# Richard Bellman's Books

Richard Bellman has written over 50 books on a wide variety of topics and at both undergraduate and at monograph level. We list many of these works below and give information such as publisher's information, extracts from the preface and extracts from a number of reviews.

**Click on a link below to go to that book**- A Survey of the Theory of the Boundedness, Stability, and Asymptotic Behavior of Solutions of Linear and Nonlinear Differential and Difference Equations (1949)

- Stability theory of differential equations (1953)

- An introduction to the theory of dynamic programming (1953)

- A survey of the mathematical theory of time-lag, retarded control, and hereditary processes (1954)

- Dynamic programming of continuous processes (1954)

- Dynamic programming (1957)

- Some aspects of the mathematical theory of control processes (1958) with I Glicksberg and O A Gross

- Introduction to matrix analysis (1960)

- A brief introduction to theta functions (1961)

- An Introduction to Inequalities (1961) with Edwin Beckenbach

- Inequalities (1961) with Edwin Beckenbach

- Adaptive control processes: A guided tour (1961)

- Applied dynamic programming (1962) with Stuart E Dreyfus

- Invariant imbedding and radiative transfer in slabs of finite thickness (1962) with R E Kalaba and M C Prestrud

- Differential-difference equations (1963) with Kenneth L Cooke

- Perturbation techniques in mathematics, physics, and engineering (1964)

- Invariant imbedding and time-dependent transport processes (1964) with H H Kagiwada, R E Kalaba and M C Prestrud

- Quasilinearization and nonlinear boundary-value problems (1965) with Robert E Kalaba

- Inequalities (2nd edition) (1965) with Edwin Beckenbach

- Dynamic programming and modern control theory (1965) with Robert E Kalaba

- Numerical inversion of the Laplace transform: Applications to biology, economics, engineering and physics (1966) with Robert E Kalaba and Jo Ann Lockett

- Introduction to the mathematical theory of control processes. Vol. 1: Linear equations and quadratic criteria (1967)

- Modern elementary differential equations (1968) with Kenneth Cooke

- Some vistas of modern mathematics. Dynamic programming, invariant imbedding, and the mathematical biosciences (1968)

- Stability theory in differential equations (reprint) (1969)

- Methods of nonlinear analysis. Vol. 1 (1970)

**1. A Survey of the Theory of the Boundedness, Stability, and Asymptotic Behavior of Solutions of Linear and Nonlinear Differential and Difference Equations (1949), by Richard Bellman.**

**1.1. From the Abstract.**

The nature of solutions of differential and difference equations is of great interest to the applied mathematician, the physicist, and the engineer. In particular, numerous problems of vital concern to the Navy in connection with the self-excitation of oscillations in electronic and mechanical systems, the stability of moving bodies, the steering and turning of ships, etc. lead to non-linear differential equations. Thus far, results in the theory of these equations have been comparatively disorganised, and this survey should be useful in the unification and further development of the field. The Office of Naval Research is therefore pleased to make this report available in accordance with its statutory function of disseminating scientific information.

**1.2. Review by: E T Copson.**

*Mathematical Reviews*MR0030662

**(11,31e)**.

Systems of ordinary differential equations, in addition to being of considerable intrinsic interest, are widely applied in various branches of theoretical physics and engineering. The subject has a vast literature, sometimes relatively inaccessible; and hence there is considerable duplication. It was hoped that a survey of the field would be a valuable book of reference and would stimulate research and prevent further duplication. The survey is concerned with the boundedness, stability and asymptotic behaviour of solutions of a system of real ordinary differential equations. In addition to the omission of a discussion of the solution in the complex plane, many other topics of considerable interest, such as the Sturm-Liouville theory, eigenfunction theory, existence and uniqueness theorems, topological methods, have had to be left out, either because they are adequately dealt with elsewhere or because such a branch would require a separate survey if it were to be adequately covered. Despite these limitations, the field covered is a large one and there is no room for proofs. Each chapter of the survey contains careful statements of the principal results with full explanations of the new ideas introduced and references to the original papers in the bibliography at the end of the chapter.

**2. Stability theory of differential equations (1953), by Richard Bellman.**

**2.1. From the Publisher of the 2008 edition.**

Suitable for advanced undergraduates and graduate students, this was the first English-language text to offer detailed coverage of boundedness, stability, and asymptotic behaviour of linear and nonlinear differential equations. It remains a classic guide, featuring material from original research papers, including the author's own studies.

The linear equation with constant and almost-constant coefficients receives in-depth attention that includes aspects of matrix theory. No previous acquaintance with the theory is necessary, since author Richard Bellman derives the results in matrix theory from the beginning. In regard to the stability of nonlinear systems, results of the linear theory are used to drive the results of Poincaré and Liapounoff. Professor Bellman then surveys important results concerning the boundedness, stability, and asymptotic behavior of second-order linear differential equations. The final chapters explore significant nonlinear differential equations whose solutions may be completely described in terms of asymptotic behaviour. Only real solutions of real equations are considered, and the treatment emphasises the behaviour of these solutions as the independent variable increases without limit.

**2.2. Review by: J J L Hinrichsen.**

*The American Mathematical Monthly*

**61**(5) (1954), 358.

This book is concerned with real solutions of real differential equations and the behaviour of these solutions as the independent variable increases without limit. The properties of the solutions of greatest interest are boundedness, asymptotic behaviour, oscillation, and stability. Very little of the material is to be found in other textbooks since most of it is either original with the author or was taken by him from research papers in the literature.

There has long been a need for an up-to-date text in English to serve as a basis for a graduate course in the theory of ordinary differential equations. This book should help to meet this need as well as to stimulate others to carry on investigations of some of the unsolved problems in this very fascinating and increasingly important area of analysis. The book is not intended to be encyclopaedic in scope but rather as introductory to the modern theory of stability and asymptotic behaviour of linear and non-linear differential equations.

The book opens with a detailed treatment of the linear system using matrix methods. The results in matrix theory required are derived from the beginning assuming no previous acquaintance with matrix theory. After the linear system with constant and almost-constant coefficients has been treated in considerable detail, the results of the linear theory are used to derive the results of Poincare and Liapounoff concerning the stability of nonlinear systems. A survey of the important results concerning the boundedness, stability, and asymptotic behaviour of the second-order, linear differential equations is presented. The last two chapters are devoted to some important nonlinear differential equations whose solutions may be completely described as far as asymptotic behaviour is concerned. Any discussion of periodic solutions of nonlinear equations is omitted since such a discussion would require the use of more advanced analytic and topological tools, and the author desires to keep the book at a somewhat more elementary level.

Considerable pain has been taken in the writing and organising of this book. Each of the seven chapters is divided into sections with appropriate headings and is closed by a bibliography classified as to the section of the chapter concerned. The various definitions, theorems, lemmas and corollaries are carefully labelled and italicised. Proofs are presented in a pleasing and direct style, and a considerable number of well chosen exercises are distributed throughout the text. The book contains a minimum number of typographical errors. It should prove to be a most valuable text for the student of analysis as well as an excellent reference book for the student interested in applications.

**2.3. Review by: Anon.**

*The Military Engineer*46 (311) (1954), 244.

Special features include material taken from original research papers, and a treatment of matrix theory required for differential equations which is not found elsewhere. It is stated to be the first book in English covering in detail that part of the modern theory of boundedness, stability, and asymptotic behaviour of linear and non-linear differential equations which is distinct from the theory of periodic solutions

**2.4. Review by: G E H Reuter.**

*Mathematical Reviews*MR0061235

**(15,794b)**.

The scope of this book is somewhat wider than its title may suggest. It is concerned with the behaviour of solutions of real differential equations as the independent variable $t$ tends to infinity.

...

