Algebra
This book has been carefully designed in accordance with the model syllabus prescribed by the University Grants Commission (UGC), India, for courses in Algebra and Linear Algebra. It is well-suited for undergraduate and postgraduate students of mathematics across Indian universities and other institutions offering similar curricula.To ensure comprehensive understanding, the book begins with a preliminary chapter on Set Theory, providing the foundational concepts necessary for a deeper grasp of abstract algebraic structures. The content then progresses to cover a wide range of essential topics including integers, groups, rings, and fields, as well as polynomials, vector spaces, linear transformations, matrices, and Boolean algebra.Written in a clear and accessible language, the book emphasizes conceptual clarity through illustrative examples and real-world applications. Special attention has been given to demonstrating how abstract algebraic concepts are relevant and useful in number theory and the theory of equations, particularly in understanding the roots of polynomials.To enhance the utility of the text, an additional chapter on fuzzy set theory has been included. This modern topic introduces students to a broader perspective on set-related concepts and logical structures, offering them insight into emerging mathematical frameworks.Overall, the book serves as a valuable and self-contained resource for academic learning and exam preparation in algebra and linear algebra.
Discrete and Algebraic Structures
This textbook presents the topics typically covered in a standard course on discrete structures. It is aimed at students of computer science and mathematics (teaching degree and Bachelor's/Master's) and is designed to accompany lectures, for self-study, and for exam preparation. Through explanatory introductions to definitions, numerous examples, counterexamples, diagrams, cross-references, and outlooks, the authors manage to present the wide range of topics concisely and comprehensibly. Numerous exercises facilitate the deepening of the material. Due to its compact presentation of all important discrete and algebraic structures and its extensive index, the book also serves as a reference for mathematicians, computer scientists, and natural scientists. Contents: From propositional and predicate logic to sets and combinatorics, numbers, relations and mappings, graphs, to the rich spectrum of algebraic structures, and a brief introduction to category theory. Additional chapters include rings and modules as well as matroids. This book is a translation of the second German edition. The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content, so the book may read stylistically differently from a conventional translation.
Ideal Theory of Commutative Rings and Monoids
This book offers a concise treatment of multiplicative ideal theory in the language of multiplicative monoids. It presents a systematic development of the theory of weak ideal systems and weak module systems on arbitrary commutative monoids. Examples of monoids that are investigated include, but are not limited to, Mori monoids, Laskerian monoids, Pr羹fer monoids and Krull monoids. An in-depth study of various constructions from ring theory is also provided, with an emphasis on polynomial rings, Kronecker function rings and Nagata rings. The target audience is graduate students and researchers in ring and semigroup theory.
Your First Calculus Journey
Your First Calculus Journey is a beginner-friendly guide that breaks down the fundamental concepts of calculus into simple, relatable explanations. Designed for those who are new to the concepts of calculus, this book aims to make complex mathematical ideas easy to grasp by using real-world examples and step-by-step explanations. Readers will explore essential topics such as functions, graphs, limits, derivatives, and integral. From understanding how a bike's speed relates to derivatives to visualizing integrals as filling a swimming pool, each chapter builds a strong foundation for deeper mathematical concepts. With clear explanations, coherent examples, and engaging visuals, Your First Calculus Journey is the perfect starting point for anyone to grasp the calculus concepts. Whether you're preparing for an upcoming class or simply curious about how calculus shapes the world, this book will guide you every step of the way.
Your First Calculus Journey
Your First Calculus Journey is a beginner-friendly guide that breaks down the fundamental concepts of calculus into simple, relatable explanations. Designed for those who are new to the concepts of calculus, this book aims to make complex mathematical ideas easy to grasp by using real-world examples and step-by-step explanations. Readers will explore essential topics such as functions, graphs, limits, derivatives, and integral. From understanding how a bike's speed relates to derivatives to visualizing integrals as filling a swimming pool, each chapter builds a strong foundation for deeper mathematical concepts. With clear explanations, coherent examples, and engaging visuals, Your First Calculus Journey is the perfect starting point for anyone to grasp the calculus concepts. Whether you're preparing for an upcoming class or simply curious about how calculus shapes the world, this book will guide you every step of the way.
