ordinary differential equations (i. g. petrovski)

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266 BOOK REVIEWS of individual processes and seem prone to miss the mathematical connections with the recently developed theories. Karlin’s work seems to bridge this gap, and in this respect the book must be highly praised, particularly because it is an introductory treatise on the theory of stochastic processes. The mathe- matical description is lucid and well organized to give an excellent fusion of theory with a good number of specialized processes, mostly from biology and engineering. The study of Markov process penetrates most parts of the book and is linked to the most specific processes. Although the book is entitled A First Course in Stochastic Processes, and ac- tually it is, it brings readers to the recent developments quickly and at the same time furnishes them with numerous exercises. The reviewer believes that this is also a good reference for the specialist. The contents are: Chapter 1, Elements of Stochastic Processes; Chapter 2, Marlcov Chains; Chapter 3, The Basic Limit Theorem of Marlcov Chains and Applications; Chapter 4, Algebraic Methods in Marlcov Chains; Chapter 5, Ratio Theorems of Transient Probabilities and Applications; Chapter 6, Sums of Inde- pendent Random Variables as a Markov Chain;Chapter 7, Classical Ezamples of Continuous Time Markov Chains; Chapter 8, Continuous Time Martcov Chains; Chapter 9, Order Statistics, Poisson Processes and Applications; Chapter 10, Brownian Motions; Chapter 11, Branching Processes; Chapter 12, Compounding Stochastic Processes; Chapter 13, Deterministic and Stochastic Genetic and Ecolog- ical Processes; Chapter 14, Queueing Processes; Appendix; Review of Matrix Analysis and Miscellaneous Problems. Chapter 4 includes special computational methods in Markov chains giving the connection with the orthogonal polynomials with a weight function. Chapter 5 covers up to the discussion of the systems of linear equations associated with a probability matrix from the viewpoint of the theory of regular, superregular and subregular sequences. Chapter 6 deals mostly with the recurrence properties of the sums of independent random variables, and a sharpening of the characteristics of right regular sequences is given for the sum. Chapter 7 gives the properties of the birth and death processes illustrating more specialized processes. In Chapter 8 the reference is made to the strong Markov property. Chapter 10 covers the property of crossing the axis of the paths. Chapters 12-14 provide excellent articles of respective processes so that the reader will be able to get good pictures of the roles of those processes in practical problems, as well as the mathematical properties of them. TATSUO KAWATA The Catholic University of America Ordinary Differential Equations. By I. G. PETROVSKI. Prentice-Hall, Inc., Engle- wood Cliffs, New Jersey, 1966. x -- 232 pp. 87.95. This translation of Petrovski’s classic text on ordinary differential equations is a very welcome addition to the literature in English on the subject. It is very clearly written and quite suitable for a one-semester course at the first-year graduate level. Downloaded 11/30/14 to 129.120.242.61. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php

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Page 1: Ordinary Differential Equations (I. G. Petrovski)

266 BOOK REVIEWS

of individual processes and seem prone to miss the mathematical connectionswith the recently developed theories. Karlin’s work seems to bridge this gap,and in this respect the book must be highly praised, particularly because itis an introductory treatise on the theory of stochastic processes. The mathe-matical description is lucid and well organized to give an excellent fusion oftheory with a good number of specialized processes, mostly from biology andengineering. The study of Markov process penetrates most parts of the bookand is linked to the most specific processes.Although the book is entitled A First Course in Stochastic Processes, and ac-

tually it is, it brings readers to the recent developments quickly and at the sametime furnishes them with numerous exercises. The reviewer believes that this isalso a good reference for the specialist.The contents are: Chapter 1, Elements of Stochastic Processes; Chapter 2,

Marlcov Chains; Chapter 3, The Basic Limit Theorem of Marlcov Chains andApplications; Chapter 4, Algebraic Methods in Marlcov Chains; Chapter 5, RatioTheorems of Transient Probabilities and Applications; Chapter 6, Sums of Inde-pendent Random Variables as a Markov Chain;Chapter 7, Classical Ezamples ofContinuous Time Markov Chains; Chapter 8, Continuous Time Martcov Chains;Chapter 9, Order Statistics, Poisson Processes and Applications; Chapter 10,Brownian Motions; Chapter 11, Branching Processes; Chapter 12, CompoundingStochastic Processes; Chapter 13, Deterministic and Stochastic Genetic and Ecolog-ical Processes; Chapter 14, Queueing Processes; Appendix; Review of MatrixAnalysis and Miscellaneous Problems.