... space is used to prove theorems several times if this exhibits several useful methods, but never to give an exhaustive catalogue of known results. There are plenty of exercises and references which the reader may use to extend his knowledge; perhaps some exercises might have been accompanied by hints for solution or detailed references, since they seem too difficult for unaided solution by non-expert readers. The book is to be judged not as a monograph but as an advanced textbook: as such, it is excellent both in its choice of material (most of which is not to be found in standard textbooks on differential equations) and in its lucid and attractive manner of presentation.

**2.5. Review by: Garrett Birkhoff.**

*Quarterly of Applied Mathematics*

**12**(4) (1955), 446-447.

This attractive little book gives an original and useful survey of various aspects of behaviour, as $t \rightarrow ∞$, of solutions of systems of ordinary differential equations. After an introductory chapter introducing the convenient vector-matrix-norm notation for treating such systems, the author define the sense in which he uses the word stability, as follows:

Definition. The solutions of

(1) $\large\frac{dy}{dt}\normalsize = A(t)y$

are stable with respect to a property $P$ and perturbations $B(t)$ of type $T$ if the solution of

$\large\frac{dz}{dt}\normalsize = (A(t) + B(t))z$

also possess property $P$. If this is not true, the solutions of (1) are said to be unstable with respect to property $P$ under perturbations of type $T$.

This definition can be applied to many problems, concerning non-linear (when rephrased) as well as linear differential equations. In many cases (e.g., if $A(t)$ is constant), the pattern followed by the author consists in showing that the qualitative asymptotic behaviour of known special solutions is unaffected by "small" perturbations of the coefficients - Liapounoff's famous stability theorem is a special case. In other cases, striking counterexamples to plausible guesses are given. In still others, ingenious isolated results are "rescued from oblivion" by displaying them in easily accessible form (e.g., the Fowler-Emden equation, to which Chap. VII is devoted).

The reader who is looking for results of the type just described should first consult Bellman's book. If he does not find them there, he will probably locate them by consulting the well-organised bibliography. On the other hand, there are a number of important topics, whose omission is not suggested by the somewhat misleading title of this book. ...

...

Perhaps because of his exclusive concern with a single problem and with formal methods, the author achieves an admirable clarity and uniformity of style. The result is a very useful survey of the qualitative asymptotic theory of ordinary differential equations.

**3. An introduction to the theory of dynamic programming (1953), by Richard Bellman.**

**3.1. From the Abstract.**

A discussion of dynamic programming, defined as a mathematical theory devoted to the study of multistage processes. These processes are composed of sequences of operations in which the outcome of those preceding may be used to guide the course of future ones. Operations of both deterministic and stochastic types are considered. Dynamic programming is a mathematical theory devoted to the study of multistage processes. The multistage processes discussed in this report are composed of sequences of operations in which the outcomes of those preceding may be used to guide the courses of future ones. Operations of both deterministic and stochastic types are discussed.

**3.2. Review by: D Blackwell.**

*Mathematical Reviews*MR0061805

**(15,887e)**.

A typical problem treated is stated as follows: "We are informed that a particle is in either state 0 or 1, and we are given initially the probability $x$ that it is in state 1. Use of the operation $A$ will reduce this probability to $ax$, where $a$ is some positive constant less than 1, whereas operation $L$, which consists in observing the particle, will tell us definitely which state it is in. If it is desired to transform the particle into state 0 in a minimum time, what is the optimal procedure?"

**4. A survey of the mathematical theory of time-lag, retarded control, and hereditary processes (1954), by Richard Bellman.**

**4.1. From the Abstract.**

A summary of the mathematical techniques required to analyse physical phenomena involving time lags, retarded control, or hereditary effects. Applications of these methods are significant in various fields (e.g., guided missile design, economics, psychology, medicine, and biology) and in theories of elasticity, magnetism, and fission processes.

**4.2. Review by: W J Trjitzinsky.**

*Mathematical Reviews*MR0062327

**(15,962a)**.

The author is largely concerned with [a specific system of differential equations] and he gives a systematic presentation of the theory of such systems in a form facilitating applications to problems such as those of automatic pilotless control, of the prediction of the course of an airplane (when velocity is supersonic) and so forth.

**4.3. Review by: Arnold Tustin.**

*Econometrica*

**23**(3) (1955), 354.

This short book gives a review or summary of the formal mathematical methods available for the solution of the type of equations that have sometimes been called "hystero-differential," and which the authors call "difference-differential," in which some terms are values at a time $t$, and others values at some other time $(t - r)$. Such equations arise when the equations represent processes in which time delays occur, and they are met with in many fields, such as studies of population growth, of ionisation and atomic fission, and of economic processes.

It appears that the authors, as part of the work sponsored by the U. S. Air Force under "Project Rand," have commenced a basic study of such situations, and as a first stage have made available this summary of the theorems that they consider to bear on the solution of such equations, together with a very detailed bibliography of the mathematical literature in this field. The preface states that the authors intend, at some future time, to extend the survey into a book that will not only contain more mathematical material, but in which "... it would be highly worthwhile to include an extensive treatment of applications, with particular regard to the underlying physical hypotheses which give rise to the equations of various types." Such a book would be more valuable than the present text, but it is commendable that the present survey should have been published, since it will certainly be of use to others who may be attempting to make progress in this field.

The book as now issued will be intelligible only to the mathematically sophisticated. It is extremely concise and formal, and assumes familiarity with contour integration and a good deal of advanced function theory. No attempt is made to ease the path of students; the treatment is unsuitable, and obviously not intended, as an introduction to the subject, except perhaps to mathematicians. It will be valuable on account of the formal consideration that is given to the conditions that are necessary and sufficient for the various theorems to be valid, and on account of its excellent bibliography.

In economics, or econometrics, equations of the type considered occur when time delays are involved. The book includes a five page chapter on one example of such an equation, namely the equation that arises from a model proposed by Kalecki in 1935, the solution of which was discussed by Frisch and Holme and others in

*Econometrica*soon afterwards. Characteristically the book under review deals slightly, if not naively, with the economic significance of the equation and its solution, and its contribution is limited to brief consideration, from a purely mathematical point of view, of the mathematical conditions under which the roots represent undamped oscillations.

Nevertheless, the book contains a great deal that is of interest, and that will be useful to those who have the mathematical preparation and the patience to dig out what they need from a survey that is somewhat over-concentrated and indigestible.

**5. Dynamic programming of continuous processes (1954), by Richard Bellman.**

**5.1. Review by: J M Danskin.**

*Mathematical Reviews*MR0071700

**(17,171a)**.

Chapters IV and V discuss "bottleneck problems," which lead to systems of linear ordinary differential equations in which the variables or linear combinations of the variables are subjected to inequality bounds. ... Here several complicated examples are worked out in detail.

Chapters VI and VII (the latter by H Osborn) contain a discussion of the relation of dynamic programming to the classical calculus of variations. ... Chapter VIII discusses scheduling problems ...

**6. Dynamic programming (1957), by Richard Bellman.**

**6.1. From the Preface.**

The purpose of this work is to provide an introduction to the mathematical theory of multi-stage decision processes. Since these constitute a somewhat formidable set of terms we have coined the term "dynamic programming" to describe the subject matter. Actually, as we shall see, the distinction involves more than nomenclature. Rather, it involves a certain conceptual framework which furnishes us a new and versatile mathematical tool for the treatment of many novel and interesting problems both in this new discipline and in various parts of classical analysis. Before expanding upon this theme, let us present a brief discussion of what is meant by a multi-stage decision process.

Let us suppose that we have a physical system $S$ whose state at any time $t$ is specified by a vector $p$, If we are in an optimistic frame of mind we can visualise the components of $p$ to be quite definite quantities such as Cartesian coordinates, or position and momentum coordinates, or perhaps volume and temperature, or if we are considering an economic system, supply and demand, or stockpiles and production capacities. If our mood is pessimistic, the components of $p$ may be supposed to be probability distributions for such quantities as position and momentum, or perhaps moments of a distribution.

