University Texts in the Mathematical Sciences – serie

Targeted at undergraduate mathematics students, this book aims to cover courses in group theory. Based on lectures in group theory, it includes many illustrations and examples, numerous solved exercises and detailed proofs of theorems. The book acts as a guide to teachers and is also useful to graduate students. The book discusses major topics in group theory such as groups and subgroups, binary operations, fundamental algebraic structure of groups, symmetric groups, cyclic groups, normal subgroups, quotient groups, homomorphisms, isomorphisms, direct product of groups, simple groups, set on a group, Sylow's theorem, finite group, Abelian groups and non-isomorphic Abelian groups.

This book presents a scholarly English translation of Henri Lebesgue’s seminal 1909 paper, Sur les Intégrales Singulières, first published in the Annales de l’Université de Toulouse. This foundational work, long known to specialists in French, is made accessible here to a broader mathematical audience through a clear, faithful and pedagogically enriched translation. In addition to the core translation, the book features an original introductory chapter and an appendix by the translator that trace the historical development of harmonic analysis from classical trigonometric series to the Zygmund–Calderón theory, with mention of Carleson’s breakthrough in almost everywhere convergence.The historical survey outlines the evolution of ideas contributed by Dirichlet, Fejér, Dini, Vallée Poussin, Jordan, Kolmogorov, Young and others, illuminating the pathway from early convergence results to modern singular integral theory. The material is presented in structured chapters and enhanced with diagrams, worked examples and graded exercises, offering an accessible yet rigorous learning experience. Supplementary readings are included to encourage deeper exploration and contextual understanding.This volume is ideal for advanced undergraduate and graduate students, as well as researchers interested in Fourier analysis, trigonometric series and the early development of harmonic analysis. It provides a rich conceptual bridge between classical analysis and modern harmonic tools. It is particularly well-suited for courses on harmonic analysis and the historical foundations of integration theory. Mathematicians, educators and historians of mathematics will find in this book both a pedagogical resource and a valuable preservation of Lebesgue’s contribution to the analytic tradition.

Targeted to advanced undergraduate and graduate students of mathematics, physics and engineering, this book discussed major topics in classical dynamics. Each chapter begins with clear statements of definitions, principles and theorems. To illustrate and amplify the theories and principles, there are graded sets of solved examples and supplementary problems in each chapter. The book highlights vector treatment, but the scalar treatment is not disregarded. Theorems and principles are solved by vector method whereas in case of solved problems, both vector and scalar treatments are utilized. It is assumed that the students have a fair working knowledge of differential and integral calculus, differential equations, vector algebra and vector calculus.

This core textbook on functional analysis is intended for senior undergraduates and graduate mathematics students. It is suitable for both classrooms and for self-study. The first in a two-volume series presents all the basic material needed for a solid foundation in the subject. It opens with a concise overview of the historical evolution of the subject and goes on to present foundational material in a clear, succinct manner, integrating original source quotes to enrich the narrative and blending the historical perspectives harmoniously with the flow of the subject. Pedagogically, the short chapters are more conducive for learning, and each chapter concludes with applications (including some unusual ones) to diverse fields and exercises to hone students’ understanding. The applications can also serve as sources for student seminars. Various formulations of the spectral theorem and their equivalence are discussed. Different approaches to some important results are presented to enrich the toolkit of the students. The style is neither terse nor verbose, requiring occasional paper-pencil work from the reader. The bibliography is rich and includes all the original works of the founding fathers. Thumbnail biographies of the mathematicians involved should pep up the readers.