Abstract Algebra
Abstract Algebra: An Interactive Approach, Third Edition is a new concept in learning modern algebra. Although all the expected topics are covered thoroughly and in the most popular order, the text offers much flexibility. Perhaps more significantly, the book gives professors and students the option of including technology in their courses. Each chapter in the textbook has a corresponding interactive Mathematica notebook and an interactive SageMath workbook that can be used in either the classroom or outside the classroom. Students will be able to visualize the important abstract concepts, such as groups and rings (by displaying multiplication tables), homomorphisms (by showing a line graph between two groups), and permutations. This, in turn, allows the students to learn these difficult concepts much more quickly and obtain a firmer grasp than with a traditional textbook. Thus, the colorful diagrams produced by Mathematica give added value to the students. Teachers can run the Mathematica or SageMath notebooks in the classroom in order to have their students visualize the dynamics of groups and rings. Students have the option of running the notebooks at home, and experiment with different groups or rings. Some of the exercises require technology, but most are of the standard type with various difficulty levels.The third edition is meant to be used in an undergraduate, single-semester course, reducing the breadth of coverage, size, and cost of the previous editions. Additional changes include: Binary operators are now in an independent section. The extended Euclidean algorithm is included. Many more homework problems are added to some sections. Mathematical induction is moved to Section 1.2. Despite the emphasis on additional software, the text is not short on rigor. All of the classical proofs are included, although some of the harder proofs can be shortened by using technology.
Introduction to Linear Algebra with Earth Science Applications
Student Solutions Manual for Gallian's Contemporary Abstract Algebra
Redesigned for the 11th edition of Contemporary Abstract Algebra, Student Solutions Manual, written by the author, has comprehensive solutions for all odd-numbered exercises and a large number of even-numbered exercises. This Manual also offers many alternative solutions to those appearing in the text. These will provide the student with a better understanding of the material. This is the only available student solutions manual prepared by the author of Contemporary Abstract Algebra, Eleventh Edition and the only official one. It is designed to supplement the text and the author's original approach to instruction.
Pragmatic Mathematics for Scientists and Engineers
This is a textbook on basic to intermediate mathematics for undergraduate students majoring in the physical sciences and engineering. Many chapters, covering topics like Green's functions, calculus of variations, and functions of a complex variable, are well-suited for graduate classes. Additionally, researchers can benefit from the book as a mathematical refresher for their professional work.The book provides readers with a fundamental understanding of underlying principles, using derivations based more on mathematical intuition rather than exposing them to multiple theorems, proofs, and lemmas. Each chapter includes highly relevant examples with detailed solutions and explanations, promoting a practical application of knowledge to real problems in the physical sciences. For the convenience of both students and instructors, there are end-of-chapter exercises with answers that can be easily utilized for assignments.The book is not a replacement for calculus textbooks, but rather a guide to the mathematics most relevant to the physical sciences and engineering.In conclusion, this book can be readily adapted for upper-level undergraduate and graduate classes, particularly those focusing on mathematical methods for students in physical sciences, applied mathematics, and engineering majors.
Space-Time Algebra of Sedeons
This book is a comprehensive guide to the space-time algebra of sixteen-component values "sedeons". This algebra is designed to provide a compact representation of equations that describe various physical systems. The book considers the symmetry of physical quantities concerning the operations of spatial and temporal inversion. This approach allows the formulation of a wide class of mathematical physics equations within a unified framework and enables the generalization of these equations for essential problems in electrodynamics, hydrodynamics, plasma physics, field theory, and quantum mechanics. In particular, it is shown that the broken symmetry between electricity and magnetism in electrodynamics equations is a result of choosing an asymmetric representation of these phenomena. The sedeonic algebra enables the formulation of Maxwell-like equations for the fields with a nonzero mass of quantum, which facilitates the calculation of energy for baryon-baryon interaction and the semi-classical interpretation of this interaction. It also allows one to generalize the hydrodynamics equations for the case of vortex turbulent flows and for a hydrodynamic two-fluid model of electron-ion plasma.
