The sign-solvability of a linear system implies that the signs of the entries of the solution are determined solely on the basis of the signs of the coefficients of the system. That it might be worthwhile and possible to investigate such linear systems was recognised by Samuelson in his classic book Foundations of Economic Analysis. Sign-solvability is part of a larger study which seeks to understand the special circumstances under which an algebraic, analytic or geometric property of a matrix can be determined from the combinatorial arrangement of the positive, negative and zero elements of the matrix. The large and diffuse body of literature connected with sign-solvability is presented as a coherent whole for the first time in this book, displaying it as a beautiful interplay between combinatorics and linear algebra. One of the features of this book is that algorithms that are implicit in many of the proofs have been explicitly described and their complexity has been commented on.

This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers in the specific field of their expertise and, in turn, written for early career researchers. As a survey of the current state of the art, this book will serve as a valuable reference and is particularly well suited as an introduction to the field of symmetries and integrability of difference equations. Therefore, the book will be welcomed by advanced undergraduate and graduate students as well as by more advanced researchers.

This is an exposition of the analytic theory of continued fractions in the complex domain with emphasis on applications and computational methods.

Vector algebra is a particularly weak point in undergraduate mathematics but seminal to understanding more advanced algebra topics. The book is meant as a primary book but might also be used as a supplement to courses in linear algebra and multivariable or vector calculus. There are no direct, current competitors at this level (undergraduate)

In the preface of this book, the authors express the view that 'a good working knowledge of injective modules is a sound investment for module theorists'. The existing literature on the subject has tended to deal with the applications of injective modules to ring theory. The aim of this tract is to demonstrate some of the applications of injective modules to commutative algebra. A number of well-known concepts and results which so far have been applicable principally to commutative rings are generalized to a non-commutative context. There are exercises and brief notes appended to each chapter to illustrate and extend the scope of the treatment in the main text. Together with the short bibliography the notes form a guide to sources of reading for students and researchers who wish to delve more exhaustively into the theory of injective modules. The tract is intended primarily for those who have some knowledge of the rudiments of commutative algebra, although these are recalled at the outset.

This book is summarizing the results of the workshop "Uniform Distribution and Quasi-Monte Carlo Methods" of the RICAM Special Semester on "Applications of Algebra and Number Theory" in October 2013. The survey articles in this book focus on number theoretic point constructions, uniform distribution theory, and quasi-Monte Carlo methods. As deterministic versions of the Monte Carlo method, quasi-Monte Carlo rules enjoy increasing popularity, with many fruitful applications in mathematical practice, as for example in finance, computer graphics, and biology. The goal of this book is to give an overview of recent developments in uniform distribution theory, quasi-Monte Carlo methods, and their applications, presented by leading experts in these vivid fields of research.

Algebra and number theory have always been counted among the most beautiful and fundamental mathematical areas with deep proofs and elegant results. However, for a long time they were not considered of any substantial importance for real-life applications. This has dramatically changed with the appearance of new topics such as modern cryptography, coding theory, and wireless communication. Nowadays we find applications of algebra and number theory frequently in our daily life. We mention security and error detection for internet banking, check digit systems and the bar code, GPS and radar systems, pricing options at a stock market, and noise suppression on mobile phones as most common examples. This book collects the results of the workshops "Applications of algebraic curves" and "Applications of finite fields" of the RICAM Special Semester 2013. These workshops brought together the most prominent researchers in the area of finite fields and their applications around the world. They address old and new problems on curves and other aspects of finite fields, with emphasis on their diverse applications to many areas of pure and applied mathematics.

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science.

Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Introduction to Finite and Infinite Dimensional Lie Algebras and Superalgebras introduces the theory of Lie superalgebras, their algebras, and their representations. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semi-simple Lie algebras. While discussing all classes of finite and infinite dimensional Lie algebras and Lie superalgebras in terms of their different classes of root systems, the book focuses on Kac-Moody algebras. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras.

The theory of local Chern characters used in commutative algebra originated in topology about thirty years ago, and from there was introduced in algebraic geometry. This book describes the theory in an algebraic setting, presenting recent research results and important algebraic applications, some of which come from the author's own work. It concentrates on the background in commutative algebra and homological algebra and describes the relations between these subjects, including extensive discussions of the homological conjectures and of the use of the Frobenius map. It will be particularly useful for graduate students and researchers.