This book introduces the theory of univalent functions, focusing on the key subclasses of starlike, convex and close-to-convex functions. These subclasses play a central role in the study of geometric function theory. The book begins with the general class of univalent functions and studies their basic properties. It then explores essential mathematical tools, including results on growth, distortion and coefficients for bounded analytic functions, as well as the theory of functions with positive real parts and differential subordination. The last three chapters delve into starlike, convex and close-to-convex functions, presenting key results on coefficient bounds, growth, distortion estimates and subordination theorems.Designed for graduate students and early-stage researchers, the book requires minimal prerequisites of a basic course in complex analysis. Detailed proofs have been provided for all results, ensuring that readers can follow the logical development of the material. Exercises given at the end of each chapter offer opportunities for further study and deeper understanding of the concepts. This book serves as both a textbook and a resource for those beginning research in geometric function theory. Its accessible style, coupled with its clear explanations and motivating examples, makes it an ideal choice for anyone looking to explore the fascinating world of univalent functions.

The basic topics of the subject that every analyst should know are discussed in the first volume. This second part ‘Select Topics’ provides glimpses of some beautiful vistas of the functional analytic terrain that a reader may like to explore and enjoy. Introductions to these topics, including those of current interest, with pointers to further edification are provided. An author, with various constraints, necessarily faces a Hamletian dilemma in choosing the topics for such a work. The choices made here are: a little Fourier analysis (Lp, Schwartz and Sobolev spaces, Fourier transform and tempered distributions, interpolation, spectral synthesis), some operator theory (compact, Hilbert-Schmidt, trace class, Fredholm and Toeplitz operators, spectral theorem, invariant subpaces), operator algebras (commutative Banach algebras, C∗algebras, von Neumann algebras and operator spaces), Gabor analysis (Hilbert frames, Gabor frames, Riesz bases, wavelets), Banach space geometry, Schauder bases and functional analytic methods in PDE (scattered throughout the book). It also gives some unusual applications. Several mini courses and projects are possible based on the book, besides an advanced course in Functional Analysis. A historical perspective blending harmoniously with the flow, a clear succinct style that is not terse nor verbose and biographic thumbnails continue. The bibliography is extensive and includes all original sources.

This book serves as a core text in discrete mathematics. It discusses topics such as symbolic logic, enumerative combinatorics, algebraic structures, graph theory, and related applications to computer science and other allied subjects. The presentation of related concepts is suitable for sophomore, junior, and senior-level undergraduate students. Exercises provided at the end of each chapter are designed to help the reader have an active learning experience throughout the study.

This book serves as a core text in discrete mathematics. It discusses topics such as symbolic logic, enumerative combinatorics, algebraic structures, graph theory, and related applications to computer science and other allied subjects. The presentation of related concepts is suitable for sophomore, junior, and senior-level undergraduate students. Exercises provided at the end of each chapter are designed to help the reader have an active learning experience throughout the study.

This book delves into the fundamental theoretical aspects of reliability analysis, focusing on various probabilistic models. These models are essential for representing random variations in underlying variables such as time-to-failure, the number of failures between consecutive repairs and similar metrics. The calculation, estimation and prediction of reliability all hinge on using appropriate probability models. The book introduces various models beneficial for researchers in the field of reliability. It also provides a comprehensive overview of the available models, highlighting their distinctive features and practical applications in a narrative format. The content of the book is designed to appeal to a broad readership. Students and researchers in the field of reliability analysis will find a comprehensive yet easily understandable summary of models applicable to their data sets of interest. It should be noted, however, As stated clearly in the preface, this book does not illustrate applications of the models discussed in terms of real-life data.

This book delves into the fundamental theoretical aspects of reliability analysis, focusing on various probabilistic models. These models are essential for representing random variations in underlying variables such as time-to-failure, the number of failures between consecutive repairs and similar metrics. The calculation, estimation and prediction of reliability all hinge on using appropriate probability models. The book introduces various models beneficial for researchers in the field of reliability. It also provides a comprehensive overview of the available models, highlighting their distinctive features and practical applications in a narrative format. The content of the book is designed to appeal to a broad readership. Students and researchers in the field of reliability analysis will find a comprehensive yet easily understandable summary of models applicable to their data sets of interest. It should be noted, however, As stated clearly in the preface, this book does not illustrate applications of the models discussed in terms of real-life data.

