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This is part one of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning--the construction of the number systems and set theory--then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are also appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twenty-five to thirty lectures each.The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.The fourth edition incorporates a large number of additional corrections reported since the release of the third edition, as well as some additional new exercises.

This is part two of a two-volume introduction to real analysis and is intended for honours undergraduates who have already been exposed to calculus. The emphasis is on rigour and on foundations. The material starts at the very beginning--the construction of the number systems and set theory--then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and finally to the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. There are also appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of twenty-five to thirty lectures each.The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.The fourth edition incorporates a large number of additional corrections reported since the release of the third edition, as well as some additional new exercises.

According to a remark attributed to Mark Kac 'Probability Theory is a measure theory with a soul'. This book with its choice of proofs, remarks, examples and exercises has been prepared taking both these aesthetic and practical aspects into account.

This volume contains refereed and updated versions of papers presented at the International Conference on Algebra and Number Theory held on the occasion of the silver jubilee of the School of Mathematics and Computer/Information Science at the University of Hyderabad, India. There are three survey articles on the cyclicity problem for division algebras, one on Ramanujan graphs and supplementary zeroes of p-adic L functions of modular forms. Three articles announce results, which are to appear elsewhere. There are eighteen original contributions.The theme in the section on Algebra centres on recent work on Quadratic Forms and Division Algebras though there are also papers on Modules of Witt vectors, the Hodge-Tate conjecture, Calabi-Yau manifolds and Moduli stacks of vector bundles.The section on Number Theory centres on Automorphic Forms and Representations; however, here again there are papers on other themes pertaining to Elliptic curves and transcendental number theory.This volume should be useful to young researchers especially those interested in recent developments in Algebra and Number Theory.

This book offers a self-contained elementary introduction to the fundamental concepts and techniques of Algebraic Geometry, leading to some gems of the subject like Bezout’s Theorem, the Fundamental Theorem of Projective Geometry, and Zariski’s Main Theorem.The book contains a detailed treatment of algebraic plane curves with a special emphasis on elliptic curves and their birational classification. The role played by elliptic curves in modern theory of cryptology is illustrated.A novel feature of the book is a discussion of the state of the art on the Jacobian Problem and its relation to the Epimorphism Theorem. The recently introduced Tame Transformation Method of Cryptosystems, is sketched.Prerequisities are limited to a knowledge of finite Galois Theory, and of commutative Noetherian rings. All the Commutative Algebra needed is presented in Chapter 1, and could form the basis for a mini course on the subject. The exposition retains classroom flavour. About 300 exercises are included, often with adequate hints.

This book arose out of courses taught by the author. It covers the traditional topics of differential manifolds, tensor fields, Lie groups, integration on manifolds and basic differential and Riemannian geometry. The author emphasizes geometric concepts, giving the reader a working knowledge of the topic. Motivations are given, exercises are included, and illuminating nontrivial examples are discussed.Important features include the following: Geometric and conceptual treatment of differential calculus with a wealth of nontrivial examples. A thorough discussion of the much-used result on the existence, uniqueness, and smooth dependence of solutions of ODEs. Careful introduction of the concept of tangent spaces to a manifold. Early and simultaneous treatment of Lie groups and related concepts. A motivated and highly geometric proof of the Frobenius theorem. A constant reconciliation with the classical treatment and the modern approach. Simple proofs of the hairy-ball theorem and Brouwer's fixed point theorem. Construction of manifolds of constant curvature à la Chern. This text would be suitable for use as a graduate-level introduction to basic differential and Riemannian geometry.

This book is an elaboration of a series of lectures given at the Harish-Chandra Research Institute. The reader will be taken through a journey on the arithmetical sides of the large sieve inequality when applied to the Farey dissection. This will reveal connections between this inequality, the Selberg sieve and other less used notions like pseudo-characters and the $\Lambda_Q$-function, as well as extend these theories. One of the leading themes of these notes is the notion of so-called\emph{local models} that throws a unifying light on the subject. As examples and applications, the authors present, among other things, an extension of the Brun-Tichmarsh Theorem, a new proof of Linnik's Theorem on quadratic residues and an equally novel one of the Vinogradov three primes Theorem; the authors also consider the problem of small prime gaps, of sums of two squarefree numbers and several other ones, some of them being new, like a sharp upper bound for the number of twin primes $p$ that are such that $p 1$ is squarefree. In the end the problem of equality in the large sieve inequality is considered and several results in this area are also proved.

