New statements of problems arose recently demanding thorough ana lysis. Notice, first of all, the statements of problems using adjoint equations which gradually became part of our life. Adjoint equations are capable to bring fresh ideas to various problems of new technology based on linear and nonlinear processes. They became part of golden fund of science through quantum mechanics, theory of nuclear reactors, optimal control, and finally helped in solving many problems on the basis of perturbation method and sensitivity theory. To emphasize the important role of adjoint problems in science one should mention four-dimensional analysis problem and solution of inverse problems. This range of problems includes first of all problems of global climate changes on our planet, state of environment and protection of environ ment against pollution, preservation of the biosphere in conditions of vigorous growth of population, intensive development of industry, and many others. All this required complex study of large systems: interac tion between the atmosphere and oceans and continents in the theory of climate, cenoses in the biosphere affected by pollution of natural and anthropogenic origin. Problems of local and global perturbations and models sensitivity to input data join into common complex system."

These proceedings collect several number theory articles, most of which were written in connection to the workshop WIN4: Women in Numbers, held in August 2017, at the Banff International Research Station (BIRS) in Banff, Alberta, Canada. It collects papers disseminating research outcomes from collaborations initiated during the workshop as well as other original research contributions involving participants of the WIN workshops. The workshop and this volume are part of the WIN network, aimed at highlighting the research of women and gender minorities in number theory as well as increasing their participation and boosting their potential collaborations in number theory and related fields.

Combinatorics and finite fields are of great importance in modern applications such as in the analysis of algorithms, in information and communication theory, and in signal processing and coding theory. This book contains survey articles on topics such as difference sets, polynomials, and pseudorandomness.

Introduction to Number Theory covers the essential content of an introductory number theory course including divisibility and prime factorization, congruences, and quadratic reciprocity. The instructor may also choose from a collection of additional topics. Aligning with the trend toward smaller, essential texts in mathematics, the author strives for clarity of exposition. Proof techniques and proofs are presented slowly and clearly. The book employs a versatile approach to the use of algebraic ideas. Instructors who wish to put this material into a broader context may do so, though the author introduces these concepts in a non-essential way. A final chapter discusses algebraic systems (like the Gaussian integers) presuming no previous exposure to abstract algebra. Studying general systems urges students realize unique factorization into primes is a more subtle idea than may at first appear; students will find this chapter interesting, fun and quite accessible. Applications of number theory include several sections on cryptography and other applications to further interest instructors and students alike.

This book is devoted to mean-square and weak approximations of solutions of stochastic differential equations (SDE). These approximations represent two fundamental aspects in the contemporary theory of SDE. Firstly, the construction of numerical methods for such systems is important as the solutions provided serve as characteristics for a number of mathematical physics problems. Secondly, the employment of probability representations together with a Monte Carlo method allows us to reduce the solution of complex multidimensional problems of mathematical physics to the integration of stochastic equations. Along with a general theory of numerical integrations of such systems, both in the mean-square and the weak sense, a number of concrete and sufficiently constructive numerical schemes are considered. Various applications and particularly the approximate calculation of Wiener integrals are also dealt with. This book is of interest to graduate students in the mathematical, physical and engineering sciences, and to specialists whose work involves differential equations, mathematical physics, numerical mathematics, the theory of random processes, estimation and control theory.

A collection of expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held at Boston University. The purpose of the conference, and indeed of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to show, at long last, that Fermats Last Theorem is true. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. In recognition of the historical significance of Fermats Last Theorem, the volume concludes by reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Indispensable for students and professional mathematicians alike.

