Number theory Books
Springer Research Directions in Number Theory
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).
£125.99
Springer Combinatorial and Additive Number Theory VI
Book SynopsisChessboard Domination: Introduction of Some New Problems.- Bounds on distinct and repeated dot product trees.- Fractal Dimension, Approximation, and Data Sets.- Towards the Gaussianity of Random Zeckendorf Games.- Linear Recurrences of Order at Most Two in Nontrivial Small Divisors and Large Divisors.-Representing Positive integers as a Sum of a Squarefree Number and a Prime.- Symmetric (not Complete Intersection) Numerical Semigroups and Syzygy Identities.- Commutative Monoid of Self-Dual Symmetric Polynomials.- Geometric Progressions in the Sets of Values of Reducible Cubic Forms.- Uniform Approximation by Polynomials with Integer Coe?cients via the Bernstein Lattice.- On the Distribution of Subset Sums of Certain Sets in Zp2 and in N2.- On a Polynomial Reciprocity Theorem of Carlitz.- Petersson-Knopp Type Identities for Generalized Dedekind-Rademacher Sums Attached to Three Dirichlet Characters.-Explicit Bounds for Large Gaps between Cubefree Integers.- The Family of a-Floor Quotient Partial Orders.-The Muirhead-Rado Inequality, 1: Vector Majorization and the Permutohedron.- The Muirhead-Rado Inequality, 2: Symmetric Means and Inequalities.- Patterns in the Iteration of an Arithmetic Function.- The Rate of Convergence for Selberg's Central Limit Theorem under the Riemann Hypothesis.- Some proofs about Sequences in the Spirit of Paul Du Bois-Reymond.- Series with Summands involving Harmonic Numbers.- Condensation and Densi?cation for Sets of Large Diameter.
£161.99
Springer Causality The padic Theory
Book SynopsisForeword.- Preface.- Background.- Part I. Basics of p-Adic Analysis.- Rings and Fields of p-Adic Numbers.- p-Adic Calculus.- p-Adic Series.- 1-Lipschitz functions.- Special Classes of 1-Lipschitz Functions.- Part II. The p-Adic Ergodic Theory.- Ergodic Theory: Preliminaries for the p-Adic Case.- The Main Ergodic Theorem for p-Adic 1-Lipschitz Maps.- 1-Lipschitz Ergodicity on Zp.- Measure-Preservation and Ergodicity of Uniformly Differentiable Functions.- 1-Lipschitz Ergodicity on Subspaces.- Plots of 1-Lipschitz Functions in Euclidean Space.- Part III. Applications.- Applications to Automata Theory.- Applications to Computer Science.- Application to Combinatorics.- Applications to Foundations of Quantum Theory.- References.- Index.
£134.99
Birkhauser Verlag AG Arithmetic Geometry over Global Function Fields
Book SynopsisThis volume collects the texts of five courses given in the Arithmetic Geometry Research Programme 2009-2010 at the CRM Barcelona. All of them deal with characteristic p global fields; the common theme around which they are centered is the arithmetic of L-functions (and other special functions), investigated in various aspects. Three courses examine some of the most important recent ideas in the positive characteristic theory discovered by Goss (a field in tumultuous development, which is seeing a number of spectacular advances): they cover respectively crystals over function fields (with a number of applications to L-functions of t-motives), gamma and zeta functions in characteristic p, and the binomial theorem. The other two are focused on topics closer to the classical theory of abelian varieties over number fields: they give respectively a thorough introduction to the arithmetic of Jacobians over function fields (including the current status of the BSD conjecture and its geometric analogues, and the construction of Mordell-Weil groups of high rank) and a state of the art survey of Geometric Iwasawa Theory explaining the recent proofs of various versions of the Main Conjecture, in the commutative and non-commutative settings.Table of ContentsCohomological Theory of Crystals over Function Fields and Applications.- On Geometric Iwasawa Theory and Special Values of Zeta Functions.- The Ongoing Binomial Revolution.- Arithmetic of Gamma, Zeta and Multizeta Values for Function Fields.- Curves and Jacobians over Function Fields.
£37.99
Birkhauser Verlag AG Correspondence of Leonhard Euler with Christian
Book SynopsisWhen Leonhard Euler first arrived at the Russian Academy of Sciences, at the age of 20, his career was supported and promoted by the Academy’s secretary, the Prussian jurist and amateur mathematician Christian Goldbach (1690-1764). Their encounter would grow into a lifelong friendship, as evinced by nearly 200 letters sent over 35 years.This exchange – Euler’s most substantial long-term correspondence – has now been edited for the first time with an English translation, ample commentary and documentary indices. These present an overview of 18th-century number theory, its sources and repercussions, many details of the protagonists’ biographies, and a wealth of insights into academic life in St. Petersburg and Berlin between 1725 and 1765.Part I includes an introduction and the original texts of the Euler-Goldbach letters, while Part II presents the English translations and documentary indices.Trade Review“The present volume is the second part of Lemmermeyer and Mattmüller's edition of the correspondence between Euler and Goldbach. … This edition of the Euler-Goldbach correspondence published in two volumes will soon be the indispensable reference. It is a pearl in the literature concerning history of mathematics and history of number theory in particular.” (Thomas Sonar, zbMATH 1361.01009, 2017)Table of ContentsPreface.- Introduction.- 1 Historical and biographical setting.- 1.1. Christian Goldbach: A short biography.- 1.2. Goldbach and Euler.- 1.3. The Euler-Goldbach correspondence – chronology and statistics.- 2 Main subjects of the correspondence.- 2.1. Number theory.- 2.2. Analytic tools in number theory.- 2.3. Algebra: roots of polynomials and transcendence.- 2.4. Analysis.- 2.5. Geometry, topology, combinatorics.- 2.6. Natural science.- 2.7. Professional life: Academies, prizes, publications.- 2.8. Personal life: family, travel, health.- 3 Editing the Euler-Goldbach correspondence.- 3.1. Description of the manuscript sources.- 3.2. Prior editions.- 3.3. Editorial principles.- Correspondence with Christian Goldbach. Original Texts.- Translations.- Indices: Synoptic Table.- Bibliography.- Systematic Subject Index.- Name Index.