A collection of problems from a competition for college students organised by the Iranian Mathematical Society. It compiles problems from these competitions between 1973 and 2007 and provides solutions to most of them. It is suitable for students of mathematics preparing for competitions and for advanced studies.

This book is suited for a first course on Functional Analysis at the masters level. Efforts have been made to illustrate the use of various results via examples taken from differential equations and the calculus of variations, either through brief sections or through exercises. So, this book will be particularly useful for students who aspire to a research career in the applications of mathematics.Special emphasis has been given to the treatment of weak topologies and their applications to notions like reflexivity, separability and uniform convexity. The chapter on Lebesgue spaces includes a section devoted to the simplest examples of Sobolev spaces. The chapter on compact operators includes the spectral theory of compact self-adjoint operators on a Hilbert space.Each chapter concludes with a large selection of exercises of varying degrees of difficulty. They often provide examples or counter-examples to illustrate the optimality of the hypotheses of various theorems proved in the text, or develop simple versions of theories not developed therein.In this (second) edition, the book has been completely overhauled, without altering its original structure. Proofs of many results have been rewritten for greater clarity of exposition. Many examples have been added to make the text more user-friendly. Several new exercises have been added.

This book serves as an advanced text for a graduate course on stochastic algorithms for the students of probability and statistics, engineering, economics and machine learning. This second edition gives a comprehensive treatment of stochastic approximation algorithms based on the ordinary differential equation (ODE) approach which analyses the algorithm in terms of a limiting ODE. It has a streamlined treatment of the classical convergence analysis and includes several recent developments such as concentration bounds, avoidance of traps, stability tests, distributed and asynchronous schemes, multiple time scales, general noise models, etc., and a category-wise exposition of many important applications. It is also a useful reference for researchers and practitioners in the field.

This is a textbook meant to be used at the advanced undergraduate or graduate level. It is an introduction to the theory of Lie groups and Lie algebras. The book treats real and p-adic groups in a unified manner. The first chapter outlines preliminary material that is used in the rest of the book. The second chapter is on analytic functions and is of an elementary nature; this material is included to cater to students who may not be familiar with p-adic fields. The third chapter introduces analytic manifolds and contains standard material; the only notable feature being that it covers both real and p-adic analytic manifolds. Chapters 4 and 5 are on Lie groups. All the standard results on Lie groups are proved here. Some of the proofs are different from those in the earlier literature. The last two chapters are on Lie algebras and cover their structure theory as found in the first of the Bourbaki volumes on the subject. Some proofs here are new.

This second edition has been enlarged and considerably rewritten. Among the new topics are infinite product spaces with applications to probability, disintegration of measures on product spaces, positive definite functions on the line, and additional information about Weyl's theorems on equidistribution. Topics that have continued from the first edition include Minkowski's theorem, measures with bounded powers, idempotent measures, spectral sets of bounded functions and a theorem of Szego, and the Wiener Tauberian theorem. Readers of the book should have studied the Lebesgue integral, the elementary theory of analytic and harmonic functions, and the basic theory of Banach spaces. The treatment is classical and as simple as possible. This is an instructional book, not a treatise. Mathematics students interested in analysis will find here what they need to know about Fourier analysis. Physicists and others can use the book as a reference for more advanced topics.

This book, taking a holistic view of geometry, introduces the reader to axiomatic, algebraic, analytic and differential geometry. Starting with an informal introduction to non-Euclidean plane geometries, the book develops the theory to put them on a rigorous footing. It may be considered as an explication of the Kleinian view of geometry a la Erlangen Programme. The treatment in the book, however, goes beyond the Kleinian view of geometry.Some noteworthy topics presented include:various results about triangles (including results on areas of geodesic triangles) in Euclidean, hyperbolic, and spherical planesaffine and projective classification of conicstwopoint homogeneity of the three planes and the fact that the set of distance preserving maps (isometries) are essentially the same as the set of lengths preserving maps of these planes.Geometric intuition is emphasized throughout the book. Figures are included wherever needed. The book has several exercises varying from computational problems to investigative or explorative open questions.

