The Jacobi group is a semidirect product of a symplectic group with a Heisenberg group. It is an important example for a non-reductive group and sets the frame within which to treat theta functions as well as elliptic functions - in particular, the universal elliptic curve. This text gathers for the first time material from the representation theory of this group in both local (archimedean and non-archimedean) cases and in the global number field case. Via a bridge to Waldspurger's theory for the metaplectic group, complete classification theorems for irreducible representations are obtained. Further topics include differential operators, Whittaker models, Hecke operators, spherical representations and theta functions. The global theory is aimed at the correspondence between automorphic representations and Jacobi forms. This volume is thus a complement to the seminal book on Jacobi forms by M. Eichler and D. Zagier. Incorporating results of the authors' original research, this exposition is meant for researchers and graduate students interested in algebraic groups and number theory, in particular, modular and automorphic forms.

Elementary Number Theory, 6th Edition, blends classical theory with modern applications and is notable for its outstanding exercise sets. A full range of exercises, from basic to challenging, helps students explore key concepts and push their understanding to new heights. Computational exercises and computer projects are also available. Reflecting many years of professor feedback, this edition offers new examples, exercises, and applications, while incorporating advancements and discoveries in number theory made in the past few years.

New to the Fourth Edition Reorganised and revised chapter seven and thirteen New exercises and examples Expanded, updated references Further historical material on figures besides Galois: Omar Khayyam, Vandermonde, Ruffini, and Abel A new final chapter discussing other directions in which Galois Theory has developed: the inverse Galois problem, differential Galois theory, and a (very) brief introduction to p-adic Galois representations.

This volume contains the proceedings of the International Conference on Number Theory and Discrete Mathematics in honour of Srinivasa Ramanujan, held at the Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, India, in October 2000, as a contribution to the International Year of Mathematics. It collects 29 articles written by some of the leading specialists worldwide. Most of the papers provide recent trends, problems and their current states as well as historical backgrounds of their subjects. Some contributions are related to Ramanujan's mathematics, which should stimulate the interest in his work.

Pell's Equation is a very simple Diophantine equation that has been known to mathematicians for over 2000 years. Even today research involving this equation continues to be very active, as can be seen by the publication of at least 150 articles related to this equation over the past decade. However, very few modern books have been published on Pell's Equation, and this will be the first to give a historical development of the equation, as well as to develop the necessary tools for solving the equation.

The authors provide a friendly introduction for advanced undergraduates to the delights of algebraic number theory via Pell's Equation. The only prerequisites are a basic knowledge of elementary number theory and abstract algebra. There are also numerous references and notes for those who wish to follow up on various topics.

"About binomial theorems I'm teeming with a lot of news, With many cheerful facts about the square on the hypotenuse. " - William S. Gilbert (The Pirates of Penzance, Act I) The question of divisibility is arguably the oldest problem in mathematics. Ancient peoples observed the cycles of nature: the day, the lunar month, and the year, and assumed that each divided evenly into the next. Civilizations as separate as the Egyptians of ten thousand years ago and the Central American Mayans adopted a month of thirty days and a year of twelve months. Even when the inaccuracy of a 360-day year became apparent, they preferred to retain it and add five intercalary days. The number 360 retains its psychological appeal today because it is divisible by many small integers. The technical term for such a number reflects this appeal. It is called a "smooth" number. At the other extreme are those integers with no smaller divisors other than 1, integers which might be called the indivisibles. The mystic qualities of numbers such as 7 and 13 derive in no small part from the fact that they are indivisibles. The ancient Greeks realized that every integer could be written uniquely as a product of indivisibles larger than 1, what we appropriately call prime numbers. To know the decomposition of an integer into a product of primes is to have a complete description of all of its divisors.

The text that comprises this volume is a collection of surveys and original works from experts in the fields of algebraic number theory, analytic number theory, harmonic analysis, and hyperbolic geometry. A portion of the collected contributions have been developed from lectures given at the "International Conference on the Occasion of the 60th Birthday of S. J. Patterson," held at the University Gottingen, July 27-29 2009. Many of the included chapters have been contributed by invited participants.

This volume presents and investigates the most recent developments in various key topics in analytic number theory and several related areas of mathematics.

The volume is intended for graduate students and researchers of number theory as well as applied mathematicians interested in this broad field."

Proceedings of the NATO Advanced Study Institute, Antwerp, Belgium, August 2-12, 1983

This book arose from a course of lectures given by the first author during the winter term 1977/1978 at the University of Munster (West Germany). The course was primarily addressed to future high school teachers of mathematics; it was not meant as a systematic introduction to number theory but rather as a historically motivated invitation to the subject, designed to interest the audience in number-theoretical questions and developments. This is also the objective of this book, which is certainly not meant to replace any of the existing excellent texts in number theory. Our selection of topics and examples tries to show how, in the historical development, the investigation of obvious or natural questions has led to more and more comprehensive and profound theories, how again and again, surprising connections between seemingly unrelated problems were discovered, and how the introduction of new methods and concepts led to the solution of hitherto unassailable questions. All this means that we do not present the student with polished proofs (which in turn are the fruit of a long historical development); rather, we try to show how these theorems are the necessary consequences of natural questions. Two examples might illustrate our objectives."