Even three decades ago, the words 'combinatorial algebra' contrasting, for in stance, the words 'combinatorial topology,' were not a common designation for some branch of mathematics. The collocation 'combinatorial group theory' seems to ap pear first as the title of the book by A. Karras, W. Magnus, and D. Solitar [182] and, later on, it served as the title of the book by R. C. Lyndon and P. Schupp [247]. Nowadays, specialists do not question the existence of 'combinatorial algebra' as a special algebraic activity. The activity is distinguished not only by its objects of research (that are effectively given to some extent) but also by its methods (ef fective to some extent). To be more exact, we could approximately define the term 'combinatorial algebra' for the purposes of this book, as follows: So we call a part of algebra dealing with groups, semi groups , associative algebras, Lie algebras, and other algebraic systems which are given by generators and defining relations {in the first and particular place, free groups, semigroups, algebras, etc. )j a part in which we study universal constructions, viz. free products, lINN-extensions, etc. j and, finally, a part where specific methods such as the Composition Method (in other words, the Diamond Lemma, see [49]) are applied. Surely, the above explanation is far from covering the full scope of the term (compare the prefaces to the books mentioned above).

In the present book, we have put together the basic theory of the units and cuspidal divisor class group in the modular function fields, developed over the past few years. Let i) be the upper half plane, and N a positive integer. Let r(N) be the subgroup of SL (Z) consisting of those matrices == 1 mod N. Then r(N)\i) 2 is complex analytic isomorphic to an affine curve YeN), whose compactifi cation is called the modular curve X(N). The affine ring of regular functions on yeN) over C is the integral closure of C j] in the function field of X(N) over C. Here j is the classical modular function. However, for arithmetic applications, one considers the curve as defined over the cyclotomic field Q(JlN) of N-th roots of unity, and one takes the integral closure either of Q j] or Z j], depending on how much arithmetic one wants to throw in. The units in these rings consist of those modular functions which have no zeros or poles in the upper half plane. The points of X(N) which lie at infinity, that is which do not correspond to points on the above affine set, are called the cusps, because of the way they look in a fundamental domain in the upper half plane. They generate a subgroup of the divisor class group, which turns out to be finite, and is called the cuspidal divisor class group."

This book collects more than thirty contributions in memory of Wolfgang Schwarz, most of which were presented at the seventh International Conference on Elementary and Analytic Number Theory (ELAZ), held July 2014 in Hildesheim, Germany. Ranging from the theory of arithmetical functions to diophantine problems, to analytic aspects of zeta-functions, the various research and survey articles cover the broad interests of the well-known number theorist and cherished colleague Wolfgang Schwarz (1934-2013), who contributed over one hundred articles on number theory, its history and related fields. Readers interested in elementary or analytic number theory and related fields will certainly find many fascinating topical results among the contributions from both respected mathematicians and up-and-coming young researchers. In addition, some biographical articles highlight the life and mathematical works of Wolfgang Schwarz.

In recent times, group theory has found wider applications in various fields of algebra and mathematics in general. But in order to apply this or that result, you need to know about it, and such results are often diffuse and difficult to locate, necessitating that readers construct an extended search through multiple monographs, articles, and papers. Such readers must wade through the morass of concepts and auxiliary statements that are needed to understand the desired results, while it is initially unclear which of them are really needed and which ones can be dispensed with. A further difficulty that one may encounter might be concerned with the form or language in which a given result is presented. For example, if someone knows the basics of group theory, but does not know the theory of representations, and a group theoretical result is formulated in the language of representation theory, then that person is faced with the problem of translating this result into the language with which they are familiar, etc. Infinite Groups: A Roadmap to Some Classical Areas seeks to overcome this challenge. The book covers a broad swath of the theory of infinite groups, without giving proofs, but with all the concepts and auxiliary results necessary for understanding such results. In other words, this book is an extended directory, or a guide, to some of the more established areas of infinite groups. Features An excellent resource for a subject formerly lacking an accessible and in-depth reference Suitable for graduate students, PhD students, and researchers working in group theory Introduces the reader to the most important methods, ideas, approaches, and constructions in infinite group theory.

Ten years after a 1989 meeting of number theorists and physicists at the Centre de Physique des Houches, a second event focused on the broader interface of number theory, geometry, and physics. This book is the first of two volumes resulting from that meeting. Broken into three parts, it covers Conformal Field Theories, Discrete Groups, and Renormalization, offering extended versions of the lecture courses and shorter texts on special topics.