£227.48
Birkhauser Verlag AG Correspondence of Leonhard Euler with Christian
Book SynopsisWhen Leonhard Euler first arrived at the Russian Academy of Sciences, at the age of 20, his career was supported and promoted by the Academy’s secretary, the Prussian jurist and amateur mathematician Christian Goldbach (1690-1764). Their encounter would grow into a lifelong friendship, as evinced by nearly 200 letters sent over 35 years.This exchange – Euler’s most substantial long-term correspondence – has now been edited for the first time with an English translation, ample commentary and documentary indices. These present an overview of 18th-century number theory, its sources and repercussions, many details of the protagonists’ biographies, and a wealth of insights into academic life in St. Petersburg and Berlin between 1725 and 1765.Part I includes an introduction and the original texts of the Euler-Goldbach letters, while Part II presents the English translations and documentary indices.Trade Review“The present book in the edition of Leonhard Euler's œuvre is concerned with the correspondence of Euler with Christian Goldbach. … The present volume is a pearl in the edition of Euler's correspondence and a must-read for historians of mathematics particularly interested in the history of number theory.” (Thomas Sonar, zbMATH 1361.01008, 2017)Table of ContentsPreface.- Introduction.- 1 Historical and biographical setting.- 1.1. Christian Goldbach: A short biography.- 1.2. Goldbach and Euler.- 1.3. The Euler-Goldbach correspondence – chronology and statistics.- 2 Main subjects of the correspondence.- 2.1. Number theory.- 2.2. Analytic tools in number theory.- 2.3. Algebra: roots of polynomials and transcendence.- 2.4. Analysis.- 2.5. Geometry, topology, combinatorics.- 2.6. Natural science.- 2.7. Professional life: Academies, prizes, publications.- 2.8. Personal life: family, travel, health.- 3 Editing the Euler-Goldbach correspondence.- 3.1. Description of the manuscript sources.- 3.2. Prior editions.- 3.3. Editorial principles.- Correspondence with Christian Goldbach. Original Texts.- Translations.- Indices: Synoptic Table.- Bibliography.- Systematic Subject Index.- Name Index.
£129.99
De Gruyter Groups of Prime Power Order. Volume 1
Book SynopsisThis is the first of three volumes of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this monograph include: (a) counting of subgroups, with almost all main counting theorems being proved, (b) regular p-groups and regularity criteria, (c) p-groups of maximal class and their numerous characterizations, (d) characters of p-groups, (e) p-groups with large Schur multiplier and commutator subgroups, (f) (p‒1)-admissible Hall chains in normal subgroups, (g) powerful p-groups, (h) automorphisms of p-groups, (i) p-groups all of whose nonnormal subgroups are cyclic, (j) Alperin's problem on abelian subgroups of small index. The book is suitable for researchers and graduate students of mathematics with a modest background on algebra. It also contains hundreds of original exercises (with difficult exercises being solved) and a comprehensive list of about 700 open problems.
£164.82
Birkhauser Verlag AG Brauer Groups and Obstruction Problems: Moduli
Book SynopsisThe contributions in this book explore various contexts in which the derived category of coherent sheaves on a variety determines some of its arithmetic. This setting provides new geometric tools for interpreting elements of the Brauer group. With a view towards future arithmetic applications, the book extends a number of powerful tools for analyzing rational points on elliptic curves, e.g., isogenies among curves, torsion points, modular curves, and the resulting descent techniques, as well as higher-dimensional varieties like K3 surfaces. Inspired by the rapid recent advances in our understanding of K3 surfaces, the book is intended to foster cross-pollination between the fields of complex algebraic geometry and number theory.Contributors:· Nicolas Addington · Benjamin Antieau · Kenneth Ascher · Asher Auel · Fedor Bogomolov · Jean-Louis Colliot-Thélène · Krishna Dasaratha · Brendan Hassett · Colin Ingalls · Martí Lahoz · Emanuele Macrì · Kelly McKinnie · Andrew Obus · Ekin Ozman · Raman Parimala · Alexander Perry · Alena Pirutka · Justin Sawon · Alexei N. Skorobogatov · Paolo Stellari · Sho Tanimoto · Hugh Thomas · Yuri Tschinkel · Anthony Várilly-Alvarado · Bianca Viray · Rong ZhouTable of ContentsThe Brauer group is not a derived invariant.- Twisted derived equivalences for affine schemes.- Rational points on twisted K3 surfaces and derived equivalences.- Universal unramified cohomology of cubic fourfolds containing a plane.- Universal spaces for unramified Galois cohomology.- Rational points on K3 surfaces and derived equivalence.- Unramified Brauer classes on cyclic covers of the projective plane.- Arithmetically Cohen-Macaulay bundles on cubic fourfolds containing a plane.- Brauer groups on K3 surfaces and arithmetic applications.- On a local-global principle for H3 of function fields of surfaces over a finite field.- Cohomology and the Brauer group of double covers.
£113.99
Springer International Publishing AG The Stair-Step Approach in Mathematics
Book SynopsisThis book is intended as a teacher’s manual and as an independent-study handbook for students and mathematical competitors. Based on a traditional teaching philosophy and a non-traditional writing approach (the stair-step method), this book consists of new problems with solutions created by the authors. The main idea of this approach is to start from relatively easy problems and “step-by-step” increase the level of difficulty toward effectively maximizing students' learning potential. In addition to providing solutions, a separate table of answers is also given at the end of the book. A broad view of mathematics is covered, well beyond the typical elementary level, by providing more in depth treatment of Geometry and Trigonometry, Number Theory, Algebra, Calculus, and Combinatorics.Trade Review“This book is original, enticing, and highly stimulating, and it is a useful addition to the competition-oriented literature.” (Stephen Rout, The Mathematical Gazette, Vol. 104 (560), July, 2020)Table of ContentsGeometry and Trigonometry.- Number Theory.- Algebra.- Calculus.- Combinatorics.- Hints.- Solutions.- Answers.