This book introduces algebraic number theory through the problem of generalizing 'unique prime factorization' from ordinary integers to more general domains. Solving polynomial equations in integers leads naturally to these domains, but unique prime factorization may be lost in the process. To restore it, we need Dedekind's concept of ideals. However, one still needs the supporting concepts of algebraic number field and algebraic integer, and the supporting theory of rings, vector spaces, and modules. It was left to Emmy Noether to encapsulate the properties of rings that make unique prime factorization possible, in what we now call Dedekind rings. The book develops the theory of these concepts, following their history, motivating each conceptual step by pointing to its origins, and focusing on the goal of unique prime factorization with a minimum of distraction or prerequisites. This makes a self-contained easy-to-read book, short enough for a one-semester course.

This proceedings volume, the fifth in a series from the Combinatorial and Additive Number Theory (CANT) conferences, is based on talks from the 19th annual workshop, held online due to the COVID-19 pandemic. Organized every year since 2003 by the New York Number Theory Seminar at the CUNY Graduate Center, the workshops survey state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. The CANT 2021 meeting featured over a hundred speakers from North and South America, Europe, Asia, Australia, and New Zealand, and was the largest CANT conference in terms of the number of both lectures and participants. These proceedings contain peer-reviewed and edited papers on current topics in number theory. Topics featured in this volume include sumsets, minimal bases, Sidon sets, analytic and prime number theory, combinatorial and discrete geometry, numerical semigroups, and a survey of expansion, divisibility, and parity. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Nestled between number theory, combinatorics, algebra and analysis lies a rapidly developing subject in mathematics variously known as additive combinatorics, additive number theory, additive group theory, and combinatorial number theory. Its main objects of study are not abelian groups themselves, but rather the additive structure of subsets and subsequences of an abelian group, i.e., sumsets and subsequence sums. This text is a hybrid of a research monograph and an introductory graduate textbook. With few exceptions, all results presented are self-contained, written in great detail, and only reliant upon material covered in an advanced undergraduate curriculum supplemented with some additional Algebra, rendering this bookusable as an entry-level text. However, it will perhaps be of even more interest to researchers already in the field.

The majority of material is not found in book form and includes many new results as well. Even classical results, when included, are given in greater generality or using new proof variations. The text has a particular focus on results of a more exact and precise nature, results with strong hypotheses and yet stronger conclusions, and on fundamental aspects of the theory. Also included are intricate results often neglected in other texts owing to their complexity. Highlights include an extensive treatment of Freiman Homomorphisms and the Universal Ambient Group of sumsets A+B, an entire chapter devoted to Hamidoune s Isoperimetric Method, a novel generalization allowing infinite summands in finite sumset questions, weighted zero-sum problems treated in the general context of viewing homomorphisms as weights, and simplified proofs of the Kemperman Structure Theorem and the Partition Theorem for setpartitions."

The second of two volumes presenting papers from an international conference on analytic number theory. The two volumes contain 50 papers, with an emphasis on topics such as sieves, related combinatorial aspects, multiplicative number theory, additive number theory, and Riemann zeta-function.

Deals with the most basic notion of linear algebra, to bring emphasis on approaches to the topic serving at the elementary level and more broadly. A typical feature is where computational algorithms and theoretical proofs are brought together. Another is respect for symmetry, so that when this has some part in the form of a matter it should also be reflected in the treatment. Issues relating to computational method are covered. These interests may have suggested a limited account, to be rounded-out suitably. However this limitation where basic material is separated from further reaches of the subject has an appeal of its own. To the `elementary operations' method of the textbooks for doing linear algebra, Albert Tucker added a method with his `pivot operation'. Here there is a more primitive method based on the `linear dependence table', and yet another based on `rank reduction'. The determinant is introduced in a completely unusual upside-down fashion where Cramer's rule comes first. Also dealt with is what is believed to be a completely new idea, of the `alternant', a function associated with the affine space the way the determinant is with the linear space, with n+1 vector arguments, as the determinant has n. Then for affine (or barycentric) coordinates we find a rule which is an unprecedented exact counterpart of Cramer's rule for linear coordinates, where the alternant takes on the role of the determinant. These are among the more distinct or spectacular items for possible novelty, or unfamiliarity. Others, with or without some remark, may be found scattered in different places.

Over the last decade, the role of computational simulations in all aspects of aerospace design has steadily increased. However, despite the many advances, the time required for computations is far too long. This book examines new ideas and methodologies that may, in the next twenty years, revolutionize scientific computing. The book specifically looks at trends in algorithm research, human computer interface, network-based computing, surface modeling and grid generation and computer hardware and architecture. The book provides a good overview of the current state-of-the-art and provides guidelines for future research directions. The book is intended for computational scientists active in the field and program managers making strategic research decisions.