Algebra
For Waldorf teachers, math is often difficult to teach. On the one hand, memories of their own school days can cloud their view of the children's developmental needs, whereas, Steiner's numerous indications do not form a cohesive structure for the math curriculum. Thus, various ways of teaching were developed during the history of Waldorf education. Such diversity underscores the responsibility teachers carries for their lessons.This guide does not intend in any way to diminish this responsibility, but attempts to contribute to a unified view of Steiner indications for a developmentally appropriate math curriculum. This approach might differ from some existing methods, mainly in directly and quickly beginning math activities and avoiding pictures when introducing the numbers.This algebra manual is for Grades 6, 7, and 8. The indications given in the Waldorf school syllabus for teaching algebra in these three grades are as follows: Grade 6-- Starting with interest and percent, proceed to simple elements of business and banking arithmetic and, from there, working from interest go over into work with literal numbersGrade 7-- Study powers, roots, negative numbers, and the theory of simple equations, relating them all to practical lifeGrade 8-- Carry the work of both arithmetic and algebra further, sustaining it with manifold applications
Algebra
Algebra is a subject we have become acquainted with during most of our mathematical education, often in connection with the solution of equations. Algebra: Groups, Rings, and Fields, Second Edition deals with developments related to their solutions.The principle at the heart of abstract algebra, a subject that enables one to deduce sweeping conclusions from elementary premises, is that the process of abstraction enables us to solve a variety of such problems with economy of effort. This leads to the glorious world of mathematical discovery.This second edition follows the original three-pronged approach: the theory of finite groups, number theory, and Galois' amazing theory of field extensions tying solvability of equations to group theory.As algebra has branched out in many directions, the authors strive to keep the text manageable while at the same time introducing the student to exciting new paths. In order to support this approach, the authors broadened the first edition, giving monoids a greater role, and relying more on matrices. Hundreds of new exercises were added.A course in abstract algebra, properly presented, could treat mathematics as an art as well as a science. In this exposition, we try to present underlying ideas, as well as the results they yield.
An Introduction to Module Theory
Module theory is a fundamental area of algebra, taught in most universities at the graduate level. This textbook, written by two experienced teachers and researchers in the area, is based on courses given in their respective universities over the last thirty years. It is an accessible and modern account of module theory, meant as a textbook for graduate or advanced undergraduate students, though it can also be used for self-study. It is aimed at students in algebra, or students who need algebraic tools in their work. Following the recent trends in the area, the general approach stresses from the start the use of categorical and homological techniques. The book includes self-contained introductions to category theory and homological algebra with applications to Module theory, and also contains an introduction to representations of quivers. It includes a very large number of examples of all kinds worked out in detail, mostly of abelian groups, modules over matrix algebras, polynomial algebras, or algebras given by bound quivers. In order to help visualise and analyse examples, it includes many figures. Each section is followed by exercises of all levels of difficulty, both computational and theoretical, with hints provided to some of them.
Completely Regular Semigroup Varieties
This book presents further developments and applications in the area of completely regular semigroup theory, beginning with applications of Pol獺k's theorem to obtain detailed descriptions of various kernel classes including the K-class covers of the kernel class of all bands. The important property of modularity of the lattice of varieties of completely regular semigroups is then employed to analyse various principal sublattices. This is followed by a study of certain important complete congruences on the lattice; the group, local and core relations. The next chapter is devoted to a further treatment of certain free objects and related word problems. There are many constructions in the theory of semigroups. Those that have played an important role in the theory of varieties of completely regular semigroups are presented as they apply in this context. The mapping that takes each variety to its intersection with the variety of bands is a complete retraction of the lattice of varieties of completely regular semigroups onto the lattice of band varieties and so induces a complete congruence for which every class has a greatest member. The sublattice generated by these greatest members is then investigated with the help of many applications of Pol獺k's theorem. The book closes with a fascinating conjecture regarding the structure of this sublattice.