In the course of time, this system is subject to changes of either deterministic or stochastic origin which, mathematically speaking, means that the variables describing the system undergo transformations. Assume now that in distinction to the above we have a process in which we have a choice of the transformations which may be applied to the system at any time. A process of this type we call a

*decision process*, with a decision equivalent to a transformation. If we have to make a single decision, we call the process a single-stage process; if a sequence of decisions, than we use the term

*multi-stage decision process*.

**6.2. Review by: S Vajda.**

*The Mathematical Gazette*

**43**(346) (1959), 307.

Dynamic Programming is a mathematical theory of multi-stage decision processes, determinate or stochastic, with routines for solving its problems. It has been almost exclusively developed by the author of this book, sometimes in collaboration with his colleagues at the Rand Corporation, a consultant body to the U.S.A.F. The techniques derive from the principle of optimality: An optimal policy has the property that whatever the initial state and initial decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. This approach leads to a formulation in functional equations, and the theory is then concerned with elucidating the structure of the solutions. About one quarter of the book consists of Exercises and Research Problems, showing the wide scope of the theory, and its connection with other branches of mathematics.

**6.3. Review by: S Vajda.**

*Econometrica*

**27**(3) (1959), 537-538.

The readers of Richard Bellman's many earlier publications on the subject of Dynamic Programming will be glad to know that a book of this title and from his pen has now been published as a RAND Corporation Research Study.

Dynamic Programming is one of those more recent branches of applied mathematics which owe their growth to the increasing awareness amongst operational research workers of the usefulness and, indeed, necessity of new techniques by which their specific problems can be tackled. It joins thus queuing theory, renewal theory, inventory theory, theory of games, and linear programming. In fact, it embraces portions of some of these and extends the scope of their applications. The author does not define Dynamic Programming, beyond saying that it is a mathematical theory of multi-stage decision processes. It is best described by saying that it has at its centre the following "principle of optimality": "An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision."

The consequences of this principle are discussed in the author's characteristic personal style, which makes the book eminently readable; only a modest technical knowledge is required (deeper mathematics are relegated to a promised second volume). The author lays stress on the principal ideas, and they are made perfectly clear and provide the unifying character of a book which deals with a great variety of problems. It must be said from a practical point of view, though, that although the formulation of a problem by an application of the principle of optimality is usually straightforward, the resulting functional equation may be a hard nut to crack.

**6.4. Review by: Preston C Hammer.**

*Science, New Series*

**127**(3304) (1958), 976.

The book under review may be considered to be a mathematics book, since the author is a well-known mathematician. However, it is not a book primarily about mathematics, as all too many mathematics books are apt to be. It is devoted to developing mathematics in response to problems arising in the social, business, military, economic, and political worlds as well as in engineering and the natural sciences.

Here, then, one may find direct statements of the applications of the mathematical theories developed, together with the construction of theories to solve specific classes of problems, such as inventory problems, depletion problems, and scheduling problems in general. The title Dynamic Programming refers to development of a dynamic optimal policy or program as a guide for the making of time-dependent decisions in complex problems involving many variables. Optimisation may refer to maximising net profit, to minimising risk probabilities, to minimising storage space, to minimising delivery times, and so on.

Dynamic Programming takes its place among the comparatively recent attempts to develop mathematics to meet problems of modern civilisation and was undertaken in somewhat the same spirit as were John Von Neumann's study of the theory of games and Abraham Wald's of the theory of sequential analysis.

While the book includes many problems indicating the scope of applications, it is not a book that can be easily read for its philosophical content alone, since the author uses concepts of advanced mathematics with ease and makes comparisons which require mathematical experience on the part of the reader.

The need for some serious attention to higher-dimensional geometry and analysis in the undergraduate curriculum is again seen in this book, which could be read with profit by leaders in a wide variety of fields if they had the capacity to assimilate its contents.

**6.5. Review by: J Kiefer.**

*Mathematical Reviews*MR0090477

**(19,820d)**.

"Dynamic Programming" is the term coined by the author for the study of multi-stage decision processes and, in particular, for the functional equation approach to finding optimum policies. This kind of approach, as the author points out, is intimately related to the invariance principle found in certain developments in physics (e.g., light scattering); more recently, the characterisation of Bayes strategies in sequential statistical decision problems by Wald and Wolfowitz (and the related optimum character of the sequential probability ratio test in statistics) was another predecessor of dynamic programming which used this approach. The approach is merely to note that an optimal policy must have the property that, whatever the initial state of the system and initial decision, relative to the resulting state (new initial state) the remaining decisions must constitute an optimal policy. This principle makes it simple to write down a functional equation for the total gain to be achieved from using an optimal policy (as a function of the initial state) for $N$ stages, in terms of the corresponding function for $N−1$ stages and appropriate transformations and costs which describe the transition from one stage to the next; in the stationary case of infinitely many periods, the functional equation relates the function to itself. Every functional equation in the book is obtained easily in this way. The real problem starts after the equation is written down: to solve it and describe the associated optimal policy. The author has been a leading developer of techniques for solving such functional equations. The book is a collection of methods, uniqueness and existence theorems, and examples, which have for the most part appeared in the literature in recent years and are due to various people, but most often to the author.

**6.6. Review by: F G Foster.**

*Economica*

**26**(104) (1959), 364-365.

This book treats multi-stage decision processes. These occur in a variety of applications: in inventory and production control, input-output analysis, investment planning, control system engineering, sequential testing and design of experiments in statistics. They also occur in card games, such as in the bidding system in contract bridge and in poker. In each case we envisage a system of some sort, characterised by being in one of a number of states, which changes as the system evolves through a succession of stages. At each stage there is a choice of a number of decisions which will affect the next state assumed by the system. The object is to maximise (or minimise) some function (the criterion function) of the succession of states, i.e., to find an optimal policy.

The unity of the book resides in the approach to these problems. The author propounds a single approach, the theory of which he originated and for which he has coined the term "dynamic programming ": it indicates a mathematical method rather than an area of application. It is based on his "principle of optimality": an optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. Since the invariant property required of processes in order that this principle may apply is not clearly stated, the principle itself could perhaps be taken as defining the class of processes under consideration. It would, however, be interesting (by way of exceptions to prove the rule) to know of some processes to which it does not apply. No such examples appear to be given.

Application of this principle yields a functional equation, the solution to which determines also an optimal policy. The book treats techniques for the solution of such functional equations occurring in the applications. The problem tackled by dynamic programming can be thought of as one of maximising a function subject to constraints, and Bellman's method (like linear programming) provides an alternative to that of Lagrange multipliers, applicable in circumstances where the latter is not feasible. It has the advantages of revealing the structure of the optimal policy, and also of providing a computational method for determining a solution numerically by successive approximations.

Chapters 1 and 2 treat two simple examples, and in Chapter 3 the approach to these is generalised. In Chapter 4 a number of general existence and uniqueness theorems for solutions of the functional equations are established. In succeeding chapters dynamic programming is applied in turn to the optimal inventory problem, "bottleneck" problems involving interdependent activities, problems in the classical calculus of variations, multi-stage games and, finally, Markov decision processes. An important feature is the large number of unworked exercises and research problems, many of which are of great interest.

The book is directed towards a wide audience: mathematicians, economists, statisticians, systems engineers and operational researchers. It is written at the advanced calculus level. The author has a lively style, and the mathematics is nowhere very deep. However, the reader will often require Bellman's special type of pioneering zeal to be able to persevere to the end of an argument-otherwise he may find the going difficult in places and (dare one admit it?) tedious. The trouble is that the type of mathematical problem posed in this book is such that it is only under excessively simplified conditions that one can ever hope for pleasing analytic solution.

**6.7. Review by: Albert Newhouse.**

*The American Mathematical Monthly*

**65**(10) (1958), 788-789.

This book brings under one cover the introduction and development of the theory of dynamic programming, which to a great extent has appeared previously in many papers scattered throughout many journals and pamphlets.