The present book is an outcome of the SERC school on Computational Statistical Physics held at the Indian Institute of Technology, Guwahati, in December 2008.Numerical experimentation has played an extremely important role in statistical physics in recent years. Lectures given at the School covered a large number of topics of current and continuing interest.Based on lectures by active researchers in the field- Bikas Chakrabarti, S Chaplot, Deepak Dhar, Sanjay Kumar, Prabal Maiti, Sanjay Puri, Purusattam Ray, Sitangshu Santra and Subir Sarkar- the nine chapters comprising the book deal with topics that range from the fundamentals of the field, to problems and questions that are at the very forefront of current research.This book aims to expose the graduate student to the basic as well as advanced techniques in computational statistical physics. Following a general introduction to statistical mechanics and critical phenomena, the various chapters cover Monte Carlo and molecular dynamics simulation methodology, along with a variety of applications. These include the study of coarsening phenomena and diffusion in zeolites.In addition, graphical enumeration techniques are covered in detail with applications to percolation and polymer physics, and methods for optimisation are also discussed. Beginning graduate students and young researchers in the area of statistical physics will find the book useful. In addition, this will also be a valuable general reference for students and researchers in other areas of science and engineering.

This book on integration theory is based on the lecture notes for courses that the author gave at the Tata Institute of Fundamental Research, Mumbai, and at ETH, Zürich. The subject matter is classical. The goal of the notes is to provide a concise, clear, and accurate treatment of the basic ideas of the subject.

This book contains the author's notes for a course that he taught at ETH, Zürich. The aim is to lead the reader to a proof of the Peter-Weyl theorem, the basic theorem in the representation theory of compact topological groups. The topological, analytical, and algebraic groundwork needed for the proof is provided as part of the course.

This book presents the previously unpublished notes from a series of lectures given by the author at the Tata Institute of Fundamental Research in 1961. Basic material on affine connections and on locally or globally Riemannian and Hermitian symmetric spaces is covered. The final chapter proves the basic theorems on maximal compact subgroups of Lie groups. Readers should be familiar with differential manifolds and the elementary theory of Lie groups and Lie algebras.

The newly developed field of Seiberg-Witten gauge theory has become a well-established part of the differential topology of four-manifolds and three-manifolds. This book offers an introduction and an up-to-date review of the state of current research.

Disordered systems are ubiquitous in nature and their study remains a profound and challenging subject of current research. Ideas and methods from the physics of Disordered systems have been fruitfully applied to several fields ranging from computer science to neuroscience. This book contains a selection of lectures delivered at the `SERC School on Disordered Systems', spanning topics from classic results to frontier areas of research in this field. Spin glasses, disordered Ising models, quantum disordered systems, structural glasses, dilute magnets, interfaces in random field systems and disordered vortex systems are among the topics discussed in the text, in chapters authored by active researchers in the field, including Bikas Chakrabarti, Arnab Das, Deepak Kumar, Gautam Menon, G. Ravikumar, Purusattam Ray, Srikanth Sastry and Prabodh Shukla. This book provides a gentle and comprehensive introduction to the physics of disordered systems and is aimed at graduate students and young scientists either working in or intending to enter this exciting field. It should also serve as a general reference for students and practicing researchers alike.

This is an introductory book on Ergodic Theory. The presentation has a slow pace and the book can be read by any person with a background in basic measure theory and metric topology. A new feature of the book is that the basic topics of Ergodic Theory such as the Poincaré recurrence lemma, induced automorphisms and Kakutani towers, compressibility and E. Hopf's theorem, the theorem of Ambrose on representation of flows are treated at the descriptive set-theoretic level before their measure-theoretic or topological versions are presented. In addition, topics around the Glimm-Effros theorem are discussed. In the third edition a chapter entitled `Additional Topics' has been added. It gives Liouville's Theorem on the existence of invariant measure, entropy theory leading up to Kolmogorov-Sinai Theorem, and the topological dynamics proof of van der Waerden's theorem on arithmetical progressions.

This book originates from a series of 10 lectures given by Professor Michel Brion at the Chennai Mathematical Institute during January 2011. The book presents a theorem due to Chevalley on the structure of connected algebraic groups, over algebraically closed fields, as the starting point of various other structure results developed in the recent past. Chevalley's structure theorem states that any connected algebraic group over an algebraically closed field is an extension of an abelian variety by a connected affine algebraic group. This theorem forms the foundation for the classification of anti-affine groups which plays a central role in the development of the structure theory of homogeneous bundles over abelian varieties and for the classification of complete homogeneous varieties. All these results are presented in this book. The book begins with an overview of the results exposed, the proofs of which constitute the rest of the book. Various open questions also have been indicated in the course of the exposition.This book assumes certain preliminary knowledge of linear algebraic groups, abelian varieties and algebraic geometry. The book is intended for graduate students and researchers in algebraic geometry.