Proceedings of the second conference on Applied Mathematics and Scientific Computing, held June 4-9, 2001 in Dubrovnik, Croatia.

The main idea of the conference was to bring together applied mathematicians both from outside academia, as well as experts from other areas (engineering, applied sciences) whose work involves advanced mathematical techniques.

During the meeting there were one complete mini-course, invited presentations, contributed talks and software presentations. A mini-course Schwarz Methods for Partial Differential Equations was given by Prof Marcus Sarkis (Worcester Polytechnic Institute, USA), and invited presentations were given by active researchers from the fields of numerical linear algebra, computational fluid dynamics, matrix theory and mathematical physics (fluid mechanics and elasticity).

This volume contains the mini-course and review papers by invited speakers (Part I), as well as selected contributed presentations from the field of analysis, numerical mathematics, and engineering applications.

"Presents the proceedings of the recently held Third International Conference on Commutative Ring Theory in Fez, Morocco. Details the latest developments in commutative algebra and related areas-featuring 26 original research articles and six survey articles on fundamental topics of current interest. Examines wide-ranging developments in commutative algebra, together with connections to algebraic number theory and algebraic geometry."

This introduction to algebraic number theory via the famous problem of "Fermats Last Theorem" follows its historical development, beginning with the work of Fermat and ending with Kummers theory of "ideal" factorization. The more elementary topics, such as Eulers proof of the impossibilty of x+y=z, are treated in an uncomplicated way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummers theory to quadratic integers and relates this to Gauss'theory of binary quadratic forms, an interesting and important connection that is not explored in any other book.

What is the "most uniform" way of distributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? Such questions are treated in geometric discrepancy theory. The book is an accessible and lively introduction to this area, with numerous exercises and illustrations. In separate, more specialized parts, it also provides a comprehensive guide to recent research. Including a wide variety of mathematical techniques (from harmonic analysis, combinatorics, algebra etc.) in action on non-trivial examples, the book is suitable for a "special topic" course for early graduates in mathematics and computer science. Besides professional mathematicians, it will be of interest to specialists in fields where a large collection of objects should be "uniformly" represented by a smaller sample (such as high-dimensional numerical integration in computational physics or financial mathematics, efficient divide-and-conquer algorithms in computer science, etc.).

The 1980 Maratea NATO Advanced Study Institute (= ASI) followed the lines of the 1976 Liege NATO ASI. Indeed, the interest of boundary problems for linear evolution partial differential equations and systems is more and more acute because of the outstanding position of those problems in the mathematical description of the physical world, namely through sciences such as fluid dynamics, elastodynamics, electro dynamics, electromagnetism, plasma physics and so on. In those problems the question of the propagation of singularities of the solution has boomed these last years. Placed in its definitive mathematical frame in 1970 by L. Hormander, this branch -of the theory recorded a tremendous impetus in the last decade and is now eagerly studied by the most prominent research workers in the field of partial differential equations. It describes the wave phenomena connected with the solution of boundary problems with very general boundaries, by replacing the (generailly impossible) computation of a precise solution by a convenient asymptotic approximation. For instance, it allows the description of progressive waves in a medium with obstacles of various shapes, meeting classical phenomena as reflexion, refraction, transmission, and even more complicated ones, called supersonic waves, head waves, creeping waves, ****** The !'tudy of singularities uses involved new mathematical concepts (such as distributions, wave front sets, asymptotic developments, pseudo-differential operators, Fourier integral operators, microfunctions, *** ) but emerges as the most sensible application to physical problems. A complete exposition of the present state of this theory seemed to be still lacking.

The present book addresses a number of specific topics in computational number theory whereby the author is not attempting to be exhaustive in the choice of subjects. The book is organized as follows. Chapters 1 and 2 contain the theory and algorithms concerning Dedekind domains and relative extensions of number fields, and in particular the generalization to the relative case of the round 2 and related algorithms. Chapters 3, 4, and 5 contain the theory and complete algorithms concerning class field theory over number fields. The highlights are the algorithms for computing the structure of (Z_K/m)^*, of ray class groups, and relative equations for Abelian extensions of number fields using Kummer theory. Chapters 1 to 5 form a homogeneous subject matter which can be used for a 6 months to 1 year graduate course in computational number theory. The subsequent chapters deal with more miscellaneous subjects. Written by an authority with great practical and teaching experience in the field, this book together with the author's earlier book will become the standard and indispensable reference on the subject.

This book provides insight into the mathematics of Galerkin finite element method as applied to parabolic equations. The revised second edition has been influenced by recent progress in application of semigroup theory to stability and error analysis, particulatly in maximum-norm. Two new chapters have also been added, dealing with problems in polygonal, particularly noncovex, spatial domains, and with time discretization based on using Laplace transformation and quadrature.