Drawing on rich classroom observations of educators teaching in China and the U.S., this book details an innovative and effective approach to teaching algebra at the elementary level, namely, "teaching through example-based problem solving" (TEPS). Recognizing young children's particular cognitive and developmental capabilities, this book powerfully argues for the importance of infusing algebraic thinking into early grade mathematics teaching and illustrates how this has been achieved by teachers in U.S. and Chinese contexts. Documenting best practice and students' responses to example-based instruction, the text demonstrates that this TEPS approach - which involves the use of worked examples, representations, and deep questions - helps students learn and master fundamental mathematical ideas, making it highly effective in developing algebraic readiness and mathematical understanding. This text will benefit post-graduate students, researchers, and academics in the fields of mathematics, STEM, and elementary education, as well as algebra research more broadly. Those interested in teacher education, classroom practice, and developmental and cognitive psychology will also find this volume of interest.

An accessible introduction to the theoretical and computational aspects of linear algebra using MapleTM

Many topics in linear algebra can be computationally intensive, and software programs often serve as important tools for understanding challenging concepts and visualizing the geometric aspects of the subject. Principles of Linear Algebra with Maple uniquely addresses the quickly growing intersection between subject theory and numerical computation, providing all of the commands required to solve complex and computationally challenging linear algebra problems using Maple. The authors supply an informal, accessible, and easy-to-follow treatment of key topics often found in a first course in linear algebra.

Requiring no prior knowledge of the software, the book begins with an introduction to the commands and programming guidelines for working with Maple. Next, the book explores linear systems of equations and matrices, applications of linear systems and matrices, determinants, inverses, and Cramer's rule. Basic linear algebra topics such as vectors, dot product, cross product, and vector projection are explained, as well as the more advanced topics of rotations in space, rolling a circle along a curve, and the TNB Frame. Subsequent chapters feature coverage of linear transformations from Rn to Rm, the geometry of linear and affine transformations, least squares fits and pseudoinverses, and eigenvalues and eigenvectors.

The authors explore several topics that are not often found in introductory linear algebra books, including sensitivity to error and the effects of linear and affine maps on the geometry of objects. The Maple software highlights the topic's visual nature, as the book is complete with numerous graphics in two and three dimensions, animations, symbolic manipulations, numerical computations, and programming. In addition, a related Web site features supplemental material, including Maple code for each chapter's problems, solutions, and color versions of the book's figures.

Extensively class-tested to ensure an accessible presentation, Principles of Linear Algebra with Maple is an excellent book for courses on linear algebra at the undergraduate level. It is also an ideal reference for students and professionals who would like to gain a further understanding of the use of Maple to solve linear algebra problems.

This is the first volume of the two-volume book on linear algebra, in the University of Tokyo (UTokyo) Engineering Course.The objective of this volume is to present, from the engineering viewpoint, the standard mathematical results in linear algebra such as those on systems of equations and eigenvalue problems. In addition to giving mathematical theorems and formulas, it explains how the mathematical concepts such as rank, eigenvalues, and singular values are linked to engineering applications and numerical computations.In particular, the following four aspects are emphasized.

Introduction to Traveling Waves is an invitation to research focused on traveling waves for undergraduate and masters level students. Traveling waves are not typically covered in the undergraduate curriculum, and topics related to traveling waves are usually only covered in research papers, except for a few texts designed for students. This book includes techniques that are not covered in those texts. Through their experience involving undergraduate and graduate students in a research topic related to traveling waves, the authors found that the main difficulty is to provide reading materials that contain the background information sufficient to start a research project without an expectation of an extensive list of prerequisites beyond regular undergraduate coursework. This book meets that need and serves as an entry point into research topics about the existence and stability of traveling waves. Features Self-contained, step-by-step introduction to nonlinear waves written assuming minimal prerequisites, such as an undergraduate course on linear algebra and differential equations. Suitable as a textbook for a special topics course, or as supplementary reading for courses on modeling. Contains numerous examples to support the theoretical material. Supplementary MATLAB codes available via GitHub.

