provide some background and introduction to the subject. These are followed by a presentation of different applications to arithmetic geometry. The final part describes the recent application of Arakelov geometry to Shimura varieties and the proof of an averaged version of Colmez's conjecture. This book thus blends initiation to fundamental tools of Arakelov geometry with original material corresponding to current research. This book will be particularly useful for graduate students and researchers interested in the connections between algebraic geometry and number theory. The prerequisites are some knowledge of number theory and algebraic geometry.Table of Contents- Introduction. - Part A Concepts of Arakelov Geometry. - Chapter I: Arithmetic Intersection. - Chapter II: Minima and Slopes of Rigid Adelic Spaces. - Chapter III : Introduction aux théorèmes de Hilbert-Samuel arithmétiques. - Chapter IV: Euclidean Lattices, Theta Invariants, and Thermodynamic Formalism. - Part B Distribution of Rational Points and Dynamics. - Chapter V: Beyond Heights: Slopes and Distribution of Rational Points. - Chapter VI: On the Determinant Method and Geometric Invariant Theory. - Chapter VII: Arakelov Geometry, Heights, Equidistribution, and the Bogomolov Conjecture. - Chapter VIII : Autour du théorème de Fekete-Szeg˝o. - Chapter IX: Some Problems of Arithmetic Origin in Rational Dynamics. - Part C Shimura Varieties. - Chapter XI: The Arithmetic Riemann–Roch Theorem and the Jacquet–Langlands Correspondence. - Chapter XII: The Height of CM Points on Orthogonal Shimura Varieties and Colmez’s Conjecture.

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Book SynopsisThis textbook offers a unique exploration of analytic number theory that is focused on explicit and realistic numerical bounds. By giving precise proofs in simplified settings, the author strategically builds practical tools and insights for exploring the behavior of arithmetical functions. An active learning style is encouraged across nearly three hundred exercises, making this an indispensable resource for both students and instructors. Designed to allow readers several different pathways to progress from basic notions to active areas of research, the book begins with a study of arithmetic functions and notions of arithmetical interest. From here, several guided “walks” invite readers to continue, offering explorations along three broad themes: the convolution method, the Levin–Faĭnleĭb theorem, and the Mellin transform. Having followed any one of the walks, readers will arrive at “higher ground”, where they will find opportunities for extensions and applications, such as the Selberg formula, Brun’s sieve, and the Large Sieve Inequality. Methodology is emphasized throughout, with frequent opportunities to explore numerically using computer algebra packages Pari/GP and Sage. Excursions in Multiplicative Number Theory is ideal for graduate students and upper-level undergraduate students who are familiar with the fundamentals of analytic number theory. It will also appeal to researchers in mathematics and engineering interested in experimental techniques in this active area.Trade Review“The book is well designed for use either in a classroom or for independent study. … References are well documented and provided at the end of every chapter. Additionally, the author regularly offers suggestions for further reading for more comprehensive dives into the topics.” (Matthew Dolan Jobrack, Mathematical Reviews, November, 2023)“What a wonderful book! If you’re a number theorist with a slight aversion to the more technical parts of analytic number theory, then this book is the proper remedy.” (Franz Lemmermeyer, zbMATH 1496.11003, 2022)“It does touch on a wealth of topics and techniques. … The book is easy to read. … the book is thoroughly footnoted, including references to the original papers and modern expositions;” (Allen Stenger, MAA Reviews, May 9, 2022)Table of ContentsApproach: Multiplicativity.- Arithmetic Convolution.- A Calculus on Arithmetical Functions.- Analytical Dirichlet Series.- Growth of Arithmetical Functions.- An "Algebraical" Multiplicative Function.- Möbius Inversions.- The Convolution Walk.- Handling a Smooth Factor.- The Convolution Method.- Euler Products and Euler Sums.- Some Practice.- The Hyperbola Principle.- The Levin-Fanleib Walk.- The Mertens Estimates.- The Levin-Fanleib Theorem.- Variations on a Theme of Chebyshev.- Primes in progressions.- A famous constant.- Euler Products with Primes in AP.- Chinese Remainder and Multiplicativity.- The Mellin Walk.- The Riemann zeta-function.- The Mellin Transform.- Proof Theorem ℓ.- Roughing up: Removing a Smoothening.- Proving the Prime Number Theorem.- Higher Ground: Applications / Extensions.- The Selberg Formula.- Rankin's Trick and Brun's Sieve.- Three Arithmetical Exponential Sums.- Convolution method / Möbius function.- The Large Sieve Inequality.- Montgomery's Sieve.