£49.49
Birkhauser Verlag AG Methods of Solving Number Theory Problems
Book SynopsisThrough its engaging and unusual problems, this book demonstrates methods of reasoning necessary for learning number theory. Every technique is followed by problems (as well as detailed hints and solutions) that apply theorems immediately, so readers can solve a variety of abstract problems in a systematic, creative manner. New solutions often require the ingenious use of earlier mathematical concepts - not the memorization of formulas and facts. Questions also often permit experimental numeric validation or visual interpretation to encourage the combined use of deductive and intuitive thinking. The first chapter starts with simple topics like even and odd numbers, divisibility, and prime numbers and helps the reader to solve quite complex, Olympiad-type problems right away. It also covers properties of the perfect, amicable, and figurate numbers and introduces congruence. The next chapter begins with the Euclidean algorithm, explores the representations of integer numbers in different bases, and examines continued fractions, quadratic irrationalities, and the Lagrange Theorem. The last section of Chapter Two is an exploration of different methods of proofs. The third chapter is dedicated to solving Diophantine linear and nonlinear equations and includes different methods of solving Fermat’s (Pell’s) equations. It also covers Fermat’s factorization techniques and methods of solving challenging problems involving exponent and factorials. Chapter Four reviews the Pythagorean triple and quadruple and emphasizes their connection with geometry, trigonometry, algebraic geometry, and stereographic projection. A special case of Waring’s problem as a representation of a number by the sum of the squares or cubes of other numbers is covered, as well as quadratic residuals, Legendre and Jacobi symbols, and interesting word problems related to the properties of numbers. Appendices provide a historic overview of number theory and its main developments from the ancient cultures in Greece, Babylon, and Egypt to the modern day. Drawing from cases collected by an accomplished female mathematician, Methods in Solving Number Theory Problems is designed as a self-study guide or supplementary textbook for a one-semester course in introductory number theory. It can also be used to prepare for mathematical Olympiads. Elementary algebra, arithmetic and some calculus knowledge are the only prerequisites. Number theory gives precise proofs and theorems of an irreproachable rigor and sharpens analytical thinking, which makes this book perfect for anyone looking to build their mathematical confidence.Table of ContentsPreface.- Numbers: Problems Involving Integers.- Further Study of Integers.- Diophantine Equations and More.- Pythagorean Triples, Additive Problems, and More.- Homework.
£42.74
Springer Fachmedien Wiesbaden Insel der Zahlen: Eine zahlentheoretische Genesis
Book SynopsisTable of Contents1 Der Stein.- 2 Symbole.- 3 Beweise.- 4 Schlechte Zahlen.- 5 Fortschritte.- 6 Der dritte Tag.- 7 Entdeckung.- 8 Addition.- 9 Die Antwort.- 10 Sätze.- 11 Der Antrag.- 12 Unheil.- 13 Wiederherstellung.- 14 Das Universum.- 15 Unendlich.- 16 Multiplikation.- Nachwort.
£49.49
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Cyclotomic Fields and Zeta Values
Book SynopsisWritten by two leading workers in the field, this brief but elegant book presents in full detail the simplest proof of the "main conjecture" for cyclotomic fields. Its motivation stems not only from the inherent beauty of the subject, but also from the wider arithmetic interest of these questions. From the reviews: "The text is written in a clear and attractive style, with enough explanation helping the reader orientate in the midst of technical details." --ZENTRALBLATT MATHTrade ReviewFrom the reviews:"The author’s aim in this book is to present a proof of the so-called Iwasawa Main Conjecture for the pth cyclotomic field … . The text is written in a clear and attractive style, with enough explanation helping the reader orientate in the midst of technical details. According to the authors, the book is intended for graduate students and the non-expert in Iwasawa theory. I think that also the expert may enjoy reading this kind of unified treatment of such a beautiful theme." (Tauno Metsänkylä, Zentralblatt MATH, Vol. 1100 (2), 2007)"This book was written to present in full detail a complete proof of the so-called ‘Main Conjecture’ in the arithmetic theory of cyclotomic fields. … The book is intended for graduate students and the non-expert in Iwasawa theory; however, the expert will find this work a valuable source in the arithmetic theory of cyclotomic fields. The book is very pleasant to read and is written with enough detail … . The authors have contributed in an important way to Iwasawa theory with this beautiful book." (Gabriel D. Villa-Salvador, Mathematical Reviews, Issue 2007 g)“The aim of this monograph is to present a detailed proof of the Main Conjecture, described by the authors as ‘the deepest result we know about the arithmetic of cyclotomic fields’. … This beautiful book will enable non-experts to study a state-of-the-art proof of the Main Conjecture. Furthermore, it might be a source of inspiration for new generations of mathematicians trying to tackle one of the many similar relations conjectured to hold in arithmetic geometry.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 154 (1), May, 2008)Table of ContentsCyclotomic Fields.- Local Units.- Iwasawa Algebras and p-adic Measures.- Cyclotomic Units and Iwasawa's Theorem.- Euler Systems.- Main Conjecture.
£66.49
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Algebraic Number Theory
Book SynopsisThis introduction to algebraic number theory discusses the classical concepts from the viewpoint of Arakelov theory. The treatment of class theory is particularly rich in illustrating complements, offering hints for further study, and providing concrete examples. It is the most up-to-date, systematic, and theoretically comprehensive textbook on algebraic number field theory available.Trade Reviewhful and unabridged reprint of the original edition of J. Neukirch’s excellent textbook on modern algebraic number theory … . this unique classic in algebraic number theory is certainly of the highest advantage for new generations of students, teachers, and researchers in German-speaking mathematical communities, and therefore more than welcome. … it will remain as one of the valuables in the legacy of an outstanding researcher and teacher in algebraic number theory forever." (Werner Kleinert, Zentralblatt MATH, Vol. 1131 (9), 2008)Table of ContentsI: Algebraic Integers.- II: The Theory of Valuations.- III: Riemann-Roch Theory.- IV: Abstract Class Field Theory.- V: Local Class Field Theory.- VI: Global Class Field Theory.- VII: Zeta Functions and L-series.
