Written by one of the subject's foremost experts, this book focuses on the central developments and modern methods of the advanced theory of abelian groups, while remaining accessible, as an introduction and reference, to the non-specialist. It provides a coherent source for results scattered throughout the research literature with lots of new proofs. The presentation highlights major trends that have radically changed the modern character of the subject, in particular, the use of homological methods in the structure theory of various classes of abelian groups, and the use of advanced set-theoretical methods in the study of un decidability problems. The treatment of the latter trend includes Shelah's seminal work on the un decidability in ZFC of Whitehead's Problem; while the treatment of the former trend includes an extensive (but non-exhaustive) study of p-groups, torsion-free groups, mixed groups and important classes of groups arising from ring theory. To prepare the reader to tackle these topics, the book reviews the fundamentals of abelian group theory and provides some background material from category theory, set theory, topology and homological algebra. An abundance of exercises are included to test the reader's comprehension, and to explore noteworthy extensions and related sidelines of the main topics. A list of open problems and questions, in each chapter, invite the reader to take an active part in the subject's further development.

The second edition of this book updates and expands upon a historically important collection of mathematical problems first published in the United States by Birkhauser in 1981. These problems serve as a record of the informal discussions held by a group of mathematicians at the Scottish Cafe in Lwow, Poland, between the two world wars. Many of them were leaders in the development of such areas as functional and real analysis, group theory, measure and set theory, probability, and topology. Finding solutions to the problems they proposed has been ongoing since World War II, with prizes offered in many cases to those who are successful. In the 35 years since the first edition published, several more problems have been fully or partially solved, but even today many still remain unsolved and several prizes remain unclaimed. In view of this, the editor has gathered new and updated commentaries on the original 193 problems. Some problems are solved for the first time in this edition. Included again in full are transcripts of lectures given by Stanislaw Ulam, Mark Kac, Antoni Zygmund, Paul Erdoes, and Andrzej Granas that provide amazing insights into the mathematical environment of Lwow before World War II and the development of The Scottish Book. Also new in this edition are a brief history of the University of Wroclaw's New Scottish Book, created to revive the tradition of the original, and some selected problems from it. The Scottish Book offers a unique opportunity to communicate with the people and ideas of a time and place that had an enormous influence on the development of mathematics and try their hand on the unsolved problems. Anyone in the general mathematical community with an interest in the history of modern mathematics will find this to be an insightful and fascinating read.

This book gathers selected contributions presented at the INdAM Meeting Structured Matrices in Numerical Linear Algebra: Analysis, Algorithms and Applications, held in Cortona, Italy on September 4-8, 2017. Highlights cutting-edge research on Structured Matrix Analysis, it covers theoretical issues, computational aspects, and applications alike. The contributions, written by authors from the foremost international groups in the community, trace the main research lines and treat the main problems of current interest in this field. The book offers a valuable resource for all scholars who are interested in this topic, including researchers, PhD students and post-docs.

This book features selected papers from The Seventh International Conference on Research and Education in Mathematics that was held in Kuala Lumpur, Malaysia from 25 - 27th August 2015. With chapters devoted to the most recent discoveries in mathematics and statistics and serve as a platform for knowledge and information exchange between experts from academic and industrial sectors, it covers a wide range of topics, including numerical analysis, fluid mechanics, operation research, optimization, statistics and game theory. It is a valuable resource for pure and applied mathematicians, statisticians, engineers and scientists, and provides an excellent overview of the latest research in mathematical sciences.

This monograph is about monotone complete C*-algebras, their properties and the new classification theory. A self-contained introduction to generic dynamics is also included because of its important connections to these algebras. Our knowledge and understanding of monotone complete C*-algebras has been transformed in recent years. This is a very exciting stage in their development, with much discovered but with many mysteries to unravel. This book is intended to encourage graduate students and working mathematicians to attack some of these difficult questions. Each bounded, upward directed net of real numbers has a limit. Monotone complete algebras of operators have a similar property. In particular, every von Neumann algebra is monotone complete but the converse is false. Written by major contributors to this field, Monotone Complete C*-algebras and Generic Dynamics takes readers from the basics to recent advances. The prerequisites are a grounding in functional analysis, some point set topology and an elementary knowledge of C*-algebras.