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Book Synopsis​This book discusses the p-adic modular forms, the eigencurve that parameterize them, and the p-adic L-functions one can associate to them. These theories and their generalizations to automorphic forms for group of higher ranks are of fundamental importance in number theory.For graduate students and newcomers to this field, the book provides a solid introduction to this highly active area of research. For experts, it will offer the convenience of collecting into one place foundational definitions and theorems with complete and self-contained proofs.Written in an engaging and educational style, the book also includes exercises and provides their solution.Trade Review“This book represented hope. If I read it carefully, maybe I would finally get to know what they were all talking about, and gain some real insight into what are obviously very important and influential ideas. While I cannot claim to be an expert by now, my first skim through, skipping all the exercises, has provided me with a satisfying foundation, and I found that revisited passages responded well to a second reading to consolidate what I had learned.” (Neil P. Dummigan, Mathematical Reviews, May, 2023)“Complete proofs (or detailed references) of all statements are given and many exercises (with their solutions or hints) are included, hence the book may be addressed to graduate students working in this beautiful area of number theory and arithmetic algebraic geometry. This is a welcome addition to the literature in a field.” (Andrzej Dąbrowski, zbMATH 1493.11002, 2022)Table of Contents- Introduction.- Part I The ‘Eigen’ Construction.- Eigenalgebras.- Eigenvarieties.- Part II Modular Symbols and L-Functions.- Abstract Modular Symbols.- Classical Modular Symbols, Modular Forms, L-functions.- Rigid Analytic Modular Symbols and p-Adic L-functions.- Part III The Eigencurve and its p-Adic L-Functions.- The Eigencurve of Modular Symbols.- p-Adic L-Functions on the Eigencurve.- The Adjoint p-Adic L-Function and the Ramification Locus of the Eigencurve.- Solutions and Hints to Exercises.

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Book SynopsisThis undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level. Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study. Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.Trade Review“The book is very nicely written and the original style and choices of the topics make it agreeable reading, and might well complement and motivate the study of other classical introductions to the theory of more general number fields.” (Alessandro Cobbe, zbMATH 1498.11003, 2022)Table of Contents1. Prehistory.- 2 Quadratic Number Fields.- 3 The Modularity Theorem.- 4 Divisibility in Integral Domains.- 5 Arithmetic in some Quadratic Number Fields.- 6 Ideals in Quadratic Number Fields.- 7 The Pell Equation.- 8 Catalan's Equation.- 9 Ambiguous Ideal Classes and Quadratic Reciprocity.- 10 Quadratic Gauss Sums.- A Computing with Pari and Sage.- B Solutions.- Bibliography.- Name Index.- Subject Index.

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Book SynopsisMahler measure, a height function for polynomials, is the central theme of this book. It has many interesting properties, obtained by algebraic, analytic and combinatorial methods. It is the subject of several longstanding unsolved questions, such as Lehmer’s Problem (1933) and Boyd’s Conjecture (1981). This book contains a wide range of results on Mahler measure. Some of the results are very recent, such as Dimitrov’s proof of the Schinzel–Zassenhaus Conjecture. Other known results are included with new, streamlined proofs. Robinson’s Conjectures (1965) for cyclotomic integers, and their associated Cassels height function, are also discussed, for the first time in a book.One way to study algebraic integers is to associate them with combinatorial objects, such as integer matrices. In some of these combinatorial settings the analogues of several notorious open problems have been solved, and the book sets out this recent work. Many Mahler measure results are proved for restricted sets of polynomials, such as for totally real polynomials, and reciprocal polynomials of integer symmetric as well as symmetrizable matrices. For reference, the book includes appendices providing necessary background from algebraic number theory, graph theory, and other prerequisites, along with tables of one- and two-variable integer polynomials with small Mahler measure. All theorems are well motivated and presented in an accessible way. Numerous exercises at various levels are given, including some for computer programming. A wide range of stimulating open problems is also included. At the end of each chapter there is a glossary of newly introduced concepts and definitions. Around the Unit Circle is written in a friendly, lucid, enjoyable style, without sacrificing mathematical rigour. It is intended for lecture courses at the graduate level, and will also be a valuable reference for researchers interested in Mahler measure. Essentially self-contained, this textbook should also be accessible to well-prepared upper-level undergraduates.Trade Review“The reader at the graduate level having enough time and energy can learn a lot from this book about the Mahler measure, conjugate sets of algebraic integers, and related results. Some chapters of the book are quite accessible to undergraduate students as well, and may serve as an introduction to their research in this area.” (Arturas Dubickas, Mathematical Reviews, May, 2023)“It contains some material that is unavailable elsewhere. Each chapter is concluded by notes and a glossary of newly introduced definitions. … The reader at the graduate level having enough time and energy from this book can learn a lot about the Mahler measure, conjugate sets of algebraic integers and related results.” (Artūras Dubickas, zbMATH 1486.11003, 2022)Table of Contents1 Mahler Measures of Polynomials in One Variable.- 2 Mahler Measures of Polynomials in Several Variables.- 3 Dobrowolski's Theorem.- 4 The Schinzel–Zassenhaus Conjecture.- 5 Roots of Unity and Cyclotomic Polynomials.- 6 Cyclotomic Integer Symmetric Matrices I: Tools and Statement of the Classification Theorem.- 7 Cyclotomic Integer Symmetric Matrices II: Proof of the Classification Theorem.- 8 The Set of Cassels Heights.- 9 Cyclotomic Integer Symmetric Matrices Embedded in Toroidal and Cylindrical Tesselations.- 10 The Transfinite Diameter and Conjugate Sets of Algebraic Integers.- 11 Restricted Mahler Measure Results.- 12 The Mahler Measure of Nonreciprocal Polynomials.- 13 Minimal Noncyclotomic Integer Symmetric Matrices.- 14 The Method of Explicit Auxiliary Functions.- 15 The Trace Problem For Integer Symmetric Matrices.- 16 Small-Span Integer Symmetric Matrices.- 17 Symmetrizable Matrices I: Introduction.- 18 Symmetrizable Matrices II: Cyclotomic Symmetrizable Integer Matrices.- 19 Symmetrizable Matrices III: The Trace Problem.- 20 Salem Numbers from Graphs and Interlacing Quotients.- 21 Minimal Polynomials of Integer Symmetric Matrices.- 22 Breaking Symmetry.- A Algebraic Background.- B Combinatorial Background.- C Tools from the Theory of Functions.- D Tables.- References.- Index.