The present monograph develops a unified theory of Steinberg groups, independent of matrix representations, based on the theory of Jordan pairs and the theory of 3-graded locally finite root systems. The development of this approach occurs over six chapters, progressing from groups with commutator relations and their Steinberg groups, then on to Jordan pairs, 3-graded locally finite root systems, and groups associated with Jordan pairs graded by root systems, before exploring the volume's main focus: the definition of the Steinberg group of a root graded Jordan pair by a small set of relations, and its central closedness. Several original concepts, such as the notions of Jordan graphs and Weyl elements, provide readers with the necessary tools from combinatorics and group theory. Steinberg Groups for Jordan Pairs is ideal for PhD students and researchers in the fields of elementary groups, Steinberg groups, Jordan algebras, and Jordan pairs. By adopting a unified approach, anybody interested in this area who seeks an alternative to case-by-case arguments and explicit matrix calculations will find this book essential.

This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions. An Introduction to Diophantine Equations: A Problem-Based Approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants - including Olympiad and Putnam competitors - as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine examples, ideas, and techniques.

This volume presents research and expository papers highlighting the vibrant and fascinating study of irregularities in the distribution of primes. Written by an international group of experts, contributions present a self-contained yet unified exploration of a rapidly progressing area. Emphasis is given to the research inspired by Maier's matrix method, which established a newfound understanding of the distribution of primes. Additionally, the book provides an historical overview of a large body of research in analytic number theory and approximation theory. The papers published within are intended as reference tools for graduate students and researchers in mathematics.

Praise for the First Edition beautiful and well worth the reading with many exercises and a good bibliography, this book will fascinate both students and teachers. Mathematics Teacher Fibonacci and Lucas Numbers with Applications, Volume I, Second Edition provides a user-friendly and historical approach to the many fascinating properties of Fibonacci and Lucas numbers, which have intrigued amateurs and professionals for centuries. Offering an in-depth study of the topic, this book includes exciting applications that provide many opportunities to explore and experiment. In addition, the book includes a historical survey of the development of Fibonacci and Lucas numbers, with biographical sketches of important figures in the field. Each chapter features a wealth of examples, as well as numeric and theoretical exercises that avoid using extensive and time-consuming proofs of theorems. The Second Edition offers new opportunities to illustrate and expand on various problem-solving skills and techniques. In addition, the book features: A clear, comprehensive introduction to one of the most fascinating topics in mathematics, including links to graph theory, matrices, geometry, the stock market, and the Golden Ratio Abundant examples, exercises, and properties throughout, with a wide range of difficulty and sophistication Numeric puzzles based on Fibonacci numbers, as well as popular geometric paradoxes, and a glossary of symbols and fundamental properties from the theory of numbers A wide range of applications in many disciplines, including architecture, biology, chemistry, electrical engineering, physics, physiology, and neurophysiology The Second Edition is appropriate for upper-undergraduate and graduate-level courses on the history of mathematics, combinatorics, and number theory. The book is also a valuable resource for undergraduate research courses, independent study projects, and senior/graduate theses, as well as a useful resource for computer scientists, physicists, biologists, and electrical engineers. Thomas Koshy, PhD, is Professor Emeritus of Mathematics at Framingham State University in Massachusetts and author of several books and numerous articles on mathematics. His work has been recognized by the Association of American Publishers, and he has received many awards, including the Distinguished Faculty of the Year. Dr. Koshy received his PhD in Algebraic Coding Theory from Boston University. Anyone who loves mathematical puzzles, number theory, and Fibonacci numbers will treasure this book. Dr. Koshy has compiled Fibonacci lore from diverse sources into one understandable and intriguing volume, [interweaving] a historical flavor into an array of applications. Marjorie Bicknell-Johnson

This volume presents a collection of results related to the BSD conjecture, based on the first two India-China conferences on this topic. It provides an overview of the conjecture and a few special cases where the conjecture is proved. The broad theme of the two conferences was "Theoretical and Computational Aspects of the Birch and Swinnerton-Dyer Conjecture". The first was held at Beijing International Centre for Mathematical Research (BICMR) in December 2014 and the second was held at the International Centre for Theoretical Sciences (ICTS), Bangalore, India in December 2016. Providing a broad overview of the subject, the book is a valuable resource for young researchers wishing to work in this area. The articles have an extensive list of references to enable diligent researchers to gain an idea of the current state of art on this conjecture.