This book systematically develops the theory of Metric Spaces while serving as a connection between classical Real Analysis and General Topology. It is designed for senior undergraduate and graduate students, providing formal definitions, theorems, proofs, examples, remarks, exercises, and explanatory notes. Instructors can use the numerous examples and miscellaneous results to structure their teaching approach. The book contains seven chapters, first of which lists primary results (without proofs) from the undergraduate Real Analysis course. Each subsequent chapter builds on cues from previous levels, adapting them to the context of Metric Spaces.Additionally, the book includes four appendix chapters. The first three are included to maintain the flow of discussion in the main chapters, relegating less relevant proofs to the appendices. The fourth appendix on the Cantor set is included to provide insight into this notable mathematical concept.

This book presents an in-depth guide designed for students, researchers and professionals seeking to harness the power of LaTeX, which is a high-quality typesetting tool widely used in academia, engineering and scientific communities for producing well-structured, complex documents. It is crafted to demystify LaTeX, equipping readers with the skills to create professional-quality documents from simple articles to complex reports, theses and books. The book covers both basic and advanced aspects of LaTeX and addresses common challenges faced by LaTeX users, such as structuring long documents, creating mathematical equations, managing bibliographies and incorporating graphics. By providing a clear progression from fundamental to advanced topics, it aims to make LaTeX accessible and practical for students, researchers and professionals across disciplines.

This book presents an in-depth guide designed for students, researchers and professionals seeking to harness the power of LaTeX, which is a high-quality typesetting tool widely used in academia, engineering and scientific communities for producing well-structured, complex documents. It is crafted to demystify LaTeX, equipping readers with the skills to create professional-quality documents from simple articles to complex reports, theses and books. The book covers both basic and advanced aspects of LaTeX and addresses common challenges faced by LaTeX users, such as structuring long documents, creating mathematical equations, managing bibliographies and incorporating graphics. By providing a clear progression from fundamental to advanced topics, it aims to make LaTeX accessible and practical for students, researchers and professionals across disciplines.

Designed for senior undergraduate and graduate students in mathematics, this textbook offers a comprehensive exploration of measure theory and integration. It acts as a pivotal link bridging the Riemann integral and the Lebesgue integral, with a primary focus on tracing the evolution of measure and integration from their historical roots. A distinctive feature of the book is meticulous guidance, providing a step-by-step journey through the subject matter, thus rendering complex concepts more accessible to beginners. A fundamental grasp of differential and integral calculus, as well as Riemann integration, is recommended to ensure a smoother comprehension of the material.This textbook comprises 10 well-structured chapters, each thoughtfully organized to lead students from fundamental principles to advanced complexities. Beginning with the establishment of Lebesgue’s measure on the real line and an introduction to measurable functions, the book then delves into exploring the cardinalities of various set classes. As readers progress, the subtleties of the Lebesgue integral emerge, showcasing its generalization of the Riemann integral and its unique characteristics in higher dimensions.One of the book’s distinctive aspects is its indepth comparison of the Lebesgue integral, improper Riemann integral, and Newton integral, shedding light on their distinct qualities and relative independence. Subsequent chapters delve into the realm of general measures, Lebesgue-Stieltje's measure, Hausdorff ’s measure, and the concept of measure and integration in product spaces. Furthermore, the book delves into function spaces, such as 𝘓𝘱 spaces, and navigates the intricacies of signed and complex measures, providing students with a comprehensive foundation in this vital area of mathematics.