£113.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Post-Quantum Cryptography
Book SynopsisQuantum computers will break today's most popular public-key cryptographic systems, including RSA, DSA, and ECDSA. This book introduces the reader to the next generation of cryptographic algorithms, the systems that resist quantum-computer attacks: in particular, post-quantum public-key encryption systems and post-quantum public-key signature systems. Leading experts have joined forces for the first time to explain the state of the art in quantum computing, hash-based cryptography, code-based cryptography, lattice-based cryptography, and multivariate cryptography. Mathematical foundations and implementation issues are included. This book is an essential resource for students and researchers who want to contribute to the field of post-quantum cryptography.Table of Contentsto post-quantum cryptography.- Quantum computing.- Hash-based Digital Signature Schemes.- Code-based cryptography.- Lattice-based Cryptography.- Multivariate Public Key Cryptography.
£113.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG The Classical Groups and K-Theory
Book SynopsisIt is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. When H. Weyl coined the words "classical groups", foremost in his mind were their connections with invariant theory, which his famous book helped to revive. Although his approach in that book was deliberately algebraic, his interest in these groups directly derived from his pioneering study of the special case in which the scalars are real or complex numbers, where for the first time he injected Topology into Lie theory. But ever since the definition of Lie groups, the analogy between simple classical groups over finite fields and simple classical groups over IR or C had been observed, even if the concept of "simplicity" was not quite the same in both cases. With the discovery of the exceptional simple complex Lie algebras by Killing and E. Cartan, it was natural to look for corresponding groups over finite fields, and already around 1900 this was done by Dickson for the exceptional Lie algebras G and E • However, a deep reason for this 2 6 parallelism was missing, and it is only Chevalley who, in 1955 and 1961, discovered that to each complex simple Lie algebra corresponds, by a uniform process, a group scheme (fj over the ring Z of integers, from which, for any field K, could be derived a group (fj(K).Table of ContentsNotation and Conventions.- 1. General Linear Groups, Steinberg Groups, and K-Groups.- 2. Linear Groups over Division Rings.- 3. Isomorphism Theory for the Linear Groups.- 4. Linear Groups over General Classes of Rings.- 5. Unitary Groups, Unitary Steinberg Groups, and Unitary K-Groups.- 6. Unitary Groups over Division Rings.- 7. Clifford Algebras and Orthogonal Groups over Commutative Rings.- 8. Isomorphism Theory for the Unitary Groups.- 9. Unitary Groups over General Classes of Form Rings.- Concluding Remarks.- Index of Concepts.- Index of Symbols.
£89.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Extremal Combinatorics: With Applications in
Book SynopsisThis book is a concise, self-contained, up-to-date introduction to extremal combinatorics for nonspecialists. There is a strong emphasis on theorems with particularly elegant and informative proofs, they may be called gems of the theory. The author presents a wide spectrum of the most powerful combinatorial tools together with impressive applications in computer science: methods of extremal set theory, the linear algebra method, the probabilistic method, and fragments of Ramsey theory. No special knowledge in combinatorics or computer science is assumed – the text is self-contained and the proofs can be enjoyed by undergraduate students in mathematics and computer science. Over 300 exercises of varying difficulty, and hints to their solution, complete the text.This second edition has been extended with substantial new material, and has been revised and updated throughout. It offers three new chapters on expander graphs and eigenvalues, the polynomial method and error-correcting codes. Most of the remaining chapters also include new material, such as the Kruskal—Katona theorem on shadows, the Lovász—Stein theorem on coverings, large cliques in dense graphs without induced 4-cycles, a new lower bounds argument for monotone formulas, Dvir's solution of the finite field Kakeya conjecture, Moser's algorithmic version of the Lovász Local Lemma, Schöning's algorithm for 3-SAT, the Szemerédi—Trotter theorem on the number of point-line incidences, surprising applications of expander graphs in extremal number theory, and some other new results.Trade ReviewFrom the reviews of the second edition:“This is an entertaining and impressive book. I say impressive because the author managed to cover a very large part of combinatorics in 27 short chapters, without assuming any graduate-level knowledge of the material. … The collection of topics covered is another big advantage of the book. … The book is ideal as reference material or for a reading course for a dedicated graduate student. One could teach a very enjoyable class from it as well … .” (Miklós Bóna, The Mathematical Association of America, May, 2012)"[R]eaders interested in any branch of combinatorics will find this book compelling. ... This book is very suitable for advanced undergraduate and graduate mathematics and computer science majors. It requires a very solid grounding in intermediate-level combinatorics and an appreciation for several proof methods, but it is well worth the study." (G.M. White, ACM Computing Reviews, May 2012)“This is the second edition of a well-received textbook. It has been extended with new and updated results. Typographical errors in the first edition are corrected. … This textbook is suitable for advanced undergraduate or graduate students as well as researchers working in discrete mathematics or theoretical computer science. The author’s enthusiasm for the subject is evident and his writing is clear and smooth. This is a book deserving recommendation.” (Ko-Wei Lih, Zentralblatt MATH, Vol. 1239, 2012)“This is an introductory book that deals with the subject of extremal combinatorics. … The book is nicely written and the author has included many elegant and beautiful proofs. The book contains many interesting exercises that will stimulate the motivated reader to get a better understanding of this area. … author’s goal of writing a self-contained book that is more or less up to date … and that is accessible to graduate and motivated undergraduate students in mathematics and computer science, has been successfully achieved.” (Sebastian M. Cioabă, Mathematical Reviews, January, 2013)Table of ContentsPreface.- Prolog: What this Book Is About.- Notation.- Counting.- Advanced Counting.- Probabilistic Counting.- The Pigeonhole Principle.- Systems of Distinct Representatives.- Sunflowers.- Intersecting Families.- Chains and Antichains.- Blocking Sets and the Duality.- Density and Universality.- Witness Sets and Isolation.- Designs.- The Basic Method.- Orthogonality and Rank Arguments.- Eigenvalues and Graph Expansion.- The Polynomial Method.- Combinatorics of Codes.- Linearity of Expectation.- The Lovász Sieve.- The Deletion Method.- The Second Moment Method.- The Entropy Function.- Random Walks.- Derandomization.- Ramseyan Theorems for Numbers.- The Hales–Jewett Theorem.- Applications in Communications Complexity.- References.- Index.