Tamari lattices originated from weakenings or reinterpretations of the familar associativity law. This has been the subject of Dov Tamari's thesis at the Sorbonne in Paris in 1951 and the central theme of his subsequent mathematical work. Tamari lattices can be realized in terms of polytopes called associahedra, which in fact also appeared first in Tamari's thesis.

By now these beautiful structures have made their appearance in many different areas of pure and applied mathematics, such as algebra, combinatorics, computer science, category theory, geometry, topology, and also in physics. Their interdisciplinary nature provides much fascination and value.

On the occasion of Dov Tamari's centennial birthday, this book provides an introduction to topical research related to Tamari's work and ideas. Most of the articles collected in it are written in a way accessible to a wide audience of students and researchers in mathematics and mathematical physics and are accompanied by high quality illustrations.

This book is intended for amateurs, students and teachers. The author presents partial results which could be obtained with exclusively elementary methods. The proofs are given in detail, with minimal prerequisites. An original feature are the ten interludes, devoted to important topics of elementary number theory, thus making the reading of this book self-contained. Their interest goes beyond Fermat's theorem. The Epilogue is a serious attempt to render accessible the strategy of the recent proof of Fermat's last theorem, a great mathematical feat.

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.

A Fibonacci-Based Pseudo-Random Number Generator.- On the Proof of GCD and LCM Equalities Concerning the Generalized Binomial and Multinomial Coefficients.- Supercube.- A Note on Fundamental Properties of Recurring Series.- Period Patterns of Certain Second-Order Linear Recurrences Modulo A Prime.- Nearly Isosceles Triangles Where the Vertex Angle Is a Multiple of the Base Angle.- The Ring of Fibonacci (Fibonacci "Numbers" With Matrix Subscript).- One-Relator Products of Cyclic Groups and Fibonacci-Like Sequences.- A Generalization of the Fibonacci Search.- Pascal's Triangle: Top Gun or Just One of the Gang?.- Conversion of Fibonacci Identities into Hyperbolic Identities Valid for an Arbitrary Argument.- Derivative Sequences of Fibonacci and Lucas Polynomials.- A Carry Theorem for Rational Binomial Coefficients.- On Co-Related Sequences Involving Generalized Fibonacci Numbers.- Fibonacci and B-Adic Trees in Mosaic Graphs.- Fibonacci Representations of Graphs.- On the Sizes of Elements in the Complement of a Submonoid of Integers.- Genocchi Polynomials.- An Application of Zeckendorf's Theorem.- A New Kind of Golden Triangle.- Terms Common to Two Sequences Satisfying the Same Linear Recurrence.- Recurrence Relations in Exponential Functions and in Damped Sinusoids and Their Applications in Electronics.- Some Basic Properties of the Fibonacci Line-Sequence.- De Moivre-Type Identities for the Tetrabonacci Numbers.- Two Generalizations of Gould's Star of David Theorem.- On Triangular Lucas Numbers.- A Fast Algorithm of the Chinese Remainder Theorem and Its Application to Fibonacci Numbers.- Generating the Pythagorean Triples Via Simple Continued Fractions.- On the Moebius Knot Tree and Euclid's Algorithm.- Generalized Fibonacci and Lucas Factorizations.- On Even Fibonacci Pseudoprimes.- Possible Restricted Periods of Certain Lucas Sequences Modulo P.- Using Matrix Techniques to Establish Properties of a Generalized Tribonacci Sequence.