Designed for senior undergraduate and graduate students in mathematics, this textbook offers a comprehensive exploration of measure theory and integration. It acts as a pivotal link bridging the Riemann integral and the Lebesgue integral, with a primary focus on tracing the evolution of measure and integration from their historical roots. A distinctive feature of the book is meticulous guidance, providing a step-by-step journey through the subject matter, thus rendering complex concepts more accessible to beginners. A fundamental grasp of differential and integral calculus, as well as Riemann integration, is recommended to ensure a smoother comprehension of the material.This textbook comprises 10 well-structured chapters, each thoughtfully organized to lead students from fundamental principles to advanced complexities. Beginning with the establishment of Lebesgue’s measure on the real line and an introduction to measurable functions, the book then delves into exploring the cardinalities of various set classes. As readers progress, the subtleties of the Lebesgue integral emerge, showcasing its generalization of the Riemann integral and its unique characteristics in higher dimensions.One of the book’s distinctive aspects is its indepth comparison of the Lebesgue integral, improper Riemann integral, and Newton integral, shedding light on their distinct qualities and relative independence. Subsequent chapters delve into the realm of general measures, Lebesgue-Stieltje's measure, Hausdorff ’s measure, and the concept of measure and integration in product spaces. Furthermore, the book delves into function spaces, such as 𝘓𝘱 spaces, and navigates the intricacies of signed and complex measures, providing students with a comprehensive foundation in this vital area of mathematics.

Primarily designed for graduate students of mathematics, this textbook delves into Naïve set theory, offering valuable insights for senior undergraduate students and researchers specializing in set theory. Commencing with a comprehensive exploration of functions and relations, the book extends its coverage to various applications of Naïve set theory across multiple mathematical branches, including real analysis, linear and abstract algebra, general topology, and introductory aspects of complex analysis and measure theory. The text meticulously introduces cardinal and ordinal numbers, along with transfinite induction, following the natural progression discovered by Cantor during his examination of trigonometric series. While this book provides a solid foundation, students intrigued by set theory for its intrinsic value should recognize that the subject extends far beyond the scope of this text.

Primarily designed for graduate students of mathematics, this textbook delves into Naïve set theory, offering valuable insights for senior undergraduate students and researchers specializing in set theory. Commencing with a comprehensive exploration of functions and relations, the book extends its coverage to various applications of Naïve set theory across multiple mathematical branches, including real analysis, linear and abstract algebra, general topology, and introductory aspects of complex analysis and measure theory. The text meticulously introduces cardinal and ordinal numbers, along with transfinite induction, following the natural progression discovered by Cantor during his examination of trigonometric series. While this book provides a solid foundation, students intrigued by set theory for its intrinsic value should recognize that the subject extends far beyond the scope of this text.

This is an essential textbook for senior undergraduate and graduate students of statistics, stochastic processes, stochastic finance, and probability theory. It covers all the important notations of probability theory and stochastic processes that are crucial for students to overcome their initial challenges during their studies. It thoroughly discusses the concepts of stochastic processes, both Markov and non-Markov processes, as well as stochastic calculus. With a special focus on finance, the book dedicates three chapters to explore the applications of stochastic processes in options, credit risk and insurance.Organized into sixteen chapters and one appendix, the book takes the readers to a well-organized learning. To fully grasp the intricacies of stochastic processes, students are expected to have a solid grounding in real analysis, linear algebra, and differential equations. Practical examples are emphasized throughout the book, carefully selected from various fields. The exercises at the end of each chapter are designed with the same objective in mind. Stochastic processes play a significant role in various scientific disciplines and real-life applications.

This is an essential textbook for senior undergraduate and graduate students of statistics, stochastic processes, stochastic finance, and probability theory. It covers all the important notations of probability theory and stochastic processes that are crucial for students to overcome their initial challenges during their studies. It thoroughly discusses the concepts of stochastic processes, both Markov and non-Markov processes, as well as stochastic calculus. With a special focus on finance, the book dedicates three chapters to explore the applications of stochastic processes in options, credit risk and insurance.Organized into sixteen chapters and one appendix, the book takes the readers to a well-organized learning. To fully grasp the intricacies of stochastic processes, students are expected to have a solid grounding in real analysis, linear algebra, and differential equations. Practical examples are emphasized throughout the book, carefully selected from various fields. The exercises at the end of each chapter are designed with the same objective in mind. Stochastic processes play a significant role in various scientific disciplines and real-life applications.