Geometric Gems (V2)
Our physical world is embedded in a geometric environment. Plane geometry has many amazing wonders beyond those that are briefly touched on at school. The quadrilateral, one of the basic instruments in geometry, has a plethora of unexpected curiosities. The authors present these in an easily understood fashion, requiring nothing more than the very basics of school geometry to appreciate these curiosities and their justifications or proofs.The book is intended to be widely appreciated by a general audience, and their love for geometry should be greatly enhanced through exploring these many unexpected relationships in geometry. Geometric Gems is also suitable for mathematics teachers, to enhance the education of their students with these highly motivating quadrilateral properties.
Geometric Gems (V2)
Our physical world is embedded in a geometric environment. Plane geometry has many amazing wonders beyond those that are briefly touched on at school. The quadrilateral, one of the basic instruments in geometry, has a plethora of unexpected curiosities. The authors present these in an easily understood fashion, requiring nothing more than the very basics of school geometry to appreciate these curiosities and their justifications or proofs.The book is intended to be widely appreciated by a general audience, and their love for geometry should be greatly enhanced through exploring these many unexpected relationships in geometry. Geometric Gems is also suitable for mathematics teachers, to enhance the education of their students with these highly motivating quadrilateral properties.
Category and Measure
Topological spaces in general, and the real numbers in particular, have the characteristic of exhibiting a 'continuity structure', one that can be examined from the vantage point of Baire category or of Lebesgue measure. Though they are in some sense dual, work over the last half-century has shown that it is the former, topological view, that has pride of place since it reveals a much richer structure that draws from, and gives back to, areas such as analytic sets, infinite games, probability, infinite combinatorics, descriptive set theory and topology. Keeping prerequisites to a minimum, the authors provide a new exposition and synthesis of the extensive mathematical theory needed to understand the subject's current state of knowledge, and they complement their presentation with a thorough bibliography of source material and pointers to further work. The result is a book that will be the standard reference for all researchers in the area.
Linear Algebra with its Applications
This book contains a detailed discussion of the matrix operation, its properties, and its applications in finding the solution of linear equations and determinants. Linear algebra is a subject that has found the broadest range of applications in all branches of mathematics, physical and social sciences, and engineering.
Exploring Linear Algebra
This text focuses on the primary topics in a first course in Linear Algebra including additional advanced topics related to data analysis, singular value decomposition and connections to differential equations. This is a lab text that would lead a class through Linear Algebra using Mathematica demonstrations and Mathematica coding
Quadratic Ideal Numbers
This book introduces quadratic ideal numbers as objects of study with applications to binary quadratic forms and other topics. The text requires only minimal background in number theory, much of which is reviewed as needed. Computational methods are emphasized throughout, making this subject appropriate for individual study or research at the undergraduate level or above.
Groups St Andrews 2022 in Newcastle
Every four years leading researchers gather to survey the latest developments in all aspects of group theory. Since 1981, the proceedings of these meetings have provided a regular snapshot of the state of the art in group theory and helped to shape the direction of research in the field. This volume contains selected papers from the 2022 meeting held in Newcastle. It includes substantial survey articles from the invited speakers, namely the mini course presenters Michel Brion, Fanny Kassel and Pham Huu Tiep; and the invited one-hour speakers Bettina Eick, Scott Harper and Simon Smith. It features these alongside contributed survey articles, including some new results, to provide an outstanding resource for graduate students and researchers.