£75.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Rational Points and Arithmetic of Fundamental Groups: Evidence for the Section Conjecture
Book SynopsisThe section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves. This monograph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and the local-to-global approach is studied in detail. The monograph concludes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by using a characteristic quotient or its birational analogue in lieu of the fundamental group extension.Trade ReviewFrom the book reviews:“The book under review, resulting from the author’s dissertation … is both a research monograph and a thorough presentation of the arithmetic and geometry of Grothendieck’s section conjecture from the foundations to the current state of the art. … It will be useful not only to specialists, as it is accessible to anyone familiar with the basics of modern algebraic geometry and the theory of algebraic fundamental groups.” (Marco A. Garuti, Mathematical Reviews, May, 2014)Table of ContentsPart I Foundations of Sections.- 1 Continuous Non-abelian H1 with Profinite Coefficients.-2 The Fundamental Groupoid.- 3 Basic Geometric Operations in Terms of Sections.- 4 The Space of Sections as a Topological Space.- 5 Evaluation of Units.- 6 Cycle Classes in Anabelian Geometry.- 7 Injectivity in the Section Conjecture.- Part II Basic Arithmetic of Sections.- 7 Injectivity in the Section Conjecture.- 8 Reduction of Sections.- 9 The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers.- Part III On the Passage from Local to Global.- 10 Local Obstructions at a p-adic Place.- 11 Brauer-Manin and Descent Obstructions.- 12 Fragments of Non-abelian Tate–Poitou Duality.- Part IV Analogues of the Section Conjecture.- 13 On the Section Conjecture for Torsors.- 14 Nilpotent Sections.- 15 Sections over Finite Fields.- 16 On the Section Conjecture over Local Fields.- 17 Fields of Cohomological Dimension 1.- 18 Cuspidal Sections and Birational Analogues.
£49.99
Springer Fachmedien Wiesbaden Elementare Zahlentheorie: Beispiele, Geschichte,
Book SynopsisDieses Buch bietet einen historisch orientierten Einstieg in die elementare Zahlentheorie. Es stellt eine solide Basis für vielfältige Ausbaumöglichkeiten dar. Besondere Zielsetzungen sind: Elementarität und Anschaulichkeit, die Berücksichtigung der historischen Entwicklung, Motivation der Begriffe und Verfahren anhand konkreter, aussagekräftiger Beispiele unter Einbezug moderner Werkzeuge (Computeralgebra Systeme, Internet). Als Zusatzmedien werden Computer- und Internet-spezifische Interaktions- und Visualisierungsmöglichkeiten (kostenlos) zur Verfügung gestellt. Das Werk wendet sich an Studierende (Bachelor/Lehramt), Lehrer(innen) sowie alle an Elementarmathematik interessierten Leser.Table of ContentsGeschichtliches zu Zahl und Zahldarstellung.- Die Division mit Rest und die Teilbarkeitsrelation.- Euklidischer Algorithmus.- Primzahlen.- Kongruenzen und Restklassen.- Stellenwertsysteme, Teilbarkeitsregeln und Rechenproben.- Die Sätze von Euler, Fermat und Wilson.- Anhänge.
£24.99
Springer Fachmedien Wiesbaden Primzahltests für Einsteiger: Zahlentheorie –
Book SynopsisIn diesem Buch geht es um den AKS-Algorithmus, den ersten deterministischen Primzahltest mit polynomieller Laufzeit. Er wurde benannt nach den Informatikern Agrawal, Kayal und Saxena, die ihn 2002 entwickelt haben. Primzahlen sind Gegenstand vieler mathematischer Probleme und spielen im Zusammenhang mit Verschlüsselungsmethoden eine wichtige Rolle. Das vorliegende Buch leitet den AKS-ALgorithmus in verständlicher Art und Weise her, ohne wesentliche Vorkenntnisse zu benötigen, und ist daher bereits für interessierte Gymnasialschüler(innen) zugänglich. Außerdem eignet sich das Buch von Studienbeginn an für Lehrveranstaltungen im Mathematik- oder Informatikstudium. Es kann schon in den ersten Semestern als Grundlage für zweistündige Vorlesungen oder (Pro-)Seminare dienen, ohne auf andere Lehrveranstaltungen (wie z. B. Zahlentheorie) zurückzugreifen, und ist daher im Bachelor- und Lehramtsstudium gut einsetzbar. Es gibt viele Aufgaben und weiterführende Anmerkungen sowie Lösungshinweise am Ende des Buches. Table of ContentsNatürliche Zahlen und Primzahlen.- Algorithmen und Komplexität.- Zahlentheoretische Grundlagen.- Primzahlen und Kryptographie.- Der Ausgangspunkt: Fermat für Polynome.- Der Satz von Agrawal, Kayal und Saxena.- Der Algorithmus.- Offene Fragen über Primzahlen.- Lösungen und Hinweise zu wichtigen Aufgaben.
£26.59
Springer Fachmedien Wiesbaden Grundbegriffe der elementaren Zahlentheorie: Von
Book SynopsisDie elementare Zahlentheorie befasst sich mit den Eigenschaften der natürlichen Zahlen und benötigt als Grundlage hierfür nur die Arithmetik. Sie ist ein unverzichtbarer Bestandteil des Bachelorstudiums Mathematik.Die Leser*innen erhalten mit diesem essential eine kompakte und auf das Wesentliche fokussierte Darstellung der elementaren Zahlentheorie, die insbesondere für einen ersten Überblick über dieses Teilgebiet, für die Prüfungsvorbereitung oder zum Nachschlagen wichtiger Definitionen und Sätze herangezogen werden kann. Table of ContentsTeilerrelation und Teilermenge.- Teilbarkeitsregeln.- Gemeinsame Teiler und Vielfache.- Primzahlen.- Primfaktorzerlegung.- Kongruenz modulo m.