Targeted at undergraduate mathematics students, this book aims to cover courses in group theory. Based on lectures in group theory, it includes many illustrations and examples, numerous solved exercises and detailed proofs of theorems. The book acts as a guide to teachers and is also useful to graduate students. The book discusses major topics in group theory such as groups and subgroups, binary operations, fundamental algebraic structure of groups, symmetric groups, cyclic groups, normal subgroups, quotient groups, homomorphisms, isomorphisms, direct product of groups, simple groups, set on a group, Sylow's theorem, finite group, Abelian groups and non-isomorphic Abelian groups.

Targeted to upper-undergraduate and graduate students of mathematics, this book discusses special integrals and their applications in finding certain series sums. It starts with the differentiation and the integration methods for summing a series that is applied to find the sum of various binomial and trigonometrical series. It also discusses methods to find the sum of series involving the variables having exponents in integral or fractional powers of 2. Complex variables are freely used to derive several theorems, which result in several special integrals and series sums. Bessel coefficients, Bessel functions, and their various generalizations are also discussed in the book. As a particular case of generalized Bessel functions, pseudo-exponential functions are defined, and their properties are studied in the book. Broadly divided into two parts—Part 1 and Part 2—the book has six chapters in Part 1, whereas Part 2 has six chapters on solutions to the problems in Part 1. To understand the topics in the book, the minimum prerequisites are the knowledge of calculus, complex analysis, and Fourier series.

This book discusses continuous and discrete nonlinear systems in systematic and sequential approaches. The unique feature of the book is its mathematical theories on flow bifurcations, nonlinear oscillations, Lie symmetry analysis of nonlinear systems, chaos theory, routes to chaos and multistable coexisting attractors. The logically structured content and sequential orientation provide readers with a global overview of the topic. A systematic mathematical approach has been adopted, featuring a multitude of detailed worked-out examples alongside comprehensive exercises. The book is useful for courses in dynamical systems and chaos and nonlinear dynamics for advanced undergraduate, graduate and research students in mathematics, physics and engineering. The second edition of the book is thoroughly revised and includes several new topics: center manifold reduction, quasi-periodic oscillations, Bogdanov–Takens, periodbubbling and Neimark–Sacker bifurcations, and dynamics on circle. The organized structures in bi-parameter plane for transitional and chaotic regimes are new active research interest and explored thoroughly. The connections of complex chaotic attractors with fractals cascades are explored in many physical systems. Chaotic attractors may attain multiple scaling factors and show scale invariance property. Finally, the ideas of multifractals and global spectrum for quantifying inhomogeneous chaotic attractors are discussed.

This book discusses continuous and discrete nonlinear systems in systematic and sequential approaches. The unique feature of the book is its mathematical theories on flow bifurcations, nonlinear oscillations, Lie symmetry analysis of nonlinear systems, chaos theory, routes to chaos and multistable coexisting attractors. The logically structured content and sequential orientation provide readers with a global overview of the topic. A systematic mathematical approach has been adopted, featuring a multitude of detailed worked-out examples alongside comprehensive exercises. The book is useful for courses in dynamical systems and chaos and nonlinear dynamics for advanced undergraduate, graduate and research students in mathematics, physics and engineering. The second edition of the book is thoroughly revised and includes several new topics: center manifold reduction, quasi-periodic oscillations, Bogdanov–Takens, periodbubbling and Neimark–Sacker bifurcations, and dynamics on circle. The organized structures in bi-parameter plane for transitional and chaotic regimes are new active research interest and explored thoroughly. The connections of complex chaotic attractors with fractals cascades are explored in many physical systems. Chaotic attractors may attain multiple scaling factors and show scale invariance property. Finally, the ideas of multifractals and global spectrum for quantifying inhomogeneous chaotic attractors are discussed.