Vectorization
Enables readers to develop foundational and advanced vectorization skills for scalable data science and machine learning and address real-world problems Offering insights across various domains such as computer vision and natural language processing, Vectorization covers the fundamental topics of vectorization including array and tensor operations, data wrangling, and batch processing. This book illustrates how the principles discussed lead to successful outcomes in machine learning projects, serving as concrete examples for the theories explained, with each chapter including practical case studies and code implementations using NumPy, TensorFlow, and PyTorch. Each chapter has one or two types of contents: either an introduction/comparison of the specific operations in the numerical libraries (illustrated as tables) and/or case study examples that apply the concepts introduced to solve a practical problem (as code blocks and figures). Readers can approach the knowledge presented by reading the text description, running the code blocks, or examining the figures. Written by the developer of the first recommendation system on the Peacock streaming platform, Vectorization explores sample topics including: Basic tensor operations and the art of tensor indexing, elucidating how to access individual or subsets of tensor elements Vectorization in tensor multiplications and common linear algebraic routines, which form the backbone of many machine learning algorithms Masking and padding, concepts which come into play when handling data of non-uniform sizes, and string processing techniques for natural language processing (NLP) Sparse matrices and their data structures and integral operations, and ragged or jagged tensors and the nuances of processing them From the essentials of vectorization to the subtleties of advanced data structures, Vectorization is an ideal one-stop resource for both beginners and experienced practitioners, including researchers, data scientists, statisticians, and other professionals in industry, who seek academic success and career advancement.
Standard and Non-Standard Methods for Solving Elementary Algebra Problems
Solving elementary algebra lies at the heart of this basic textbook. Some of the topics addressed include inequalities with rational functions, equations and inequalities with modules, exponential, irrational, and logarithmic equations and inequalities, and problems with trigonometric functions. Special attention is paid to methods for solving problems containing parameters.The book takes care to introduce topics with a description of the basic properties of the functions under study, as well as simple, typical tasks necessary for the initial study of the subject. Each topic concludes with problems for readers to solve, some of which may require serious effort and solutions are provided in all cases. Many of these problems were specifically created for this book and are set at university entrance exam or mathematical Olympiad level.The authors both have extensive experience in conducting and compiling tasks for exams and Olympiads. They seek to continue and share the traditions of Russian mathematical schools with schoolchildren, math teachers, and everyone who loves to solve problems.
Standard and Non-Standard Methods for Solving Elementary Algebra Problems
Solving elementary algebra lies at the heart of this basic textbook. Some of the topics addressed include inequalities with rational functions, equations and inequalities with modules, exponential, irrational, and logarithmic equations and inequalities, and problems with trigonometric functions. Special attention is paid to methods for solving problems containing parameters.The book takes care to introduce topics with a description of the basic properties of the functions under study, as well as simple, typical tasks necessary for the initial study of the subject. Each topic concludes with problems for readers to solve, some of which may require serious effort and solutions are provided in all cases. Many of these problems were specifically created for this book and are set at university entrance exam or mathematical Olympiad level.The authors both have extensive experience in conducting and compiling tasks for exams and Olympiads. They seek to continue and share the traditions of Russian mathematical schools with schoolchildren, math teachers, and everyone who loves to solve problems.
The Art of Working with the Mathieu Group M24
The Leech lattice Λ, the Conway group ∙O, and the Monster group M are immensely famous structures. They each grow out of the Mathieu group M24 and its underlying combinatorial structure, and play an important role in various branches of mathematics and in theoretical physics. Written by an expert in the field, this book provides a new generation of mathematicians with the intimate knowledge of M24 needed to understand these beautiful objects, and many others. It starts by exploring Steiner systems, before introducing the Miracle Octad Generator (MOG) as a device for working with the Steiner system S(5,8,24). Emphasizing how theoretical and computational approaches complement one another, the author describes how familiarity with M24 leads to the concept of 'symmetric generation' of groups. The final chapter brings together the various strands of the book to produce a nested chain of groups culminating in the largest Conway simple group Co1.
Lectures on Lie Algebras
No detailed description available for "Lectures on Lie Algebras".
Enveloping Algebras
No detailed description available for "Enveloping Algebras".