Chapter 4 includes special computational methods in Markov chains givingthe connection with the orthogonal polynomials with a weight function. Chapter 5covers up to the discussion of the systems of linear equations associated with aprobability matrix from the viewpoint of the theory of regular, superregular andsubregular sequences. Chapter 6 deals mostly with the recurrence properties ofthe sums of independent random variables, and a sharpening of the characteristicsof right regular sequences is given for the sum. Chapter 7 gives the properties ofthe birth and death processes illustrating more specialized processes. In Chapter 8the reference is made to the strong Markov property. Chapter 10 covers theproperty of crossing the axis of the paths. Chapters 12-14 provide excellentarticles of respective processes so that the reader will be able to get good picturesof the roles of those processes in practical problems, as well as the mathematicalproperties of them.

TATSUO KAWATAThe Catholic University of America

Ordinary Differential Equations. By I. G. PETROVSKI. Prentice-Hall, Inc., Engle-wood Cliffs, New Jersey, 1966. x -- 232 pp. 87.95.This translation of Petrovski’s classic text on ordinary differential equations

is a very welcome addition to the literature in English on the subject. It is veryclearly written and quite suitable for a one-semester course at the first-yeargraduate level.

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Page 2: Ordinary Differential Equations (I. G. Petrovski)

BOOK REVIEWS 267

Except for an absence of any material concerning boundary value problems,the material covered is fairly standard and complete. The general outline is toproceed from the scalar first order equation to systems of first order equations.A separate chapter is devoted to linear systems and another to" linear systemswith constant coefficients, in a few places, there are references to, or discussionof, somewhat more specialized and advanced topics than is usual at this level,for example, singular perturbations and Lyapunov’s second method.The author first briefly treats the usual elementary first order differential

equations, for example, separated variables and homogeneous equations, in avery engaging geometrical manner. This geometrical flavor reappears at severalplaces in the text and greatly enhances the discussion. There are also, however,such nongeometrie matters discussed as the Cauehy theorem for analytic equa-tions, which is established by the method of majorizing series.

Pieard’s method of successive approximations for proving the usual existencetheorem concerning equations satisfying a Lipsehitz condition is used to moti-vate a discussion of contraction mappings. The latter is then applied to integralequations and an implicit function theorem. A geometric interpretation of theprinciple of contraction mappings is also given. The last chapter deals withautonomous systems. There, besides the usual two-dimensional material, onefinds some discussion of n-dimensional systems including several applications ofBrouwer’s fixed-point theorem.

JACOB J. LEVINUniversity of Wisconsin

OTHER BOOKS RECEIVED

1. Optimal Shutdown Control of Nuclear Reactors. By MILTON ASH. AcademicPress, New York, 1966. xiv -t- 169 pp. $8.50.

2. Elements of Probability Theory. By J. BAss. Academic Press, New York,1966. xiv - 249 pp. $9.75.

3. Elementary Partial Differential Equations. By PAUL W. BERG and JAMES L.McGREGOR. HoldeR-Day, Inc., San Francisco, 1966. xv 421 pp. $11.75.

4. Topology Seminar Wisconsin, 1965. Edited by R. H. BING and R. J. BEAN.Princeton University Press, Princeton, New Jersey, 1966. vi -t- 246 pp.$5.00.

5. Mathematical Methods in the Physical Sciences. By MAnY L. BOAS. JohnWiley and Sons, Inc., New York, 1966. xix -- 778 pp. $11.95. A text forphysicists.

6. Differential and Difference Equations. By Lous BRAD. John Wiley andSons, Inc., New York, 1966. xvi + 698 pp. $11.95.

7. Square Summable Power Series. By Louis DE BRANGES and JAMES ROVNYAK.Holt, Rinehart and Winston, Inc., New York, 1966. viii nu 104 pp. $3.95.

8. Calculus and Analytic Geometry. By JACK R. BRITTON, t. BEN KRIEGH andLEON W. RVTLND. W. H. Freeman and Co., San Francisco, 1966. xiii +1069 pp. $12.00.

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