£11.77
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Pi und die Primzahlen: Eine Entdeckungsreise in die Mathematik
Book SynopsisSpaß an der Mathematik haben? Ja, das geht wirklich, wie dieses Buch zeigt! Es erzählt wie ein Roman eine „mathematische Geschichte“. Man könnte behaupten, diese recht verworrene Geschichte drehe sich um eine umständliche Entwicklung einer Formel, mit deren Hilfe man die Kreiszahl Pi berechnen kann. Aber eigentlich geht es um etwas ganz anderes: Das Buch nimmt den Leser an der Hand, fordert ihn aber durch eingestreute Fragen immer wieder zum Innehalten und Mitdenken auf. Dank der behutsamen Heranführung an die Themen können diese Fragen von jedem, der die Herausforderung annimmt, mit Schulkenntnissen gemeistert werden. Man bekommt so einen Einblick in „echte“ Mathematik zwischen Geometrie, Algebra, Analysis und Zahlentheorie. Man sieht, wie man an mathematische Fragestellungen herangehen kann. Und man erfährt, warum Mathematik früher ganz anders als heute war und wie sie sich erst mühsam entwickeln musste. Anekdoten über die Menschen hinter der Mathematik gibt's auch, denn der Autor plaudert gerne, philosophiert auch ab und zu und liebt Abschweifungen. Und das Schönste ist: Am Ende wartet keine Prüfung – der Leser kann sich einfach auf die Freude am Forschen und Verstehen einlassen.Table of ContentsAb in den Dschungel.- Nicht von Pythagoras .- Was beweisen Beweise?.- Die Kreativen.- Menschenwerk.- Nichts.- Die Diva.- Gibt es Pi überhaupt?.- Der Plan.- Millimeterpapier.- Die Atome der Mathematik.- Der Gott aus der Maschine.- Reste.- Der Amateur und die Windmühlen.- Die Badeanstalt.- Der erste Algorithmus.- Komplexes Intermezzo.- Außerirdische Mathematik.- Einfaches Sudoku.- Der letzte Brief.- Der schmale Rand.- Einfach die Regeln ändern.- Fünfzehntausend Seiten.- Endlich Punkte zählen!.- Dominoeffekte.- Noch eine Hypothese.- Von Fröschen und Mäusen.- Butterkeks.- Offenes Ende.- Epilog.- Anmerkungen.- Inhalt.- Index.
£24.99
Springer Fachmedien Wiesbaden Einführung in die Zahlentheorie und Algebra
Book SynopsisEine kombinierte Einführung in die Algebra bis zur Galoistheorie und ihren klassischen Anwendungen sowie in die Zahlentheorie: Dabei profitiert die Algebra von den Motivationen und dem reichen Beispielmaterial der Zahlentheorie; letztere gewinnt an Klarheit und Kürze durch Strukturen und Sätze der Algebra. Es wird solides Grundwissen für beide Gebiete vermittelt und gleichzeitig die Brücke zu neuesten Entwicklungen geschlagen (z. B. diophantische Probleme, Faktorisierungsmethoden, inverses Problem der Galoistheorie). Die Neuauflage enthält neben Korrekturen und Aktualisierungen Lösungshinweise zu den Aufgaben. Neu ist ein umfangreiches Kapitel über Gitter, die Brücke zur Algebraischen Zahlentheorie und zu vielen Anwendungen von Algebra und Zahlentheorie in der Diskreten Mathematik.Table of ContentsGanze Zahlen - Teilbarkeit - Gruppen - Ringe - Arithmetik modulo n - Primzahltests und Primfaktorzerlegung - Körper und Körpererweiterungen - Galoistheorie - Gitter
£26.59
Birkhauser Verlag AG On Some Applications of Diophantine
Book SynopsisThis book consists mainly of the translation, by C. Fuchs, of the 1929 landmark paper "Über einige Anwendungen diophantischer Approximationen" by C.L. Siegel. The paper contains proofs of most important results in transcendence theory and diophantine analysis, notably Siegel’s celebrated theorem on integral points on algebraic curves. Many modern versions of Siegel’s proof have appeared, but none seem to faithfully reproduce all features of the original one. This translation makes Siegel’s original ideas and proofs available for the first time in English. The volume also contains the original version of the paper (in German) and an article by the translator and U. Zannier, commenting on some aspects of the evolution of this field following Siegel’s paper. To end, it presents three modern proofs of Siegel’s theorem on integral points.Trade Review“This book contains both Siegel’s original paper in German and an English translation. … this is a fundamental paper, well worth reading. … It will be of great interest to mathematicians working in transcendence theory and Diophantine approximation, and to anyone interested in the history of mathematics in the early 20th century.” (Fernando Q. Gouvêa, MAA Reviews, June, 2015)Table of ContentsPreface by C. Fuchs and U. Zannier.- On some applications of Diophantine approximations (a translation by C. Fuchs of C.L. Siegel’s Über einige Anwendungen diophantischer Approximationen).- Über einige Anwendungen diophantischer Approximationen (by C.L. Siegel).- Integral points on curves: Siegel’s theorem after Siegel’s proof, by C. Fuchs and U. Zannier.
£22.79
Hindustan Book Agency Introduction to the Theory of Standard Monomials
Book SynopsisThe aim of this book is to give an introduction to what has come to be known as Standard Monomial Theory (SMT). SMT deals with the construction of nice bases of finite dimensional irreducible representations of semi-simple algebraic groups or, in geometric terms, nice bases of coordinate rings of flag varieties (and their Schubert subvarieties) associated to these groups. Besides its intrinsic interest, SMT has applications to the study of the geometry of Schubert varieties. SMT has its origin in the work of Hodge, giving bases of the coordinate rings of the Grassmannian and its Schubert subvarieties by ""standard monomials"". In its modern form, SMT was developed by the author in a series of papers written in collaboration with V. Lakshmibai and C. Musili.This book is a reproduction of a course of lectures given by the author in 1983-84 which appeared in the Brandeis Lecture Notes series. The aim of this course was to give an introduction to the series of papers by concentrating on the case of the full linear group. In recent years, there has been great progress in SMT due to the work of Peter Littelmann. Seshadri's course of lectures (reproduced in this book) remains an excellent introduction to SMT.In the second edition, Conjectures of a Standard Monomial Theory (SMT) for a general semi-simple (simply-connected) algebraic group, due to Lakshmibai, have been added as Appendix C. Many typographical errors have been corrected, and the bibliography has been revised.
