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Analytic Perturbation Theory and Its Applications

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Konstantin E. Avrachenkov, Jerzy A. Filar, and Phil G. Howlett


2014 / x + 369 pages / Hardcover / ISBN: 978-1-611973-13-6 / List Price $95.00 / SIAM Member Price $66.50 / Order Code OT135

Keywords: analytic perturbations, singular perturbations, power series, asymptotic expansions, Markov processes, optimization

SIREV Book Review

We live in an era in which mathematical models—or systems—are used to describe complex phenomena (climate change dynamics, stock markets, the Internet, logistics, etc.). These systems typically depend on one or more parameters that are assigned nominal values based on current understanding of the phenomena. Because these values are usually estimates, it is important to know how even small deviations from them affect the behavior of the system. Single-parameter deviations pose significant technical challenges, but they constitute a natural starting point, especially since much progress has been made in analyzing the asymptotic behavior of these deviations in many special settings in the sciences, engineering, and economics.

This book considers systems that can be disturbed to varying degrees by changing the value of a single perturbation parameter. The difference between the actual and nominal values of this key parameter, the perturbation, is small but unknown in most applications, so it is important to understand the behavior of the solutions as the perturbation tends to zero. Many interesting applications contain an apparent discontinuity in the limiting behavior that complicates the analysis. These are the so-called singularly perturbed problems.

Analytic Perturbation Theory and Its Applications includes

  • comprehensive treatment of analytic perturbations of matrices, linear operators, and polynomial systems, particularly the singular perturbation of inverses, generalized inverses, and polynomial systems, topics not covered in other books;
  • original applications in Markov chains, Markov decision processes, optimization, and applications to Google PageRank™ and the Hamiltonian cycle problem as well as input retrieval in linear control systems; and
  • a Problem section in every chapter to aid in course preparation.

This text is appropriate for applied and pure mathematicians, researchers, and engineers interested in systems and control. It is also suitable for advanced undergraduate, first-year graduate, and advanced graduate one-semester classes covering perturbation theory in various mathematical uses.

About the Authors
Konstantin E. Avrachenkov is Director of Research, INRIA Sophia Antipolis, France. He is an associate editor of the International Journal of Performance Evaluation and has published more than 40 journal and 50 refereed conference articles. His main research interests are Markov processes, singular perturbation theory, queueing theory, mathematical programming, game theory, and performance evaluation of communication networks.

Jerzy A. Filar is Director of Flinders Mathematical Sciences Laboratory, Flinders University, Australia. He co-authored (with K. Vrieze) Competitive Markov Decision Process (Springer, 1996) and has authored or co-authored approximately 100 refereed research papers. He is a Fellow of the Australian Mathematical Society. His research interests span both theoretical and applied topics in operations research, optimization, game theory, applied probability, and environmental modeling.

Phil G. Howlett is Emeritus Professor of Industrial and Applied Mathematics, University of South Australia. He is a member and former Leader of the Scheduling and Control Group (SCG) in the Centre for Industrial and Applied Mathematics (CIAM) and a member of the Barbara Hardy Institute. He was Director of CIAM, 1998–2004; Director of the Australian Mathematics-in-Industry Study Group, 2000–2003; and Chair of ANZIAM (Australia and New Zealand Industrial and Applied Mathematics), 2008–2009. He has worked on the development of optimal driving strategies for trains and solar-powered racing cars and on railway operations efficiency as well as other areas of applied mathematics, including recent work on management of water supply systems, rainfall modeling, and singular perturbations of linear operators.

Reproduction, including downloading of Magritte works, is prohibited by copyright laws and international conventions without the express written permission of Artists Rights Society (ARS), New York.


ISBN: 9781611973136

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