Designed for senior undergraduate and graduate courses in mathematics and engineering, this self-contained textbook discusses key topics in linear algebra with real-life applications. Split into two parts—theory in part I and solved problems in part II—the book makes both theoretical and applied linear algebra easily accessible. Topics such as sets and functions, vector spaces, linear transformations, eigenvalues and eigenvectors, normed spaces, and inner product spaces are discussed in part I; while in part II, over 500 meticulously solved problems show how to use linear algebra in real-life situations. A must-have book for linear algebra courses; it also serves as valuable supplementary material.

Designed for senior undergraduate and graduate courses in mathematics and engineering, this self-contained textbook discusses key topics in linear algebra with real-life applications. Split into two parts—theory in part I and solved problems in part II—the book makes both theoretical and applied linear algebra easily accessible. Topics such as sets and functions, vector spaces, linear transformations, eigenvalues and eigenvectors, normed spaces, and inner product spaces are discussed in part I; while in part II, over 500 meticulously solved problems show how to use linear algebra in real-life situations. A must-have book for linear algebra courses; it also serves as valuable supplementary material.

This revised book provides an accessible presentation of concepts from probability theory, statistical methods, the design of experiments, and statistical quality control. It is shaped by the experience of the two teachers teaching statistical methods and concepts to engineering students. Practical examples and end-of-chapter exercises are the highlights of the text, as they are purposely selected from different fields. Statistical principles discussed in the book have a great relevance in several disciplines like economics, commerce, engineering, medicine, health care, agriculture, biochemistry, and textiles to mention a few.Organised into 16 chapters, the revised book discusses four major topics—probability theory, statistical methods, the design of experiments, and statistical quality control. A large number of students with varied disciplinary backgrounds need a course in basics of statistics, the design of experiments and statistical quality control at an introductory level to pursue their discipline of interest. No previous knowledge of probability or statistics is assumed, but an understanding of calculus is a prerequisite. The whole book also serves as a master level introductory course in all the three topics, as required in textile engineering or industrial engineering.

This revised book provides an accessible presentation of concepts from probability theory, statistical methods, the design of experiments, and statistical quality control. It is shaped by the experience of the two teachers teaching statistical methods and concepts to engineering students. Practical examples and end-of-chapter exercises are the highlights of the text, as they are purposely selected from different fields. Statistical principles discussed in the book have a great relevance in several disciplines like economics, commerce, engineering, medicine, health care, agriculture, biochemistry, and textiles to mention a few.Organised into 16 chapters, the revised book discusses four major topics—probability theory, statistical methods, the design of experiments, and statistical quality control. A large number of students with varied disciplinary backgrounds need a course in basics of statistics, the design of experiments and statistical quality control at an introductory level to pursue their discipline of interest. No previous knowledge of probability or statistics is assumed, but an understanding of calculus is a prerequisite. The whole book also serves as a master level introductory course in all the three topics, as required in textile engineering or industrial engineering.

This book serves as an illuminating introduction to the intricacies of the geometry of numbers. It commences by exploring basic concepts of convex sets and lattices in Euclidean space and goes on to delve into Minkowski’s fundamental theorem for convex bodies and its applications. It discusses critical determinants and successive minima before explaining the core results of packings and coverings. The text goes on to delve into the significance of renowned conjectures such as Minkowski’s conjecture regarding the product of linear forms, Watson’s conjecture, and the conjecture of Bambah, Dumir, and Hans-Gill concerning non-homogeneous minima of indefinite quadratic forms. Dedicated to Prof. R.P. Bambah on his 98th birthday, a living legend of number theory in India, this comprehensive book addresses both homogeneous and non-homogeneous problems, while sprinkling in historical insights and highlighting unresolved questions in the field. It is ideally suited for beginnersembarking on self-study as well as for use as a text for a one- or two-semester introductory course.