Some of the central topics in number theory, presnted in a simple and concise fashion. The author covers an amazing amount of material, despite a leisurely pace and emphasis on readability. His heartfelt enthusiasm enables readers to see what is magical about the subject. All the topics are presented in a refreshingly elegant and efficient manner with clever examples and interesting problems throughout. The text is suitable for a graduate course in analytic number theory.

When looking for applications of ring theory in geometry, one first thinks of algebraic geometry, which sometimes may even be interpreted as the concrete side of commutative algebra. However, this highly de veloped branch of mathematics has been dealt with in a variety of mono graphs, so that - in spite of its technical complexity - it can be regarded as relatively well accessible. While in the last 120 years algebraic geometry has again and again attracted concentrated interes- which right now has reached a peak once more -, the numerous other applications of ring theory in geometry have not been assembled in a textbook and are scattered in many papers throughout the literature, which makes it hard for them to emerge from the shadow of the brilliant theory of algebraic geometry. It is the aim of these proceedings to give a unifying presentation of those geometrical applications of ring theo y outside of algebraic geometry, and to show that they offer a considerable wealth of beauti ful ideas, too. Furthermore it becomes apparent that there are natural connections to many branches of modern mathematics, e. g. to the theory of (algebraic) groups and of Jordan algebras, and to combinatorics. To make these remarks more precise, we will now give a description of the contents. In the first chapter, an approach towards a theory of non-commutative algebraic geometry is attempted from two different points of view."

In this book, we first review the history and current situation of the perfect number problem, including the origin story of the Mersenne primes, and then consider the history and current situation of the Fibonacci sequence. Both topics include results from our own research. In the later sections, we define the square sum perfect numbers, and describe for the first time the secret relationships connecting the square sum perfect numbers, the Fibonacci sequence, the Lucas sequence, the twin prime conjecture, and the Fermat primes. Throughout, we raise various interesting questions and conjectures.

The present book was conceived as an introduction for the user of universal algebra, rather than a handbook for the specialist, but when the first edition appeared in 1965, there were practically no other books entir ly devoted to the subject, whether introductory or specialized. Today the specialist in the field is well provided for, but there is still a demand for an introduction to the subject to suit the user, and this seemed to justify a reissue of the book. Naturally some changes have had to be made; in particular, I have corrected all errors that have been brought to my notice. Besides errors, some obscurities in the text have been removed and the references brought up to date. I should like to express my thanks to a number of correspondents for their help, in particular C. G. d'Ambly, W. Felscher, P. Goralcik, P. J. Higgins, H.-J. Hoehnke, J. R. Isbell, A. H. Kruse, E. J. Peake, D. Suter, J. S. Wilson. But lowe a special debt to G. M. Bergman, who has provided me with extensive comments. particularly on Chapter VII and the supplementary chapters. I have also con sulted reviews of the first edition, as well as the Italian and Russian translations."

If numbers were objects, how could there be human knowledge of number? Numbers are not physical objects: must we conclude that we have a mysterious power of perceiving the abstract realm? Or should we instead conclude that numbers are fictions? This book argues that numbers are not objects: they are magnitude properties. Properties are not fictions and we certainly have scientific knowledge of them. Much is already known about magnitude properties such as inertial mass and electric charge, and much continues to be discovered. The book says the same is true of numbers. In the theory of magnitudes, the categorial distinction between quantity and individual is of central importance, for magnitudes are properties of quantities, not properties of individuals. Quantity entails divisibility, so the logic of quantity needs mereology, the a priori logic of part and whole. The three species of quantity are pluralities, continua and series, and the book presents three variants of mereology, one for each species of quantity. Given Euclid's axioms of equality, it is possible without the use of set theory to deduce the axioms of the natural, real and ordinal numbers from the respective mereologies of pluralities, continua and series. Knowledge and the Philosophy of Number carries out these deductions, arriving at a metaphysics of number that makes room for our a priori knowledge of mathematical reality.