This volume contains the proceedings of the International Colloquium on Representations and L-functions organized by the Tata Institute of Fundamental Research in January 2012, one of a series of colloquia that began in 1956. It covers a wide spectrum of mathematics, including classifications of representations, arithmeticity, the average size of the 2-Selmer group of Jacobians for certain hyperelliptic curves, certain Kuznetsov formula for symmetry types of families of L-functions, sub-convexity bounds in the level aspect, Linnik's ergodic method, beyond endoscopy, and harmonic analysis for the relative trace formula.This volume contains refereed articles by leading experts in these fields and includes original results as well as expository materials in these areas.

Like the first volume, this is a special collection of articles describing the work of some of the best-known mathematicians from India. It contains eight articles written by experts, each of whom has chosen one major research contribution by an Indian mathematician and explained its context, significance, and impact. This is done in a way that makes the main ideas accessible to someone whose own research interests might be in a different area. Included here are commentaries on important works by:R. C. Bose, S. S. Shrikhande and E. T. ParkerH. Cramer and C. R. RaoV. B. Mehta and A. RamanathanR. NarasimhaK. R. Parthasarathy, R. Ranga Rao, and V. S. VaradarajanR. ParthasarathyD. N. VermaN. Wiener and P. R. Masani

The aim of this book is to give an introduction to what has come to be known as Standard Monomial Theory (SMT). SMT deals with the construction of nice bases of finite dimensional irreducible representations of semi-simple algebraic groups or, in geometric terms, nice bases of coordinate rings of flag varieties (and their Schubert subvarieties) associated to these groups. Besides its intrinsic interest, SMT has applications to the study of the geometry of Schubert varieties. SMT has its origin in the work of Hodge, giving bases of the coordinate rings of the Grassmannian and its Schubert subvarieties by ""standard monomials"". In its modern form, SMT was developed by the author in a series of papers written in collaboration with V. Lakshmibai and C. Musili.This book is a reproduction of a course of lectures given by the author in 1983-84 which appeared in the Brandeis Lecture Notes series. The aim of this course was to give an introduction to the series of papers by concentrating on the case of the full linear group. In recent years, there has been great progress in SMT due to the work of Peter Littelmann. Seshadri's course of lectures (reproduced in this book) remains an excellent introduction to SMT.In the second edition, Conjectures of a Standard Monomial Theory (SMT) for a general semi-simple (simply-connected) algebraic group, due to Lakshmibai, have been added as Appendix C. Many typographical errors have been corrected, and the bibliography has been revised.

The material presented in this book is suited for a first course in Functional Analysis which can be followed by masters students. While covering all the standard material expected of such a course, efforts have been made to illustrate the use of various theorems via examples taken from differential equations and the calculus of variations, either through brief sections or through exercises. In fact, this book will be particularly useful for students who would like to pursue a research career in the applications of mathematics.The book includes a chapter on weak and weak*topologies and their applications to the notions of reflexivity, separability and uniform convexity. The chapter on the Lebesgue spaces also presents the theory of one of the simplest classes of Sobolev spaces. The book includes a chapter on compact operators and the spectral theory for compact self-adjoint operators on a Hilbert space.Each chapter has large collection of exercises at the end. These illustrate the results of the text, show the optimality of the hypotheses of various theorems via examples or counterexamples, or develop simple versions of theories not elaborated upon in the text.

This book introduces the reader to the fascinating world of modular forms through a problem-solving approach. As such, it can be used by the undergraduate and graduate student for self-instruction.The topics covered include q-series, the modular group, the upper half-plane, modular forms of level one and higher level, the Ramanujan ? -function, the Petersson inner product, Hecke operators, Dirichlet series attached to modular forms and further special topics. It can be viewed as a gentle introduction for a deeper study of the subject. Thus, it is ideal for non-experts seeking an entry into the field.

This book’s principle goals are (i) to present the spectral theorem as a statement on the existence of a unique continuous and measurable functional calculus, (ii) to present a proof without digressing into a course on the Gelfand theory of commutative Banach algebras, (iii) to introduce the reader to the basic facts concerning the various von Neumann-Schatten ideals, the compact operators, the trace-class operators and all bounded operators, and finally, (iv) to serve as a primer on the theory of bounded linear operators on separable Hilbert space.