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Book SynopsisIn the two-volume set ‘A Selection of Highlights’ we present basics of mathematics in an exciting and pedagogically sound way. This volume examines fundamental results in Algebra and Number Theory along with their proofs and their history. In the second edition, we include additional material on perfect and triangular numbers. We also added new sections on elementary Group Theory, p-adic numbers, and Galois Theory. A true collection of mathematical gems in Algebra and Number Theory, including the integers, the reals, and the complex numbers, along with beautiful results from Galois Theory and associated geometric applications. Valuable for lecturers, teachers and students of mathematics as well as for all who are mathematically interested.

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Book SynopsisOrdinary differential equations (ODEs) and differential-algebraic equations (DAEs) are widely used to model control systems in engineering, natural sciences, and economy. Optimal control plays a central role in optimizing such systems and to operate them effi ciently and safely. The intention of this textbook is to provide both, the theoretical and computational tools that are necessary to investigate and to solve optimal control problems with ODEs and DAEs. An emphasis is placed on the interplay between the optimal control problem, which typically is defi ned and analyzed in a Banach space setting, and discretizations thereof, which lead to finite dimensional optimization problems. The theoretical parts of the book require some knowledge of functional analysis, the numerically oriented parts require knowledge from linear algebra and numerical analysis. Practical examples are provided throughout the book for illustration purposes. The book addresses primarily master and PhD students as well as researchers in applied mathematics, but also engineers or scientists with a good background in mathematics. The book serves as a reference in research and teaching and hopefully helps to advance the state-of-the-art in optimal control.

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Book SynopsisThe primary audience for this book is students and the young researchers interested in the core of the discipline. Commutative algebra is by and large a self-contained discipline, which makes it quite dry for the beginner with a basic training in elementary algebra and calculus. A stable mathematical discipline such as this enshrines a vital number of topics to be learned at an early stage, more or less universally accepted and practiced. Naturally, authors tend to turn these topics into an increasingly short and elegant list of basic facts of the theory. So, the shorter the better. However, there is a subtle watershed between elegance and usefulness, especially if the target is the beginner. From my experience throughout years of teaching, elegance and terseness do not do it, except much later in the carrier. To become useful, the material ought to carry quite a bit of motivation through justification and usefulness pointers. On the other hand, it is difficult to contemplate these teaching devices in the writing of a short book. I have divided the material in three parts. starting with more elementary sections, then carrying an intermezzo on more difficult themes to make up for a smooth crescendo with additional tools and, finally, the more advanced part, versing on a reasonable chunk of present-day steering of commutative algebra. Historic notes at the end of each chapter provide insight into the original sources and background information on a particular subject or theorem. Exercises are provided and propose problems that apply the theory to solve concrete questions (yes, with concrete polynomials, and so forth).