The class of problems which gave impetus to the development of the dynamic programming approach are multi-stage decision processes as they appear in economics, game theory, logistics, etc.

To give a greatly over-simplified idea of the subject, suffice it to say that "$n$" successive one-dimensional decisions are substituted for one "$n$" dimensional decision.

The author's aim of providing "an introduction to the mathematical theory" certainly has been accomplished. However, his suggestion to use the book as the text for "a course on the advanced calculus level" seems to be an application of the author's "principle of wishful thinking," since throughout the book there is a disturbing disconnectedness. Due to the scope of the subject covered this may have been unavoidable.

There are an abundant number of exercises at the end of each chapter. These, as well as the examples in the text, cover the wide range of the applicability of dynamic programming. Since the answers to most problems are not given, the designation of the problem sections as "Exercises and Research Problems" should prove rather confusing to the reader as to which is exercise and which is research problem.

Topics covered include: Chapter I: A discussion of a multi-stage allocation process, including existence and uniqueness theorems; the properties of the solution. Chapter II: A similar discussion of a stochastic multi-stage decision process. Chapter III: A more general discussion of dynamic programming processes; mathematical formulation of these. Chapter IV: Existence and uniqueness theorems, convergence, and stability. Chapter V: Formulation and discussion of the optimal inventory equation and its ramification. Chapters VI and VII: Bottleneck problems, the dual problem, and discussion of several examples. Chapter VIII: A continuous stochastic decision process; several approaches to its solution. Chapter IX: Dynamic programming approaches to several classical and other calculus-of-variation problems. Chapter X: Multi-stage games, including some discussion of nonzero-sum games. Chapter XI: Markovian decision processes.

This book will certainly prove extremely useful to the mathematicians, economists, statisticians, engineers, and operation analysts alike. The mathematician especially will look forward to the "contemplated second volume on a higher mathematical level" in which the author promises to rectify some omissions of the present volume.

**6.8. Review by: Eric Nixon.**

*Science Progress (1933-)*

**47**(187) (1959), 580-581.

The term "Dynamic Programming" was coined by the author for the study of multi-stage decision processes, and in particular for the functional equation approach to the finding of optimal policies. This approach is based on the premise that, in an optimal policy, regardless of preceding states and decisions, the remaining decisions must themselves constitute an optimal policy for the present state. Such a principle allows one to write down a recurrence relation for an $N$-stage policy in terms of the corresponding policy for an $N - 1$ stage process.

...

The author is not able to consider more than a few particular examples including, as well as finite and infinite processes, stochastic processes in which the outcomes have given probabilities. Emphasis must therefore be laid upon the 370 extremely important Exercise and Research Problems which supplement the chapters, the majority of which are extremely interesting and free from artificiality.

The book does not contain many mathematically deep conceptions, being well within the grasp of a good first- or second-year student, but in spite of this there are a number of things in it to interest the research worker. The volume will definitely be appreciated by the engineer or economist wanting to know how problems of this kind can be tackled; the mathematician will regard it as an appetiser for the promised, more mathematical, second volume.

**6.9. Review by: Martin J Beckmann.**

*Quarterly of Applied Mathematics*

**18**(1) (1960), 14; 30.

The general plan of the book is as follows. The basic ideas are introduced in terms of a specific problem concerning the repeated allocation of a single resource, an example that may well represent the simplest non-trivial case of a Dynamic Program. Following a second more complicated example involving probability (see below) the general structure of Dynamic Programming is exhibited, existence and uniqueness theorems are developed, the sensitivity of solutions and such properties of the solutions as convexity and continuity are discussed. An important tool in these investigations is the method of successive approximation through iteration. It is shown in particular that a suitable choice of the initial approximation will always result in monotone convergence. The remaining part of the book studies in detail various cases of a more advanced nature involving respectively, integral operators, systems of differential equations, partial differential equations, Min Max operators and Markov chains.

Apart from the opening problem which has little intrinsic interest, the applications developed at length are essentially of three kinds. The first of these is illustrated by the so-called gold mining problem, - obviously a military mission problem in disguise - where the object is the repeated assignment of a given piece of equipment to alternative uses subject to constant (but different) probabilities of ruin and to diminishing returns. This problem is studied in both a discrete and a continuous version (chapters II and VIII) for two and for more alternatives, and its very simple and reasonable solution is given in explicit form.

...

As a demonstration of the theoretical power of Dynamic Programming, the book includes three interesting applications to mathematics proper: to the calculus of variations with special reference to cases involving inequalities as constraints (ch. IX); to sequential games, in particular to so-called games of survival (ch. X); and to the study of Markov chains (ch. XI). In ch. IX a set of partial differential equations is obtained representing the more conventional Euler-Lagrange equations in an integrated form. The method is then applied to the determination of the eigenvalues for the differential equation of the vibrating string.

Among the most interesting and useful parts of the book are the collections of exercises and research problems at the end of each chapter which give an indication of the truly enormous range of potential applications, including such disparate items as a comparison of the arithmetic and geometric mean, Fibonaccian search, the design of missiles and the crossing of a desert by jeep.

The presentation is keyed to the level of ordinary calculus and is kept informal to the point where important proofs are left to the reader or are deferred to a second volume which is to be on a more technical level. These further developments will be looked forward to with keen expectation. Meanwhile it is a pleasure to recommend to anyone whose job is the application of mathematical tools in any context whatsoever the study of this most stimulating and rewarding book.

**6.10. Review by: John M Danskin.**

*Operations Research*

**7**(4) (1959), 536-540.

In this volume Dr Bellman of the Rand Corporation gives an extended exposition of the subject he invented about 1950.

A common military problem around that time was that of the allocation of bombers among the various strikes of a multistrike campaign against a target complex. It is easy to see that it is not practicable to give the allocations for each strike before the campaign begins. This would have two main disadvantages: First, the maximisation process, whatever the criterion used, would be carried out over a multi- or infinite-dimensional space, and it would be extremely difficult; second, it would have to be based on expected values given in advance of the campaign, thus freezing the strategy into one not taking account of the course of events as they happen. "Dynamic Programming" began when Bellman realised that the way to solve this kind of problem was not to try to give the whole strategy in advance, but to give it in the form of an answer, at each stage, to the question: "What do I do now?"

To answer this question, Bellman applies the "Principle of Optimality." What it means in this example is that the planner concerned with a given strike makes the assumption that from the next strike onward he will pursue an optimal policy. Then he allocates to this strike an amount of bombers with the following property: The sum of the return from the given strike plus the expected return from surviving and remaining bombers is a maximum. ...

...

The book has a fault of a sort which is very common in works on operations research. There is a tendency to claim too much, not only in matters of easily checked fact, as in the case of the transversality conditions above, but also as to the power and scope of dynamic programming in general. There are, for instance, a great many places in this book where the problem is stated at the beginning as if it were solved. On reading, one finds the dynamic-programming equations set up. Then one finds a discussion which says that it would be possible to solve these with a large enough machine with appropriate approximations. Indeed, there are a great many places where the book gets through to an answer, too; but it would have been better if those unfinished general problems whose discussion does not throw light on other problems had been left out.

This book is the most important book on mathematical methods in operations research to appear in several years. A great many problems do fall within its scope, or could be formulated by a person familiar with it. I think that it should be in the hands of any operations researcher who pretends to tackle any of the mathematical problems of OR except the most simple.

**7. Some aspects of the mathematical theory of control processes (1958), by R E Bellman, I Glicksberg and O A Gross.**

**7.1. From the Publisher.**

A presentation of (1) various representative mathematical problems that arise in the modern theory of the control of economic, industrial, engineering, and military systems, and (2) the types of mathematical theories and techniques that can be used to treat these problems. These problems are variational in the sense that the aim is to maximise or minimise some function which is being used as a criterion of the performance of the system under consideration. What distinguishes these problems form those encountered in the classical calculus of variations is the presence of constraints of physical origin, the consideration of random effects, or, alternatively, the introduction of the theory of games, and finally, the emphasis on computational solution. An attempt is make to indicate the new types of mathematical problems that have arisen from the more realistic description of old problems and from the pressure of new problems, and to illustrate the applicability of a wide variety of mathematical tools to their solution.