This book is divided into two parts, one theoretical and one focusing on applications, and offers a complete description of the Canonical Groebner Cover, the most accurate algebraic method for discussing parametric polynomial systems. It also includes applications to the Automatic Deduction of Geometric Theorems, Loci Computation and Envelopes. The theoretical part is a self-contained exposition on the theory of Parametric Groebner Systems and Bases. It begins with Weispfenning's introduction of Comprehensive Groebner Systems (CGS) in 1992, and provides a complete description of the Groebner Cover (GC), which includes a canonical discussion of a set of parametric polynomial equations developed by Michael Wibmer and the author. In turn, the application part selects three problems for which the Groebner Cover offers valuable new perspectives. The automatic deduction of geometric theorems (ADGT) becomes fully automatic and straightforward using GC, representing a major improvement on all previous methods. In terms of loci and envelope computation, GC makes it possible to introduce a taxonomy of the components and automatically compute it. The book also generalizes the definition of the envelope of a family of hypersurfaces, and provides algorithms for its computation, as well as for discussing how to determine the real envelope. All the algorithms described here have also been included in the software library "grobcov.lib" implemented in Singular by the author, and serve as a User Manual for it.

This book introduces the fundamental concepts, methods, and applications of Hausdorff calculus, with a focus on its applications in fractal systems. Topics such as the Hausdorff diffusion equation, Hausdorff radial basis function, Hausdorff derivative nonlinear systems, PDE modeling, statistics on fractals, etc. are discussed in detail. It is an essential reference for researchers in mathematics, physics, geomechanics, and mechanics.

This proceedings volume covers a range of research topics in algebra from the Southern Regional Algebra Conference (SRAC) that took place in March 2017. Presenting theory as well as computational methods, featured survey articles and research papers focus on ongoing research in algebraic geometry, ring theory, group theory, and associative algebras. Topics include algebraic groups, combinatorial commutative algebra, computational methods for representations of groups and algebras, group theory, Hopf-Galois theory, hypergroups, Lie superalgebras, matrix analysis, spherical and algebraic spaces, and tropical algebraic geometry. Since 1988, SRAC has been an important event for the algebra research community in the Gulf Coast Region and surrounding states, building a strong network of algebraists that fosters collaboration in research and education. This volume is suitable for graduate students and researchers interested in recent findings in computational and theoretical methods in algebra and representation theory.

The most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature - from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world. With admirable clarity, two veteran math educators take us on a fascinating tour of the many ramifications of the Fibonacci numbers. They begin with a brief history of a distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples). In art, architecture, the stock market, and other areas of society and culture, they point out numerous examples of the Fibonacci sequence as well as its derivative, the "golden ratio." And of course in mathematics, as the authors amply demonstrate, there are almost boundless applications in probability, number theory, geometry, algebra, and Pascal's triangle, to name a few.Accessible and appealing to even the most math-phobic individual, this fun and enlightening book allows the reader to appreciate the elegance of mathematics and its amazing applications in both natural and cultural settings.

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This fourth volume collects authoritative chapters covering several applications of fractional calculus in physics, including classical and continuum mechanics.

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This sixth volume collects authoritative chapters covering several applications of fractional calculus in control theory, including fractional controllers, design methods and toolboxes, and a large number of engineering applications of control.

This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This fifth volume collects authoritative chapters covering several applications of fractional calculus in physics, including electrodynamics, statistical physics and physical kinetics, and quantum theory.

This book begins with the fundamentals of the generalized inverses, then moves to more advanced topics. It presents a theoretical study of the generalization of Cramer's rule, determinant representations of the generalized inverses, reverse order law of the generalized inverses of a matrix product, structures of the generalized inverses of structured matrices, parallel computation of the generalized inverses, perturbation analysis of the generalized inverses, an algorithmic study of the computational methods for the full-rank factorization of a generalized inverse, generalized singular value decomposition, imbedding method, finite method, generalized inverses of polynomial matrices, and generalized inverses of linear operators. This book is intended for researchers, postdocs, and graduate students in the area of the generalized inverses with an undergraduate-level understanding of linear algebra.

This collaborative book presents recent trends on the study of sequences, including combinatorics on words and symbolic dynamics, and new interdisciplinary links to group theory and number theory. Other chapters branch out from those areas into subfields of theoretical computer science, such as complexity theory and theory of automata. The book is built around four general themes: number theory and sequences, word combinatorics, normal numbers, and group theory. Those topics are rounded out by investigations into automatic and regular sequences, tilings and theory of computation, discrete dynamical systems, ergodic theory, numeration systems, automaton semigroups, and amenable groups. This volume is intended for use by graduate students or research mathematicians, as well as computer scientists who are working in automata theory and formal language theory. With its organization around unified themes, it would also be appropriate as a supplemental text for graduate level courses.

This book offers an introduction to applications prompted by tensor analysis, especially by the spectral tensor theory developed in recent years. It covers applications of tensor eigenvalues in multilinear systems, exponential data fitting, tensor complementarity problems, and tensor eigenvalue complementarity problems. It also addresses higher-order diffusion tensor imaging, third-order symmetric and traceless tensors in liquid crystals, piezoelectric tensors, strong ellipticity for elasticity tensors, and higher-order tensors in quantum physics. This book is a valuable reference resource for researchers and graduate students who are interested in applications of tensor eigenvalues.