£40.76
Springer Verlag, Singapore A First Course in Group Theory
Book SynopsisThis textbook provides a readable account of the examples and fundamental results of groups from a theoretical and geometrical point of view. Topics on important examples of groups (like cyclic groups, permutation groups, group of arithmetical functions, matrix groups and linear groups), Lagrange’s theorem, normal subgroups, factor groups, derived subgroup, homomorphism, isomorphism and automorphism of groups have been discussed in depth. Covering all major topics, this book is targeted to undergraduate students of mathematics with no prerequisite knowledge of the discussed topics. Each section ends with a set of worked-out problems and supplementary exercises to challenge the knowledge and ability of the reader.Trade Review“Advanced school students and well-motivated undergraduates can profitably read it, and it is a very useful general reference for the history of substantial parts of mathematics, placed in the context of contemporary social and political events. … as a readable … and refreshingly detailed account of the whole sweep of ‘infinitesimal methods’ from antiquity to the 1990s, this book is highly recommended.” (Peter Giblin, The Mathematical Gazette, Vol. 107 (570), November, 2023)“Davvaz's book, on the other hand, features many excellent discussions of groups of matrices. Indeed, matrix groups are used not just as examples of groups, but to help clarify and add depth to Davvaz's discussion of other families of groups. … . It is also, in my opinion, the highlight of the book.” (Benjamin Linowitz, MAA Reviews, February 20, 2022)Table of ContentsPreliminaries Notions.- Symmetries of Shapes.- Binary Operations.- Cyclic Groups.- Inverse Functions and Permutations.- Group of Arithmetical Functions.- Matrix Groups.- Translation and Scaling Matrices.- Cosets of Subgroups and Lagrange’s Theorem.- Normal Subgroups and Factor Groups.- Some Special Subgroups.- Commutators and Derived Subgroups.- Maximal Subgroups.- Group Homomorphisms.- Homomorphisms and Their Properties.- Cayley’s Theorem.- Another View of Linear Groups.
£42.74
Springer Verlag, Singapore A Textbook of Algebraic Number Theory
Book SynopsisThis self-contained and comprehensive textbook of algebraic number theory is useful for advanced undergraduate and graduate students of mathematics. The book discusses proofs of almost all basic significant theorems of algebraic number theory including Dedekind’s theorem on splitting of primes, Dirichlet’s unit theorem, Minkowski’s convex body theorem, Dedekind’s discriminant theorem, Hermite’s theorem on discriminant, Dirichlet’s class number formula, and Dirichlet’s theorem on primes in arithmetic progressions. A few research problems arising out of these results are mentioned together with the progress made in the direction of each problem. Following the classical approach of Dedekind’s theory of ideals, the book aims at arousing the reader’s interest in the current research being held in the subject area. It not only proves basic results but pairs them with recent developments, making the book relevant and thought-provoking. Historical notes are given at various places. Featured with numerous related exercises and examples, this book is of significant value to students and researchers associated with the field. The book also is suitable for independent study. The only prerequisite is basic knowledge of abstract algebra and elementary number theory. Trade Review“A Textbook of Algebraic Number Theory is intended to be used as a 2-term textbook for an algebraic number theory graduate course. … As a graduate course textbook, this would be an excellent resource. … I would definitely recommend this book for a graduate course following a thorough abstract algebra sequence. The topics covered are the foundations of the study of algebraic number theory.” (McKenzie West, MAA Reviews, October 9, 2023)“This wonderful textbook will be of great help to everybody interested in algebraic number theory … . The book is an essence of a two-semester course on algebraic number theory held several times by the author to postgraduate students. … Readers will enjoy the presentation of the book together with interesting illustrations of historical notes.” (István Gaál, zbMATH 1500.11001, 2023)Table of Contents1. Algebraic Integers, Norm and Trace.-2. Integral Basis and Discriminant.-3. Properties of the Ring of Algebraic Integers.- 4. Splitting of Rational Primes and Dedekind’s Theorem.-5. Dirichlet’s Unit Theorem.- 6. Prime Ideal Decomposition in Relative Extensions.- 7. Relative Discriminant and Dedekind’s Theorem on Ramified.- 8. Ideal Class Group.-9. Dirichlet’s Class Number Formula and its Applications.- 10. Simplified Class Number Formula for Cyclotomic, Quadratic Fields.
£37.99
Springer Modular Relations and Parity in Number Theory
Book Synopsis
£125.99
Springer Lecture Notes on Geometry of Numbers
Book Synopsis1. Preliminaries.- 2. Minkowski's Fundamental Theorem and its Applications.- 3. Lattices.- 4. Minima of Positive De nite Quadratic Forms.- 5. Critical Determinant.- 6. Successive Minima.- 7. Packings Density.- 8. Coverings.- 9. Homogeneous Minimum.- 10. Inhomogeneous Problems.