**7.2. Review by: L A MacColl.**

*Mathematical Reviews*MR0094281

**(20 #800)**.

The stated purpose of this book is to provide a taste of the mathematical theory of control processes, both in formulation and in solution. On the whole, this purpose is well achieved. It is to be noted that the authors' concept of a control process is abstract and general, and includes many processes which the typical reader is probably unaccustomed to regard as control processes. On the other hand, many considerations, such as that of "feedback", which are prominent in most discussions of control systems are here disregarded, or are treated only in an involved implicit fashion. It is certain that the authors have formulated a large number of interesting and novel problems, and have solved many of these more or less completely. However, it is difficult to judge the extent to which the theory actually gives useful solutions of realistic practical problems.

...

In general, the make-up of the book is very good. However, in some places reading is made difficult by what appear to be typographical errors or careless and unannounced variations of the formulations of the problems under consideration. In the opinion of the reviewer, the greatest defect lies in the comparative dearth of concrete illustrative examples to aid the reader in clarifying and fixing his knowledge.

**7.3. Review by: J P LaSalle.**

*Quarterly of Applied Mathematics*

**17**(3) (1959), 321-322.

The purpose of the book is to provide "a taste of the mathematical theory of control processes", and this has been accomplished by a blending of techniques and considerable mixing of a variety of topics. The authors selected the ingredients from their own contributions while at the same time giving references to more complete discussions and to the work of others. The book is far superior to and much more interesting than a survey. The authors have purposely not gone deeply into the mathematical theory nor have they emphasised applications. The reader interested in either of these aspects can consult the references, and the authors so advise him.

The most interesting feature of the book is the emphasis placed on the formulation of control problems and the techniques available for their solution and the illustration that by another formulation of the same problem new techniques become applicable. The techniques are those of differential equations, variational calculus, linear (Hilbert) spaces, dynamic programming and game theory.

...

The authors provide, as they say, a taste of some aspects of control theory in which they have specialised and do so remarkably well. The book contains some misprints, some errors and a few false statements, none of which should cause the intelligent reader great concern. They have covered a wide variety of topics in a vast and exciting field of research with great skill.

**8. Introduction to matrix analysis (1960), by Richard Bellman.**

**8.1. From the Preface.**

Our aim in this volume is to introduce the reader to the study of matrix theory, a field which with a great deal of justice may be called the arithmetic of higher mathematics.

Although this is a rather sweeping claim, let us see if we can justify it. Surveying any of the classical domains of mathematics, we observe that the more interesting and significant parts are characterised by an interplay of factors. This interaction between individual elements manifests itself in the appearance of functions of several variables and correspondingly in the shape of variables which depend upon several functions. The analysis of these functions leads to transformations of multidimensional type.

It soon becomes clear that the very problem of describing the problems that arise is itself of formidable nature. One has only to refer to various texts of one hundred years ago to be convinced that at the outset of any investigation there is a very real danger of being swamped by a sea of arithmetical and algebraical detail. And this is without regard of many conceptual and analytic difficulties that multidimensional analysis inevitably conjures up.

It follows that at the very beginning a determined effort must be made to devise a useful, sensitive, and perceptive notation. Although it would certainly be rash to attempt to assign a numerical value to the dependence of successful research upon well-conceived notation, it is not difficult to cite numerous examples where the solutions become apparent when the questions are appropriately formulated. Conversely, a major effort and great ingenuity would be required were a clumsy and unrevealing notation employed. Think, for instance, of how it would be to do arithmetic or algebra in terms of Roman numerals.

A well-designed notation attempts to express the essence of the underlying mathematics without obscuring or distracting.

With this as our introduction, we can now furnish a very simple syllogism. Matrices represent the most important of transformations, the linear transformations, transformations lie at the heart of mathematics, consequently, our first statement.

This volume, the first of a series of volumes devoted to an exposition of the results and methods of modern matrix theory, is intended to acquaint the reader with the fundamental concepts of matrix theory. Subsequent volumes will expand the domain in various directions. Here we shall pay particular attention to the field of analysis, both from the standpoint of motivation and application.

**8.2. Review by: Anon.**

*The Military Engineer*

**52**(347) (1960), 258.

Symmetric matrices and quadratic forms, matrices and differential equations, and positive matrices and their use in probability theory and mathematical economics are covered in this text.

**8.3. Review by: M R Hestenes.**

*Mathematical Reviews*MR0122820

**(23 #A153)**.

The present volume serves as an introduction to the fundamental concepts of matrix theory. The contents are specifically slanted toward the needs of analysts, mathematical physicists, engineers and mathematical economists.

The aspects of matrix theory presented by the author fall into three categories: the theory of symmetric matrices and quadratic form; matrices and differential equations; positive matrices.

The book contains a wealth of material. It has abundant exercises and references to papers in the literature. It is particularly useful to persons interested in applications. The book has an excellent preface, not only giving a description of the material in the book, but also its relation to further books on matrices that will be forthcoming. An attempt has been made throughout to motivate the ideas presented. In this the author has been very successful. A book of this type will broaden the student's horizon.

Although in principle the student needs little or no familiarity with matrix theory and analysis before reading this book, the reviewer is of the opinion that the student would be well advised to obtain an elementary introduction to these topics beforehand. One of the features of the book is that it integrates algebra and analysis. Too often these fields are separated and the student fails to see the connections between them. There is a great need for books of this type. It is hoped that more books of the calibre of the present book will be forthcoming.

**8.4. Review by: R A Willoughby.**

*SIAM Review*

**2**(3) (1960), 225-226.

This book is an important addition to the existing literature on matrices. The presentation is purposely slanted toward motivating and providing insight into those aspects of matrix analysis which arise in many current applications. Each chapter ends with a guided tour through the pertinent literature and the reader will find the brief comments which accompany the listing of the references a pleasant and useful feature.

The book is modest in size, but it contains a greater abundance of information than the size would indicate because of the miscellaneous exercises at the end of each chapter. These problems, which vary greatly in content and difficulty, contain many results extracted from the periodical literature. In order to under- stand and to benefit from the book, the reader needs to work with pencil and paper as he reads, and the broad coverage of the exercises provides a stimulating basis for gaining the necessary insights.

The author consistently develops the concepts and formulas in the simplest framework which adequately conveys the essential points. Great care is taken in the proofs to avoid tedious and unimportant details, especially in the case of inductive arguments. Where additional insight can be thereby gained, one or more alternate proofs of important results are often provided.

In the introduction, it is made clear that the numerical and computational aspects of matrix theory are not covered in the present volume, but are to be developed in a subsequent volume to be authored by G E Forsythe. What is developed here are the analytical characteristics of matrices especially as they apply to such areas as dynamic programming, ordinary differential equations, probability theory, and economics. Knowledge of the topics in this book is essential as background and motivation for the numerical analysis aspects of matrix theory. Also, the book is invaluable as an aid in properly formulating problems in matrix language.

The analytical structure of matrices is intimately tied to the concept of eigenvalues and eigenvectors of matrices. A beautiful and thoroughly developed portion of this theory is that of real symmetric matrices and their associated quadratic forms with prime importance being attached to the positive definite case. The first eight chapters are devoted to the structure of real symmetric matrices.

In Chapters 10 and 11, some of the structure of real nonsymmetric matrices is introduced in connection with ordinary differential equations. Certain special types of real matrices such as Kronecker products, Markoff matrices and positive matrices are treated in the last quarter of the book.