This edited volume features a curated selection of research in algebraic combinatorics that explores the boundaries of current knowledge in the field. Focusing on topics experiencing broad interest and rapid growth, invited contributors offer survey articles on representation theory, symmetric functions, invariant theory, and the combinatorics of Young tableaux. The volume also addresses subjects at the intersection of algebra, combinatorics, and geometry, including the study of polytopes, lattice points, hyperplane arrangements, crystal graphs, and Grassmannians. All surveys are written at an introductory level that emphasizes recent developments and open problems. An interactive tutorial on Schubert Calculus emphasizes the geometric and topological aspects of the topic and is suitable for combinatorialists as well as geometrically minded researchers seeking to gain familiarity with relevant combinatorial tools. Featured authors include prominent women in the field known for their exceptional writing of deep mathematics in an accessible manner. Each article in this volume was reviewed independently by two referees. The volume is suitable for graduate students and researchers interested in algebraic combinatorics.

A consistent and near complete survey of the important progress made in the field over the last few years, with the main emphasis on the rigidity method and its applications. Among others, this monograph presents the most successful existence theorems known and construction methods for Galois extensions as well as solutions for embedding problems combined with a collection of the existing Galois realizations.

This book presents tensors and tensor analysis as primary mathematical tools for engineering and engineering science students and researchers. The discussion is based on the concepts of vectors and vector analysis in three-dimensional Euclidean space, and although it takes the subject matter to an advanced level, the book starts with elementary geometrical vector algebra so that it is suitable as a first introduction to tensors and tensor analysis. Each chapter includes a number of problems for readers to solve, and solutions are provided in an Appendix at the end of the text. Chapter 1 introduces the necessary mathematical foundations for the chapters that follow, while Chapter 2 presents the equations of motions for bodies of continuous material. Chapter 3 offers a general definition of tensors and tensor fields in three-dimensional Euclidean space. Chapter 4 discusses a new family of tensors related to the deformation of continuous material. Chapter 5 then addresses constitutive equations for elastic materials and viscous fluids, which are presented as tensor equations relating the tensor concept of stress to the tensors describing deformation, rate of deformation and rotation. Chapter 6 investigates general coordinate systems in three-dimensional Euclidean space and Chapter 7 shows how the tensor equations discussed in chapters 4 and 5 are presented in general coordinates. Chapter 8 describes surface geometry in three-dimensional Euclidean space, Chapter 9 includes the most common integral theorems in two- and three-dimensional Euclidean space applied in continuum mechanics and mathematical physics.

This book introduces the fundamental concepts, techniques and results of linear algebra that form the basis of analysis, applied mathematics and algebra. Intended as a text for undergraduate students of mathematics, science and engineering with a knowledge of set theory, it discusses the concepts that are constantly used by scientists and engineers. It also lays the foundation for the language and framework for modern analysis and its applications. Divided into seven chapters, it discusses vector spaces, linear transformations, best approximation in inner product spaces, eigenvalues and eigenvectors, block diagonalisation, triangularisation, Jordan form, singular value decomposition, polar decomposition, and many more topics that are relevant to applications. The topics chosen have become well-established over the years and are still very much in use. The approach is both geometric and algebraic. It avoids distraction from the main theme by deferring the exercises to the end of each section. These exercises aim at reinforcing the learned concepts rather than as exposing readers to the tricks involved in the computation. Problems included at the end of each chapter are relatively advanced and require a deep understanding and assimilation of the topics.

This unique collection of research papers offers a comprehensive and up-to-date guide to algebraic approaches to rough sets and reasoning with vagueness. It bridges important gaps, outlines intriguing future research directions, and connects algebraic approaches to rough sets with those for other forms of approximate reasoning. In addition, the book reworks algebraic approaches to axiomatic granularity. Given its scope, the book offers a valuable resource for researchers and teachers in the areas of rough sets and algebras of rough sets, algebraic logic, non classical logic, fuzzy sets, possibility theory, formal concept analysis, computational learning theory, category theory, and other formal approaches to vagueness and approximate reasoning. Consultants in AI and allied fields will also find the book to be of great practical value.

This monograph provides a modern introduction to the theory of quantales. First coined by C.J. Mulvey in 1986, quantales have since developed into a significant topic at the crossroads of algebra and logic, of notable interest to theoretical computer science. This book recasts the subject within the powerful framework of categorical algebra, showcasing its versatility through applications to C*- and MV-algebras, fuzzy sets and automata. With exercises and historical remarks at the end of each chapter, this self-contained book provides readers with a valuable source of references and hints for future research. This book will appeal to researchers across mathematics and computer science with an interest in category theory, lattice theory, and many-valued logic.

