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BOOK REVIEWS 207

A. Jeffrey: Mathematics for Engineers and Scientists, 4th Edn, Van Nostrand Reinhold, 1989.

In this review, the author's words will be cited rather often. All citations will be quoted. The first edition of the book appeared in 1969. It was reprinted many times. This fact is very significant. "This book has evolved from an introductory course in mathematics given to engineering students at the University of Newcastle-upon-Tyne, It represents the author's attempt to offer the engineering students, and the science student who is not majoring in a mathematical aspect of this subject, a broad and modern account of those parts of mathematics that are finding increasingly important application in the everyday development of his subject".

It is impossible for the reviewer to judge the necessity of the collection of topics contained in the book. It depends on the 'direction' of education in the concrete university. Maybe for other universities such topics as linear programming, queueing theory, partial differential equations, etc., are more important. So, it is better to consider the method of presentation of the suggested material.

I agree with the author that "every effort has been made to integrate the various chapters into a description of mathematics as a single subject". Let us list the titles of the chapters: 1. Introduction to Sets and Numbers; 2. Variables, Functions and Mappings; 3. Sequences, Limits, and Continuity; 4. Complex Numbers and Vectors; 5 Differentiation of Functions of One or More Real Variables; 6. Exponential, t~yperbolic and Logarithmic Functions; 7. Fundamentals of Integration; 8. Systematic Integration; 9. Matrices and Linear Transformation; 10. Functions of a Complex Variable; 11. Scalars, Vectors, and Fields; 12. Series, Taylor's Theorem and its Uses; 13. Differential Equations and Geometry; 14. First Order D~fferential Equations; 15. Higher Order Differential Equations; 16. Fourier Series; 17. Numerical Analysis; 18. Probability and Statistics.

"Of necessity, much of the material in this book is standard, though the emphasis and manner of introduction and presentation frequently differs from that found elsewhere...", namely the author rejects the idea (which is common for similar books) to omit proofs. He considers that "knowledge of the proof of a result is often as essential as its subsequent application, and the modern student needs and merits both". I do agree with this belief because the knowledge of the logical structure of mathematics often gives rise to new facts in applied fields. This thesis can be confirmed by a proverb: "The formula is more clever than me". It is impossible to supply a lot of facts (collected in the book) with proofs. Of course, the author is forced not to do so everywhere. I am not sure that the decision to give or not to give the proof is well- grounded always. For example there are no proofs in Chapter 18.,

Besides, there are some doubts about the logic in the sequence of Chapters. So, I believe that Chapters 4, 9, 10, 18 are not connected with their environment~ Maybe they demand another place in the book or another book.

Nevertheless, the book is well-organized. It contains a lot of problems in each

Acta Applicandae M athematicae 24: (1991).

208 BOOK REVIEWS

chapter which contain more relevant information. I am sure that this edition is not the last one.

Institute for Systems Studies, Moscow, U.S.S.R.

V. KALASHNIKOV

V. Paulauskas and A. Rackauskas: Approximation Theory in the Central Limit Theorem. Exact Results in Banach Spaces (MIA Series), Kluwer Academic Publishers, Dordrecht, Boston, London, 1989, 176pp., ISBN 90-277-2875-9, Dfl154.00/US$79.00.

Central limit theorem (c.l.t.) is one of the most popular topics in probability theory. For many years, different generations of mathematicians have tried to generalize corresponding classical results to many directions, e.g. they refused the independence property for summands, supposed that the summands take values from general spaces (not only from real-line or Eucledian space), and so on. During the past two decades, a lot of results have appeared which considered a convergence rate problem in c.l.t., i.e. what is the difference between the limit and prelimit laws depending on the number of summands (tending to infinity). It is worth mentioning here the results of Vo Zolotarev, J. Kuelbs, V. Sazonov, V. Bentkus, and many others.

The authors belong to the Lithuanian mathematical school which traditionally paid a lot of attention to the above-mentioned problem. The book is intended mainly for the study of two groups of problems: convergence-rate estimates on sets with a nonsmooth boundary in general Banach spaces and those in terms of different probability metrics. The presented topic is a new one for monographs.

The authors restricted themselves to the simple case of c.l.t, when sums consist of independent identically distributed (i.i.d.) random elements taking values from general Banach space. It turns out that even under this restriction, there is the difference between finite- and infinite-dimensional cases.

The main part of the book contains five chapters. The first chapter is introductory. It is devoted to the definitions, concepts, and statements of the theory of probability distributions in Banach spaces.

The obtained estimates are based on the so-called method of composition. Its idea is rather simple. Suppose that the difference of two probability measures F (distribu- tion of a normed sum) and G (limit distribution) on a certain subset A belonging to a Banach space ~ is to be estimated:

IF(A) - G(A)I = IEeza - EGZAt,

where Za is the indicator of the set A and E r (or Er) is the expectation, respectively, the measure F (or G). The first problem here is: 'How to substitute the discontinuous function ;(a by some sufficiently smooth function ga,~ coinciding with Za everywhere with the exception the e-neighbourhood (t3A), of the boundary t~A of the subset A? The

Acta Applicandae Mathematicae 24: (1991).