The style and subject matter of the book make it highly desirable as a textbook for a matrix theory course at the advanced undergraduate or first year graduate level. The author points out in the introduction that it is best for the beginner in matrix theory to read the first five chapters, then jump to Chapters 10 and 11 and wind up the first bout with Chapters 14 and 16. This is good advice both for the reader and also for a teacher organising a first course in matrix theory.

The reader should be warned that there are a large number of misprints and the exercises have more than their share. Fortunately, most of the misprints are easy to spot by context.

**9. A brief introduction to theta functions (1961), by Richard Bellman.**

**9.1. From the Publisher.**

Brief but intriguing, this monograph on the theory of elliptic functions was written by one of America's most prominent and widely read mathematicians. Richard Bellman encompasses a wealth of material in a succession of short chapters, spotlighting high points of the fundamental regions of elliptic functions and illustrating powerful and versatile analytic methods.

Suitable for advanced undergraduates and graduate students in mathematics, this introductory treatment is largely self-contained. Topics include Fourier series, sufficient conditions, the Laplace transform, results of Doetsch and Kober-Erdelyi, Gaussian sums, and Euler's formulas and functional equations. Additional subjects include partial fractions, mock theta functions, Hermite's method, convergence proof, elementary functional relations, multidimensional Poisson summation formula, the modular transformation, and many other areas.

**9.2. Review by: W J LeVeque.**

*Mathematical Reviews*MR0125252

**(23 #A2556)**.

This is an unorthodox book; perhaps its character is best described in the author's own words. "Despite our unqualified title, our aim is not so ambitious as to present any complete, or even partially complete theory of theta functions. ... Instead, we wish to use three principal results in the theory of elliptic functions, all expressible in terms of theta functions, as a stage upon which to parade some of the general factota of analysis, and as an excuse to discuss some intimately related results of great mathematical elegance. Our aim is to indicate the applicability and versatility of analytic techniques that should be part of the hope chest of every young mathematician." ... The surrounding material is a loosely-woven account of some of the diverse domains of analysis and number theory on which theta functions impinge, together with the derivation of many of the elementary identities involving theta functions. No mention is made of the connection between elliptic functions and quotients of theta functions, however. There are many references to the literature.

**9.3. Review by: Fritz Steinhardt.**

*The American Mathematical Monthly*

**71**(2) (1964), 228-229.

Dedicated "To devotees of analytic number theory," this book is written by one of today's most versatile research mathematicians. There are seventy sections, some only a few lines long. A partial listing may give some idea of the far-flung contents: § 2: The Four Types of Theta Functions. §§ 4, 9: The Transformation Formula for $\theta_{3} (z, t)$. § 10: Numerical Application. § 11: Modular Functions and Eisenstein Series. § 13: The Heat Equation. § 17: The [Laplace]-Transformed Transformation Formula. § 24: The Riemann Zeta Function. § 26: The Riemann Hypothesis. § 27: The Poisson Summation Formula and the Zeta Function. § 29: Gaussian Sums. § 30: Polya's Derivation. § 32: A Fundamental Infinite Product [for $\theta_{4}$.]. § 42: Mock Theta Functions. § 47: Landen's Formula. § 50: Functional Equations. § 52: Entire Solutions. § 61: Multidimensional Theta Functions. § 69: The Modular Transformation.

These sections are grouped, to quote from the Introduction, about " ... three principal results in the theory of elliptic functions ... as a stage upon which to parade some of the general factota of analysis, and as an excuse to discuss some intimately related results of great mathematical elegance. Our aim is to indicate the applicability and versatility of analytic techniques that should be part of the hope chest of every young mathematician." The first forty-one pages present various proofs, and applications, of the transformation formula for $\theta_{3} (z, t)$. The next sixteen pages develop the infinite product expansion of $\theta_{4} (z, t)$ and related material. The final portion centres around the theta functions as entire solutions of certain simple functional equations. Interrelations abound; the final portion, for example, contains yet another proof of the fruitful transformation formula for $\theta_{3}$ (with one of the very few, minor misprints occurring in its restatement on page 61).

It is difficult to take the title of the book at face value. A prior acquaintance with, say, the chapters on Elliptic Functions and on the Theta Functions in Whittaker and Watson's

*A Course of Modern Analysis*would seem to constitute the necessary and sufficient background for the general reader to derive inspiration, rather than desperation, from Bellman's enthusiastic collection of gems. His over-generous use of "elegant" and similar adjectives cannot take the place of such essential background and substantial motivation. Much more helpful are the many Comments and References at the end of most sections. To a reader who is well prepared, or willing to follow up the main references actively, Bellman's short monograph could be an exciting tour guide to a subject he refers to, in his Foreword, as "the fairyland of mathematics."

**10. An Introduction to Inequalities (1961), by Edwin Beckenbach and Richard Bellman.**

**10.1. From the Publisher.**

Most people, when they think of mathematics, think first of numbers and equations - this number $(x)$ = that number $(y)$. But professional mathematicians, in dealing with quantities that can be ordered according to their size, often are more interested in unequal magnitudes that are equal. This book provides an introduction to the fascinating world of inequalities, beginning with a systematic discussion of the relation "greater than" and the meaning of "absolute values" of numbers, and ending with descriptions of some unusual geometries. In the course of the book, the reader will encounter some of the most famous inequalities in mathematics.

**10.2. From the Preface.**

Mathematics has been called the science of tautology; that is to say, mathematicians have been accused of spending their time proving that things are equal to themselves. This statement (appropriately by a philosopher) is rather inaccurate on two counts. In the first place, mathematics, although the language of science, is not a science. Rather, it is a creative art. Secondly, the fundamental results of mathematics are often inequalities rather than equalities.

In the pages that follow, we have presented three aspects of the theory of inequalities. First, in Chapters I, 2, and 3, we have the axiomatic aspect. Secondly, in Chapter 4, we use the products of the preceding chapters to derive the basic inequalities of analysis, results that are used over and over by the practicing mathematician. In Chapter 5, we show how to use these results to derive a number of interesting and important maximum and minimum properties of the elementary symmetric figures of geometry: the square, cube, equilateral triangle, and so on . Finally, in Chapter 6, some properties of distance are studied and some unusual distance functions are exhibited.

There is thus something for many tastes, material that may be read consecutively or separately. Some readers will want to understand the axiomatic approach that is basic to higher mathematics. They will enjoy the first three chapters. In addition, in Chapter 3 there are many illuminating graphs associated with inequalities. Other readers will prefer for the moment to take these results for granted and tum immediately to the more analytic results. They will find Chapter 4 to their taste. There will be some who are interested in the many ways in which the elementary inequalities can be used to solve problems that ordinarily are treated by means of calculus. Chapter 5 is intended for these. Readers interested in generalising notions and results will enjoy the analysis of some strange non-Euclidean distances described in Chapter 6.

**10.3. Review by: Editors.**

*Mathematical Reviews*MR0130141

**(24 #A8)**.

This book is a careful, elementary treatment of inequalities written for the Monograph Project of the School Mathematics Study Group. This project is directed at able high school students.

Chapters 1, 2, and 3 emphasise axiomatics. In Chapter 4 several classical inequalities of analysis are established, for example, the relationship between the arithmetic mean and geometric mean, the Cauchy, Hölder, and Minkowski inequalities, and the triangle inequality. Chapter 5 deals with applications to geometric maximum and minimum problems, and Chapter 6 considers some special distance functions.

This is not a light, popular treatment of the subject. Students with a superficial interest in mathematics will soon lose interest as they are confronted with details of proofs. But the excellent student who works through the book carefully will acquire valuable skills and understandings.

**10.4. Review by: Nicholas D Kazarinoff.**

*The American Mathematical Monthly*

**70**(2) (1963), 228.

This is a monograph on topics that "have particularly delighted and intrigued" the authors, and "to the study of which" they "have contributed." Particular emphasis is given to inequalities related to positive definite matrices, characteristic roots, positive matrices, moment spaces, resonance theorems and the positivity of operators. The contents are presented in over two hundred short sections (many less than half a page in length). A skeletal outline of many fields is given, the reader being asked to consult original sources for details. A great many references are provided.