This volume examines mathematics as a product of the human mind and analyzes the language of "pure mathematics" from various advanced-level sources. Through analysis of the foundational texts of mathematics, it is demonstrated that math is a complex literary creation, containing objects, actors, actions, projection, prediction, planning, explanation, evaluation, roles, image schemas, metonymy, conceptual blending, and, of course, (natural) language. The book follows the narrative of mathematics in a typical order of presentation for a standard university-level algebra course, beginning with analysis of set theory and mappings and continuing along a path of increasing complexity. At each stage, primary concepts, axioms, definitions, and proofs will be examined in an effort to unfold the tell-tale traces of the basic human cognitive patterns of story and conceptual blending. This book will be of interest to mathematicians, teachers of mathematics, cognitive scientists, cognitive linguists, and anyone interested in the engaging question of how mathematics works and why it works so well.

This book addresses the mathematical aspects of modern image processing methods, with a special emphasis on the underlying ideas and concepts. It discusses a range of modern mathematical methods used to accomplish basic imaging tasks such as denoising, deblurring, enhancing, edge detection and inpainting. In addition to elementary methods like point operations, linear and morphological methods, and methods based on multiscale representations, the book also covers more recent methods based on partial differential equations and variational methods. Review of the German Edition: The overwhelming impression of the book is that of a very professional presentation of an appropriately developed and motivated textbook for a course like an introduction to fundamentals and modern theory of mathematical image processing. Additionally, it belongs to the bookcase of any office where someone is doing research/application in image processing. It has the virtues of a good and handy reference manual. (zbMATH, reviewer: Carl H. Rohwer, Stellenbosch)

A 2-book bundle that adds up to success in basic math and pre-algebra People of all ages -- from students in a math class to adults using math in the real world -- often dislike ever having to use arithmetic, fractions, decimals, percentages, geometry, and so on. Math is all around us, but it doesn't have to be hard. Basic Math & Pre-Algebra For Dummies and Basic Math & Pre-Algebra Workbook For Dummies with Online Practice is a two-book bundle designed to help you improve your math skills whether you're a student tackling math homework or preparing for a test or an adult who wants to improve their own skills or help someone else learn to love numbers. Basic Math & Pre-Algebra For Dummies provides mathematical and algebraic concepts into easy-to-understand sections. You'll learn to: Master addition, subtraction, multiplication, and division Wrangle the three E's of math: equations, expressions, and evaluation Never fear a word problem again (even ones with fractions, decimals, and percents) Know your prime numbers from composite numbers Get your mind around factors and multiples Learn the parts of a fraction and how to add, subtract, multiply, and divide fractions Understand scientific notation and how to use it Discover the X in algebra And so much more Basic Math & Pre-Algebra Workbook For Dummies covers negative numbers, fractions, decimals, geometry, graphing, basic algebra (and more), with plenty of problems so you can practice until your math is perfect. The workbook is divided into four sections, and you can work through each one or skip to any area you like: Part 1 covers basic math and pre-algebra, including addition, subtraction, multiplication, and division; operations; negative numbers; and expressions. Part 2 keeps things interesting with fractions, decimals, and percents. Part 3 dives into headier topics, like scientific notation; weights and measures; geometry; and Xy-graphs. Part 4 is all about algebraic expressions. Each topic is briefly introduced, and you're shown how to solve the problems in that section. You can review sample questions with answers (showing how to solve the problem) and then complete practice problems on your own. Don't worry -- the answers are provided (with step-by-step details), so you can grade your work. Additionally, online practice comes free with this workbook, with extra practice questions corresponding with each chapter. The workbook itself has instruction on how to register to access and use the online practice. Together these two books pack a powerful one-two punch. Grab this bundle to help you wrangle any math problem you encounter.

This book provides a comprehensive, in-depth overview of elementary mathematics as explored in Mathematical Olympiads around the world. It expands on topics usually encountered in high school and could even be used as preparation for a first-semester undergraduate course. This third and last volume covers Counting, Generating Functions, Graph Theory, Number Theory, Complex Numbers, Polynomials, and much more. As part of a collection, the book differs from other publications in this field by not being a mere selection of questions or a set of tips and tricks that applies to specific problems. It starts from the most basic theoretical principles, without being either too general or too axiomatic. Examples and problems are discussed only if they are helpful as applications of the theory. Propositions are proved in detail and subsequently applied to Olympic problems or to other problems at the Olympic level. The book also explores some of the hardest problems presented at National and International Mathematics Olympiads, as well as many essential theorems related to the content. An extensive Appendix offering hints on or full solutions for all difficult problems rounds out the book.