Vivek S. Borkar's Books

Probability Theory

An Advanced Course

By: Vivek S. Borkar

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This book presents a selection of topics from probability theory. Essentially, the topics chosen are those that are likely to be the most useful to someone planning to pursue research in the modern theory of stochastic processes. The prospective reader is assumed to have good mathematical maturity. In particular, he should have prior exposure to basic probability theory at the level of, say, K.L. Chung's 'Elementary probability theory with stochastic processes' (Springer-Verlag, 1974) and real and functional analysis at the level of Royden's 'Real analysis' (Macmillan, 1968). The first chapter is a rapid overview of the basics. Each subsequent chapter deals with a separate topic in detail. There is clearly some selection involved and therefore many omissions, but that cannot be helped in a book of this size. The style is deliberately terse to enforce active learning. Thus several tidbits of deduction are left to the reader as labelled exercises in the main text of each chapter. In addition, there are supplementary exercises at the end. In the preface to his classic text on probability ('Probability', Addison Wesley, 1968), Leo Breiman speaks of the right and left hands of probability.

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Page Count

149

Publisher

Springer Science & Business Media

Published Date

ISBN 10

1461207916

ISBN 13

9781461207917

Ergodic Control of Diffusion Processes

By: Ari Arapostathis,Vivek S. Borkar,Mrinal K. Ghosh

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Ergodic Control of Diffusion Processes

By: Ari Arapostathis,Vivek S. Borkar,Mrinal K. Ghosh

The first comprehensive account of controlled diffusions with a focus on ergodic or 'long run average' control.

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Page Count

341

Publisher

Cambridge University Press

Published Date

ISBN 10

0521768403

ISBN 13

9780521768405

Elementary Convexity with Optimization

By: Vivek S. Borkar,K. S. Mallikarjuna Rao

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Elementary Convexity with Optimization

By: Vivek S. Borkar,K. S. Mallikarjuna Rao

This book develops the concepts of fundamental convex analysis and optimization by using advanced calculus and real analysis. Brief accounts of advanced calculus and real analysis are included within the book. The emphasis is on building a geometric intuition for the subject, which is aided further by supporting figures. Two distinguishing features of this book are the use of elementary alternative proofs of many results and an eclectic collection of useful concepts from optimization and convexity often needed by researchers in optimization, game theory, control theory, and mathematical economics. A full chapter on optimization algorithms gives an overview of the field, touching upon many current themes. The book is useful to advanced undergraduate and graduate students as well as researchers in the fields mentioned above and in various engineering disciplines.

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Page Count

148

Publisher

Springer Nature

Published Date

ISBN 10

9819916526

ISBN 13

9789819916528

Hamiltonian Cycle Problem and Markov Chains

By: Vivek S. Borkar,Vladimir Ejov,Jerzy A. Filar,Giang T. Nguyen

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Hamiltonian Cycle Problem and Markov Chains

By: Vivek S. Borkar,Vladimir Ejov,Jerzy A. Filar,Giang T. Nguyen

This research monograph summarizes a line of research that maps certain classical problems of discrete mathematics and operations research - such as the Hamiltonian Cycle and the Travelling Salesman Problems - into convex domains where continuum analysis can be carried out. Arguably, the inherent difficulty of these, now classical, problems stems precisely from the discrete nature of domains in which these problems are posed. The convexification of domains underpinning these results is achieved by assigning probabilistic interpretation to key elements of the original deterministic problems. In particular, the approaches summarized here build on a technique that embeds Hamiltonian Cycle and Travelling Salesman Problems in a structured singularly perturbed Markov decision process. The unifying idea is to interpret subgraphs traced out by deterministic policies (including Hamiltonian cycles, if any) as extreme points of a convex polyhedron in a space filled with randomized policies. The above innovative approach has now evolved to the point where there are many, both theoretical and algorithmic, results that exploit the nexus between graph theoretic structures and both probabilistic and algebraic entities of related Markov chains. The latter include moments of first return times, limiting frequencies of visits to nodes, or the spectra of certain matrices traditionally associated with the analysis of Markov chains. However, these results and algorithms are dispersed over many research papers appearing in journals catering to disparate audiences. As a result, the published manuscripts are often written in a very terse manner and use disparate notation, thereby making it difficult for new researchers to make use of the many reported advances. Hence the main purpose of this book is to present a concise and yet easily accessible synthesis of the majority of the theoretical and algorithmic results obtained so far. In addition, the book discusses numerous open questions and problems that arise from this body of work and which are yet to be fully solved. The approach casts the Hamiltonian Cycle Problem in a mathematical framework that permits analytical concepts and techniques, not used hitherto in this context, to be brought to bear to further clarify both the underlying difficulty of NP-completeness of this problem and the relative exceptionality of truly difficult instances. Finally, the material is arranged in such a manner that the introductory chapters require very little mathematical background and discuss instances of graphs with interesting structures that motivated a lot of the research in this topic. More difficult results are introduced later and are illustrated with numerous examples.

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Page Count

205

Publisher

Springer Science & Business Media

Published Date

ISBN 10

1461432324

ISBN 13

9781461432326

Stochastic Approximation

A Dynamical Systems Viewpoint

By: Vivek S. Borkar

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Stochastic Approximation

A Dynamical Systems Viewpoint

By: Vivek S. Borkar

This simple, compact toolkit for designing and analyzing stochastic approximation algorithms requires only a basic understanding of probability and differential equations. Although powerful, these algorithms have applications in control and communications engineering, artificial intelligence and economic modeling. Unique topics include finite-time behavior, multiple timescales and asynchronous implementation. There is a useful plethora of applications, each with concrete examples from engineering and economics. Notably it covers variants of stochastic gradient-based optimization schemes, fixed-point solvers, which are commonplace in learning algorithms for approximate dynamic programming, and some models of collective behavior.

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Page Count

176

Publisher

Cambridge University Press

Published Date

ISBN 10

0521515920

ISBN 13

9780521515924

Book's by Vivek S. Borkar

Probability Theory

Probability Theory

Ergodic Control of Diffusion Processes

Ergodic Control of Diffusion Processes

Elementary Convexity with Optimization

Elementary Convexity with Optimization

Hamiltonian Cycle Problem and Markov Chains

Hamiltonian Cycle Problem and Markov Chains

Stochastic Approximation

Stochastic Approximation

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