Particularly good is the discussion of the inequality between arithmetic and geometric means; twelve proofs are given that illustrate several important concepts. ...

The authors are to be commended for presenting Caplygin's theorem on comparison of solutions of two differential equations. This result has been exploited almost exclusively by Soviet mathematicians and has yet to find its way into U.S. texts on differential equations, although it appears in Lusin's

*Calculus*.

The two chapters on matrices, characteristic roots, and moment spaces constitute the strong point of the book. Their pages are full of fascinating theorems.

The final chapter is on inequalities for differential operators. While many portions of other chapters, for example, the section on Garding's Inequality, are relevant to the theory of partial differential equations, to head a section of one third of a page in this chapter "Partial Differential Equations" is misleading. The gap between the authors' recognition in the Preface of the role of inequalities in the modern theory of partial differential equations and their treatment of it in the text is disappointing.

This book is not particularly suited for the student. It is rather for the research mathematician. He will often find it an invaluable source.

**10.5. Review by: J L Botsford.**

*The American Mathematical Monthly*

**69**(5) (1962), 445.

This book will be a real challenge to the superior high school student, as well as to the better college underclassman. The reader will start in with what seems elementary, almost trivial, mathematics and see it developed into a complicated and fascinating presentation of the same ideas in more intricate form. Students at different ability levels may undertake the study of this book, stopping off at different points, but all profiting by the exposure. The writers initiate the study with the discussion of the ordinary "less than" and "greater than" symbols, take up the various definitions of absolute value with its triangular inequality, and then follow with an elementary but difficult deduction of the Cauchy, Holder, Triangle, and Minkowski inequalities. These inequalities are used in solving certain maximum and minimum problems. The book ends in a discussion of different definitions of distance that satisfy the distance axioms.

**10.6. Review by: W Eugene Ferguson.**

*Science New Series*

**134**(3489) (1961), 1515.

*An Introduction to Inequalities*is a book that all seniors capable of college work should know quite a lot about. This is material not found in most "traditional" programs, but found in all of the newer mathematics programmes.

Chapter 1, "Fundamentals," chapter 2, "Tools," and chapter 3, "Absolute value," which gives the axiomatic aspect of inequalities, should be read from grade 9 through 12. Chapter 4, "The classical inequalities," requires junior and senior sophistication in mathematics. Chapter 5, "Maximisation and minimisation problems," and chapter 6, "Properties of distance," are interesting and will require the algebraic facility of the better students. This book should challenge the student during all of his high school year.

**10.7. Review by: Donald F Devine.**

*The Mathematics Teacher*

**55**(5) (1962), 401.

This book appears to be one which will help to answer many questions about inequalities in the minds of students, teachers, and interested lay persons. The approach made in this book should be very helpful to both the high-school student and to teachers who will be presenting materials involving inequalities in their classes. As a matter of fact, the book could meet the demands of a variety of interests. The first three chapters present the axiomatic aspects of inequalities. In chapter four some of the basic inequalities of analysis are derived. In chapter five these results are used to derive maximum and minimum properties of elementary symmetric figures of geometry, and in the final chapter some properties of distance are studied. Problem sections are included throughout the book and answers are given at the back of the book.

...

It would seem to me that the first three chapters of the book would be of great value to secondary-school pupils. However, I suspect that the materials concerning classical in equalities and the applications of them will have a more limited reading public. These materials would require somewhat more mathematic maturity than is possessed by the average high school student to be fully appreciated. They are certainly worth having available for this select group.

**10.8. Review by: Harry M Gehman.**

*Mathematics Magazine*

**35**(3) (1962), 182-183.

The volume by Beckenbach and Bellman will probably prove to be a difficult book for high school students and laymen to read and understand. The first three chapters emphasise the axiomatic basis of inequalities. Theorems on inequalities, on the properties of the real number system, and on absolute value are proved. The fourth chapter uses these theorems to derive many of the classical inequalities of analysis. In chapter five various problems in maximisation and minimisation are solved by using inequalities rather than by the more conventional methods of calculus. Inequalities are even used to obtain the equations of tangents to an ellipse! The final chapter is devoted to generalisations of the concept of Euclidean distance. Here the triangle inequality is of interest.

**10.9. Review by: Morris**Kline

**.**

*Scientific American*206 (1) (1962), 158.

An

*Introduction to Inequalities*by Beckenbach and Bellman, both fine mathematicians, pursues the subject of inequalities far beyond the needs of young students and laymen and is difficult reading for the purported level of the series. In addition, little is done to show where even mathematicians use such inequalities. The few applications to geometry overlap those of Kazarinoff's, but here the reliance is on arithmetic and algebraic proof, and so the applications are less perspicuous. Analytic geometry is presupposed. The final chapter, which discusses concepts of distance between two points, may serve to demonstrate the use of the inequalities, but it sheds no light on what one does with various concepts of distance.

The first three chapters are intended to teach the axiomatic method, but if the form is axiomatic, the substance is confused. In their treatment of inequalities the authors allow themselves the use of all properties of the ordinary real numbers. Hence they presuppose far more than they state explicitly in their axioms and call freely on the properties of real numbers whenever they need them. Just how the idea of the axiomatic method can emerge from such a presentation is not clear. Some of the confusion is illustrated in the following. From their axioms the authors prove that the product of a positive and a negative number is negative-but the proof appeals to "the usual algebraic rules for interchanging parentheses and minus signs: -[a(-b)] = -[-(ab)] = ab." Where do all these rules come from? This presentation shows once again that any attempt to be axiomatic about the real numbers (except on the professional mathematician's level) produces axioms, definitions, theorems and obscurity.

Equally annoying are frequent references to advanced ideas such as "the limiting processes that are used in defining irrational numbers" and "the completely ordered field of real numbers ." The treatment of absolute values of numbers through inequalities is a complex distortion of a simple idea.

**10.10. Review by: R L Goodstein.**

*The Mathematical Gazette*

**46**(356) (1962), 156-157.

Beckenbach and Bellman treat the arithmetic-geometric mean inequality in easy stages, Cauchy's inequality, the triangle inequality, and the Holder and Minkowski's inequalities. The book concludes with an interesting account of some non-Euclidean distance functions and a proof that for any bounded convex point set $S$, symmetrical about the origin, there is a distance function for which $S$ and its boundary form the unit circle.

**11. Inequalities (1961), by Edwin Beckenbach and Richard Bellman.**

**11.1. Review by: J Aczél.**

*Mathematical Reviews*MR0158038

**(28 #1266)**.

It is difficult to get complete information about inequalities from the classic work of G H Hardy, J E Littlewood and G Pólya. One cannot expect completeness from the present work, especially since an enormous amount has been published about inequalities in the narrower sense in the last 30 years. In addition, the authors' understanding of this concept is so broad that practically the entire subject of analysis belongs to it, and finally, as is mentioned in several places in this book, they intend to write a second volume on inequalities. The present work thus becomes an interesting presentation of the author's areas of interest with many references to cross-connections. The wealth and diversity of the topics covered makes it a hopeless undertaking to attempt to reproduce the content of the total of 183 paragraphs here, even cursorily.

...

The referee is convinced that every mathematician can read this work with benefit and enjoyment.

**12. Adaptive control processes: A guided tour (1961), by Richard Bellman.**

**12.1. From the Publisher.**

The aim of this work is to present a unified approach to the modern field of control theory and to provide a technique for making problems involving deterministic, stochastic, and adaptive processes of both linear and nonlinear type amenable to machine solution. ... Psychologists interested in learning theory, workers in the field of communication and prediction, statisticians working in decision theory, operations research and systems analysts, and biologists studying pest control and optimal dosage will all find the book of use. ... Due to its emphasis upon ideas and concepts, it is equally suited for the pure and applied mathematician.