£999.99
Taylor & Francis Ltd Semialgebraic Statistics and Latent Tree Models
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£58.89
Taylor & Francis Ltd Sums of Squares of Integers
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Taylor & Francis Ltd Quadratics
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Taylor & Francis Ltd Introduction to Geometric Algebra Computing
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£42.74
Taylor & Francis Ltd Differential Equations in Engineering
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Taylor & Francis Ltd An Invitation to the RogersRamanujan Identities
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£45.59
Taylor & Francis Ltd Infinite Groups
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Taylor & Francis Ltd Algebraic Number Theory
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£52.24
Taylor & Francis Ltd Differential Geometry and Its Visualization
Book SynopsisDifferential Geometry and Its Visualization is suitable for graduate level courses in differential geometry, serving both students and teachers. It can also be used as a supplementary reference for research in mathematics and the natural and engineering sciences.Differential geometry is the study of geometric objects and their properties using the methods of mathematical analysis. The classical theory of curves and surfaces in three-dimensional Euclidean space is presented in the first three chapters. The abstract and modern topics of tensor algebra, Riemannian spaces and tensor analysis are studied in the last two chapters. A great number of illustrating examples, visualizations and genuine figures created by the authors' own software are included to support the understanding of the presented concepts and results, and to develop an adequate perception of the shapes of geometric objects, their properties and the relations between them.FeatuTable of Contents1. Curves in Three–dimensional Euclidean Space. 1.1. Points and Vectors. 1.2. Vector–valued Functions of a Real Variable. 1.3. The General Concept of Curves. 1.4. Some Examples of Planar Curves. 1.5. The Arc Length of a Curve. 1.6. The Vectors of the Trihedron of a Curve. 1.7. Frenet’s Formulae. 1.8. The Geometric Significance of Curvature and Torsion. 1.9. Osculating Circles and Spheres. 1.10. Involutes and Evolutes. 1.11. The Fundamental Theorem of Curves. 1.12. Lines of Constant Slope. 1.13. Spherical Images of a Curve. 2. Surfaces in Three–dimensional Euclidean Space. 2.1. Surfaces and Curves on Surfaces. 2.2. The Tangent Planes and Normal Vectors of a Surface. 2.3. The Arc Length, Angles and Gauss’s First Fundamental Coefficients. 2.4. the Curvature of Curves on Surfaces, Geodesic and Normal Curvature. 2.5. The Normal, Principal, Gaussian and Mean Curvature. 2.6. The Shape of a Surface in the Neighbourhood of a Point. 2.7. Dupin’s Indicatrix. 2.8. Lines of Curvature and Asymptotic Lines. 2.9. Triple Orthogonal Systems. 2.10. the Weingarten Equations. 3. The Intrinsic Geometry of Surfaces. 3.1. the Christoffel Symbols. 3.2. Geodesic Lines. 3.3. Geodesic Lines on Surfaces with Orthogonal Parameters. 3.4. Geodesic Lines on Surfaces of Revolution. 3.5. the Minimum Property of Geodesic Lines. 3.6. Orthogonal and Geodesic Parameters. 3.7. Levi–civitá Parallelism. 3.8. Theorema Egregium. 3.9. Maps Between Surfaces. 3.10. the Gauss–bonnet Theorem. 3.11. Minimal Surfaces. 4. Tensor Algebra and Riemannian Geometry. 4.1. Differentiable Manifolds. 4.2. Transformation of Bases. 4.3. Linear Functionals and Dual Spaces. 4.4. Tensors of Second Order. 4.5. Symmetric Bilinear Forms and Inner Products. 4.6. Tensors of Arbitary Order. 4.7. Symmetric and Anti–symmetric Tensors. 4.8. Riemann Spaces. 4.9. the Christoffel Symbols. 5. Tensor Analysis. 5.1. Covariant Differentiation. 5.2. the Covariant Derivative of an (R, S)–tensor. 5.3. the Interchange of Order for Covariant Differentiation and Ricci’s Identity. 5.4. Bianchi’s Identities for the Covariant Derivative of the Tensors of Curvature. 5.5. Beltrami’s Differentiators. 5.6. a Geometric Meaning of the Covariant Differentiation, the Levi–civitá Parallelism. 5.7. The Fundamental Theorem for Surfaces. 5.8. A Geometric Meaning of the Riemann Tensor of Curvature. 5.9. Spaces With Vanishing Tensor of Curvature. 5.10. An Extension of Frenet’s Formulae. 5.11. Riemann Normal Coordinates and the Curvature of Spaces.
£80.74
Cambridge University Press The Discrepancy Method
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£51.29
Cambridge University Press A Brief Guide to Algebraic Number Theory 50 London Mathematical Society Student Texts Series Number 50
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£44.64
Cambridge University Press A Primer of Analytic Number Theory
Book SynopsisThis 2003 undergraduate-level introduction describes those mathematical properties of prime numbers that can be deduced with the tools of calculus. The capstone of the book is a brief presentation of the Riemann zeta function and of the significance of the Riemann Hypothesis.Trade Review'… excellent background reading for undergraduates at any stage of their course.' Zentralblatt für Mathematik'… this is a well-written book at the level of senior undergraduates.' Society for Industrial and Applied Mathematics'The book constitutes an excellent undergraduate introduction to classical analytical number theory. The author develops the subject from the very beginning in an extremely good and readable style. Although a wide variety of topics are presented in the book, the author has successfully placed a rich historical background to each of the discussed themes, which makes the text very lively … the text contains a rich supplement of exercises, brief sketches of more advanced ideas and extensive graphical support. The book can be recommended as a very good first introductory reading for all those who are seriously interested in analytical number theory.' EMS Newsletter'… a very readable account.' Mathematika'The general style is user-friendly and interactive … a well presented and stimulating informal introduction to a wide range of topics …'. Proceedings of the Edinburgh Mathematical SocietyTable of Contents1. Sums and differences; 2. Products and divisibility; 3. Order and magnitude; 4. Counterexamples; 5. Averages; 6. Prime number theorems; 7. Series; 8. The Basel problem; 9. Euler's product; 10. The Riemann zeta function; 11. Pell's equation; 12. Elliptic curves; 13. Symmetry; 14. Explicit formula.
£45.59
Cambridge University Press Spectral Thry Riemann ZetaFunction 127 Cambridge Tracts in Mathematics Series Number 127
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£49.39
Cambridge University Press Duality in Analytic Number Theory Cambridge Tracts in Mathematics 122 Cambridge Tracts in Mathematics Series Number 122
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Cambridge University Press University Teaching in Focus A learningcentred approach
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Cambridge University Press Spectral Decomposition A Paraphrase of the Scriptures Cambridge Tracts in Mathematics Series Number 113
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Cambridge University Press Modular Forms and Galois Cohomology 69 Cambridge Studies in Advanced Mathematics Series Number 69
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Cambridge University Press Divisors
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Cambridge University Press Irregularities of Distribution 89 Cambridge Tracts in Mathematics Series Number 89
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Cambridge University Press Sets of Multiples 118 Cambridge Tracts in Mathematics Series Number 118
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