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Book SynopsisThe aim of this work is to offer a concise and self-contained 'lecture-style' introduction to the theory of classical rigid geometry established by John Tate, together with the formal algebraic geometry approach launched by Michel Raynaud. These Lectures are now viewed commonly as an ideal means of learning advanced rigid geometry, regardless of the reader's level of background. Despite its parsimonious style, the presentation illustrates a number of key facts even more extensively than any other previous work.This Lecture Notes Volume is a revised and slightly expanded version of a preprint that appeared in 2005 at the University of Münster's Collaborative Research Center "Geometrical Structures in Mathematics".Trade Review“Its aim is to offer a rapid and mostly self-contained ‘lecture-style’ introduction to the theory of classical rigid geometry established by Tate, together with the formal algebraic geometry approach set up by Raynaud. Furthermore, the volume provides enlightening examples of rigid spaces and points out analogies with and differences from the theory of schemes. The book is suitable for a first course on formal and rigid geometry, but it can be used equally well for personal study.” (Alessandra Bertapelle, Mathematical Reviews, March, 2016)“All notions introduced are discussed thoroughly, proofs are lucid and elegant, and the hypotheses made and their relevance are clear throughout the text. … The reader comes away from the text with a thorough understanding of the internal motivations of the theory of formal and rigid spaces. The book is an extremely readable introduction to its subject, as well as to the techniques of modern geometry in general.” (Jeroen Sijsling, zbMATH 1314.14002, 2015)Table of ContentsClassical Rigid Geometry.- Tate Algebras.- Affinoid Algebras and their Associated Spaces.- Affinoid Functions.- Towards the Notion of Rigid Spaces.- Coherent Sheaves on Rigid Spaces.- Formal Geometry.- Adic Rings and their Associated Formal Schemes.- Raynaud's View on Rigid Spaces.- More Advanced Stuff.- Appendix.- References.- Index.

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Book SynopsisThis undergraduate textbook provides an approachable and thorough introduction to the topic of algebraic number theory, taking the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform.The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the first time that the number field sieve has been considered in a textbook at this level.Trade Review“Undergraduate mathematics students need both to develop facility with numerical and symbolic calculation and comfort with abstraction. Algebraic number theory offers an ideal context for encountering the synthesis of these goals. One could compile a shelf of graduate-level expositions of algebraic number theory, and another shelf of undergraduate general number theory texts that culminate with a first exposure to it. … Summing Up: Highly recommended. Upper-division undergraduates.” (D. V. Feldman, Choice, Vol. 52 (8), April, 2015)“In this book, the author leads the readers from the theorem of unique factorization in elementary number theory to central results in algebraic number theory. … This book is designed for being used in undergraduate courses in algebraic number theory; the clarity of the exposition and the wealth of examples and exercises (with hints and solutions) also make it suitable for self-study and reading courses.” (Franz Lemmermeyer, zbMATH, Vol. 1303, 2015)Table of ContentsUnique factorisation in the natural numbers.- Number fields.- Fields, discriminants and integral bases.- Ideals.- Prime ideals and unique factorisation.- Imaginary quadratic fields.- Lattices and geometrical methods.- Other fields of small degree.- Cyclotomic fields and the Fermat equation.- Analytic methods.- The number field sieve.

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Book SynopsisThis text aims to provide graduate students with a self-contained introduction to topics that are at the forefront of modern algebra, namely, coalgebras, bialgebras and Hopf algebras. The last chapter (Chapter 4) discusses several applications of Hopf algebras, some of which are further developed in the author’s 2011 publication, An Introduction to Hopf Algebras. The book may be used as the main text or as a supplementary text for a graduate algebra course. Prerequisites for this text include standard material on groups, rings, modules, algebraic extension fields, finite fields and linearly recursive sequences.The book consists of four chapters. Chapter 1 introduces algebras and coalgebras over a field K; Chapter 2 treats bialgebras; Chapter 3 discusses Hopf algebras and Chapter 4 consists of three applications of Hopf algebras. Each chapter begins with a short overview and ends with a collection of exercises which are designed to review and reinforce the material. Exercises range from straightforward applications of the theory to problems that are devised to challenge the reader. Questions for further study are provided after selected exercises. Most proofs are given in detail, though a few proofs are omitted since they are beyond the scope of this book.Trade Review“The goal of the book under review is to introduce graduate students to some basic results on coalgebras, bialgebras, Hopf algebras, and their applications. The book may be used as the main text or as a supplementary text for a graduate course. … This book should be very useful as a first introduction for someone who wants to learn about Hopf algebras and their applications.” (Jörg Feldvoss, zbMATH 1341.16034, 2016)Table of ContentsPreface.- Notation.- 1. Algebras and Coalgebras.- 2. Bialgebras.- 3. Hopf Algebras.- 4. Applications of Hopf Algebras.- Bibliography.

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Table of ContentsThree basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean value theorems.- Large moduli theorems.- Further results and conjectures concerning mean and large moduli.- Mean moduli of L-functions.- Zero-free regions and the proliferation of zeros.- Distribution of zeros of L-functions.- Least character non-residues and arg L(12+it, x).- The prime number theorems of Hoheisel and Selberg.- The bombieri — Vinogradov theorem.- A lemma in additive prime number theory.- The mean value theorem of Barban.

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