Linear Algebra
This book is a comprehensive guide to Linear Algebra and covers all the fundamental topics such as vector spaces, linear independence, basis, linear transformations, matrices, determinants, inner products, eigenvectors, bilinear forms, and canonical forms. It also introduces concepts such as fields, rings, group homomorphism, and binary operations early on, which gives students a solid foundation to understand the rest of the material. Unlike other books on Linear Algebra that are either too theory-oriented with fewer solved examples or too problem-oriented with less good quality theory, this book strikes a balance between the two. It provides easy-to-follow theorem proofs and a considerable number of worked examples with various levels of difficulty. The fundamentals of the subject are explained in a methodical and straightforward way. This book is aimed at undergraduate and graduate students of Mathematics and Engineering Mathematics who are studying Linear Algebra. It is also a useful resource for students preparing for exams in higher education competitions such as NET, GATE, lectureships, etc. The book includes some of the most recent and challenging questions from these exams.
A First Course on Orthogonal Polynomials
A First Course on Orthogonal Polynomials: Classical Orthogonal Polynomials and Related Topics provides an introduction to orthogonal polynomials and special functions aimed at graduate students studying these topics for the first time. A large part of its content is essentially inspired by the works of Pascal Maroni on the so-called algebraic theory of orthogonal polynomials, which distinguishes it from other contributions in the field. Features Suitable for a graduate course in orthogonal polynomials Can be used for a short course on the algebraic theory of orthogonal polynomials and its applicability to the study of the "old" classical orthogonal polynomials Includes numerous exercises for each topic Real and complex analysis are the only prerequisites
Select Topics in Signal Analysis
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.
Linear Algebra
This book is written to give instructors a tool to teach students to develop a mathematical concept from first principles. The text is organized around and offers the standard topics expected in a first undergraduate course in linear algebra.
Fundamentals of Ramsey Theory
This up-to-date book introduces the field of Ramsey theory from several different viewpoints. The book covers integer, graph, and Euclidean Ramsey theory with many proofs being combinatorial in nature. The author motivates topics and discussion, rather than just a list of theorems and proofs.
Non-commutative Algebras. Pseudo-BCK Algebras versus m-pseudo-BCK Algebras
This monograph is devoted mainly to the author's results in her research on non-commutative algebras related to logic started on October 17, 2022, results never published. It would not be written in so little time and with so many important results and examples without the help of the computer program Prover9-Mace4, developed by William W. McCune (1953 - 2011).There exist a frame-work of non-commutative algebras of logic, having in its `center' the pseudo-BCK algebra.In this monograph, the author mainly has generalized to the non-commutative case the m-BCK algebra and its related algebras, as particular cases of unital magmas, thus creating a new frame-work of non-commutative algebras, having in its `center' the new m-pseudo-BCK algebra. The pseudo-MV algebras are particular cases of m-pseudo-BCK algebras, the groups belong to this new frame-work. But, the goal of her research was to define and study the quantum-pseudo-MV algebra, the non-commutative generalization of quantum-MV algebra. She was able to reach her goal only because she has discovered the`principle' that governs the non-commutative algebras, called `transposition' principle (`m-transposition' principle, for magmas). She has also introduced and studied other non-commutative generalizations of quantum algebras: the bounded involutive pseudo-lattices, the pseudo-De Morgan algebras and the ortho-pseudo-lattices. The book has 18 chapters, divided into three parts: Part I (centered on pseudo-BCK algebras: Chapters 1 - 7), Part II (the core of the monograph, centered on m-pseudo-BCK algebras: Chapters 8 - 16) and Part III (`bridge' theorems: Chapters 17, 18).
Elliptic Partial Differential Equations Elementary Viewpoint
This is a textbook that covers several selected topics in the theory of elliptic partial differential equations which can be used in an advanced undergraduate or graduate course.The book considers many important issues such as existence, regularity, qualitative properties, and all the classical topics useful in the wide world of partial differential equations. It also includes applications with interesting examples.The structure of the book is flexible enough to allow different chapters to be taught independently.The book is friendly, welcoming, and written for a newcomer to the subject.It is essentially self-contained, making it easy to read, and all the concepts are fully explained from scratch, combining intuition and rigor, and therefore it can also be read independently by students, with limited or no supervision.