Hyperidentities are important formulae of second-order logic, and research in hyperidentities paves way for the study of second-order logic and second-order model theory.This book illustrates many important current trends and perspectives for the field of hyperidentities and their applications, of interest to researchers in modern algebra and discrete mathematics. It covers a number of directions, including the characterizations of the Boolean algebra of n-ary Boolean functions and the distributive lattice of n-ary monotone Boolean functions; the classification of hyperidentities of the variety of lattices, the variety of distributive (modular) lattices, the variety of Boolean algebras, and the variety of De Morgan algebras; the characterization of algebras with aforementioned hyperidentities; the functional representations of finitely-generated free algebras of various varieties of lattices and bilattices via generalized Boolean functions (De Morgan functions, quasi-De Morgan functions, super-Boolean functions, super-De Morgan functions, etc); the structural results for De Morgan algebras, Boole-De Morgan algebras, super-Boolean algebras, bilattices, among others.While problems of Boolean functions theory are well known, the present book offers alternative, more general problems, involving the concepts of De Morgan functions, quasi-De Morgan functions, super-Boolean functions, and super-De Morgan functions, etc. In contrast to other generalized Boolean functions discovered and investigated so far, these functions have clearly normal forms. This quality is of crucial importance for their applications in pure and applied mathematics, especially in discrete mathematics, quantum computation, quantum information theory, quantum logic, and the theory of quantum computers.

This EMS volume consists of two parts, written by leading scientists in the field of operator algebras and non-commutative geometry. The first part, written by M.Rordam, is on Elliott's classification program for nuclear C*-algebras. The emphasis is on the work of Kirchberg and the spectacular results by Kirchberg and Phillips giving a nearly complete classification, in terms of K-theoretic invariants, in the purely infinite case. This part of the program is described with almost full proofs beginning with Kirchberg's tensor product theorems and Kirchberg's embedding theorem for exact C*-algebras. The classification of finite simple C*-algebras starting with AF-algebras, and continuing with AT- and AH-algebras is covered, but mostly without proofs. The second part, written by E.Stormer, is a survey of the theory of of noncommutative entropy of automorphisms of C*-algebras and von Neumann algebras from its initiation by Connes and Stormer in 1975 till 2001.

For courses in algebra & trigonometry. "Your world is profoundly mathematical." Bob Blitzercontinues to support and inspire students with his engaging approach, makingthis text beloved year after year by students and instructors alike. Blitzer'sunique background in mathematics and behavioral science informs a wide range ofapplications, drawn from pop culture and up-to-date references, that appeal tostudents of all majors and connect math to students' lives.

This book developed from a course given by the author to undergraduate and postgraduate students. It takes up Matrix Theory, Antenna Theory, and Probability Theory in detail. The first chapter on matrix theory discusses in reasonable depth the theory of Lie Algebras leading upto Cartan's Classification Theory. It also discusses some basic elements of Functional Analysis and Operator Theory in infinite dimensional Banach and Hilbert spaces. The second chapter discusses Basic Probability Theory and the topics discussed find applications to Stochastic Filtering Theory for differential equations driven by white Gaussian noise. The third chapter is on Antenna Theory with a focus on Modern Quantum Antenna Theory. The book will be a valuable resource to students and early career researchers in the field of Mathametical Physics.

This is a collection of papers dedicated to Leonid Kantorovich and his work. Kantorovich was a Russian mathematician and economist who published in the three areas covered in this book: functional analysis, optimization, and mathematical economics. Kantorovich is credited as being amongst the first inventors of linear programming, the primary technique of optimization. Linear programming consists of creating a matrix of parameters relevant to a system and maximizing the unknown variables using those constraints. Kantorovich then applied this theory to optimal macroeconomic planning in a socialist economy, for which he received the Nobel prize. The present book is dedicated to the memory of Kantorovich, who died in 1986, but is more than a Festschrift. It contains original contributions from several researchers in the USSR never before seen in the US, which enhances the value of the volume. It is organized in a logical sequence from the mathematics to the applications of the theories to concrete problems.

This contributed volume brings together the highest quality expository papers written by leaders and talented junior mathematicians in the field of Commutative Algebra. Contributions cover a very wide range of topics, including core areas in Commutative Algebra and also relations to Algebraic Geometry, Algebraic Combinatorics, Hyperplane Arrangements, Homological Algebra, and String Theory. The book aims to showcase the area, especially for the benefit of junior mathematicians and researchers who are new to the field; it will aid them in broadening their background and to gain a deeper understanding of the current research in this area. Exciting developments are surveyed and many open problems are discussed with the aspiration to inspire the readers and foster further research.

This is the first monograph devoted to clean ring and matrix theory. It aims to study a theory of expressing an element in a ring as the sum of some special ones, such as idempotents, units, nilpotents, tripotents, involutions, etc. A matrix over such rings is thereby expressed as the sum of some special matrices. Also another topics on the behaviors of topological properties and *-properties of such rings are investigated.The book is based on the results of various published papers, particularly, by the authors'. It is accessible for students familiar with general abstract algebra, while the topics are interesting for researchers in the field of ring, matrix and operator theory.

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.

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science.