**12.2. From the Preface.**

To the mathematician we would like to present a vista of vast new regions requiring exploration, where significant problems can be simply stated but require new ideas for their solution. In almost every direction that we have taken, practically everything remains to be done. There are concepts to make precise, arguments to make rigorous, equations to solve, and computational algorithms to derive.

We feel that the theory of control processes has emerged as a mature part of the mathematical discipline. Having spent its fledgling years in the shade of the engineering world, it is now like potential theory, the theory of the heat equation, the theory of statistics, and many similar studies - a mathematical theory which can exist independent of its applications.

To the engineer we offer a new method of thinking about control problems, a new and systematic way of formulating them in analytic terms, and lastly, with the aid of a digital computer, a new and occasionally improved way of generating numerical answers to numerical questions.

**12.3. Review by: D Morgenstern.**

*Econometrica*

**30**(3) (1962), 599-600.

The processes studied by the author may describe dynamical (i.e., the time dependent) mechanical, chemical, biological, economic, or psychological systems. The unifying interest is in systems that exhibit change over time as a result of the operation of internal and external influences, the formulation of which in mathematical terms is often a problem in itself (but not one studied in this book). The simplest case, is described, of course, by an ordinary differential equation $\large\frac{dx}{dt}\normalsize = g(x)$ with initial condition $x(0) = c$ for the single scalar quantity $x(t)$. $x$ may be location, speed, dilution, temperature, drug-intensity, stockpile, price, or the like. In feed-back or control-theory a certain influence of the operator or manager is admitted, and this is represented in this simple model upon modifying the governing equation to $\large\frac{dx}{dt}\normalsize = g(x,y(t)), x(0) = c$, where $y$ represents the control. One wishes to choose $y$, perhaps subject to restrictions, in order that $x(t)$ behave in some prescribed fashion. Two criteria are considered: (1) It is required that $x(t)$ shall be as close as possible to a given $z(t)$, in the sense that

$\int h(x(t), z(t)) dt$,

where $h$ is a given loss function, is minimal. (2) It is required that the final value $x(T)$, the terminal value, be minimal.

These problems can be considered for vector-valued quantities $x, y, z$, the case most common for mechanical systems (phase space). For practical reasons, another simplification is achieved by replacing the differential equation by its discrete analogue, the difference equation, thus giving a recursive formula as the governing equation.

It is obvious that these problems formally belong to the realm of the classical calculus of variations. But in many cases degenerate problems or other inconveniences arise. For such problems, the author has developed (see his book, Dynamic programming, Princeton University Press, 1957) an independent theory which also gives computation. That theory is independent of Euler equations, etc. A more complex theory introduces random disturbances and their probability distributions. The highest peak in this edifice is the concept of adaptive control processes where these probabilities are unknown and have to be estimated in the course of time (e.g., the manager learns about this influence) or must be taken into account in other ways (cf. Wald's theory of games). The present book is mainly an introduction to all these concepts, methods, and problems. The mathematical prerequisites are only the calculus and an introduction to differential equations, which a scholar in any of the fields mentioned above should be acquainted with nowadays. The book is written in a lucid style, incorporating poetry and folksayings and because of its many references it is especially useful as an introduction.

The examples given are easy, taken mainly from fields other than econometrics, but the book leaves no doubt about the usefulness of its subject matter in econometrics. As the subtitle indicates, the book is essentially introductory and does not, therefore, always give complete proofs; many things are only mentioned, yet the book is more stimulating than would be one that only presented standard methods

for applications to specific problems.

...

Since this book contains little mathematics (compared with

*Dynamic Programming*), I hope that many non-specialists not only will follow this guided tour but will use it as a key for further reading, thinking, and research in this vast field of control theory, servomechanisms, or cybernetics.

**12.4. Review by: Henry S Churchill.**

*Scientific American*

**205**(1) (1961), 180.

A series of lectures which attempt a unified approach to control theory and seek to provide a technique for making various problems amenable to machine solution. Feedback control, dynamical systems, decision processes and dynamic programming, boundary-value problems, sequential machines and the synthesis of logical systems, uncertainty and random processes, adaptive processes, stochastic control processes, theory of games and pursuit processes, communication theory-all are within the author's net. Bellman is more than a good mathematician; he has humour, verve and style, all of which, unlikely as it may seem, he applies to these topics.

**12.5. Review by: E M Wright.**

*The Mathematical Gazette*

**46**(356) (1962), 160-161.

This book had its origin in a series of lectures given at the Hughes Aircraft Company. It provides a brilliant introduction to and survey of the mathematics of adaptive control or automation, based largely on the method of dynamic programming which owes much of its development to the author. The eighteen chapters cover so wide a variety of topics that none can be at all deeply treated, but each chapter concludes with carefully annotated references which enable the reader to pursue further any particular aspect of the subject in which he is interested.

...

We can commend this book unreservedly to a wide circle of readers. But this is unnecessary; in this field this author's name suffices to arouse our interest.

**12.6. Review by: A A Mullin.**

*American Scientist*

**49**(3) (1961), 306A; 308A.

Immanuel Kant has said that "... our diverse modes of knowledge must not be permitted to be a mere rhapsody, but must form a system." René Descartes also emphasised the unity of science and thought with the first of his twenty-one Rules of Method. With such a spirit in mind the author of the book under review makes use of his results on dynamic programming [Richard Bellman,

*Dynamic Programming*1957], among many other things, to sketch the bare outline of a broad physico-mathematical theory of control processes. Special emphasis is placed on various kinds of extremal problems, e.g., as concern least-time trajectories in some specified phase space; and special emphasis is placed on problems dealing with optimal policy, e.g., as associated with "bang-bang" or impulsive control processes.

The book is written in such a manner that one must have a fair familiarity with the technical jargon of the field of control processes in order fully to appreciate the ambitious and integrative nature of its scope.

**12.7. Review by: Marshall Freimer.**

*Journal of the American Statistical Association*

**60**(309) (1965), 383-384.

The title of this book is aptly chosen. As a guided tour of a World's Fair, which tells him something about each of the exhibits and then invites him to visit them for himself, the reader of this book finds a large number of ideas thrown at him, with references to the sources in which he can find all the details. Done well, as it is here,. this results in a desire to see it all at first hand.

One reason for this highly novel approach to mathematics is that, unlike Bellman's earlier book

*Dynamic Programming*(1957), this is not meant to be a traditional mathematics text. Instead, it attempts to instil in the reader something of the author's philosophy of mathematical research. Mathematics is not to be done for its own sake. The underlying problem, physical, geometric, or what have you, must always be kept in mind. "Only this constant interchange between the real and abstract ... keeps mathematics vital. Without it there are the dangers of sterility, atrophy, and ultimately, decadence."

A second, and more specific, reason for the guided tour approach is to be able to present a large number of applications. These illustrate the use of dynamic programming to provide a unified treatment of control processes. Bellman considers three broad classes of control processes: deterministic, in which all aspects are known a priori; stochastic, in which some relevant parameters are random variables with known probability distributions; and adaptive, in which random variables with unknown distributions enter in. The latter terminology arises from the fact that in a multistage decision process of this type there is the possibility of learning more about the random variables than was initially known, and thus adapting to the situation. ...

This is an ideal book for the reader who wishes to learn something about the use of dynamic programming without becoming involved with a treatise on the subject. It is not nearly so good as a general introduction to modern control theory. For example, the important works of Kalman and Pontrjagin are mentioned only in passing and that of Breakwell, Bryson, and Kelley not at all.

Because this book is very likely to lead students down individual paths of study, I would not attempt to use it as a text in a lecture course. It could be used for a reading course or a seminar or, as Bellman originally used it, for a series of lectures to a group of professional engineers and scientists.

**12.8. Review by: E S Page.**

*Journal of the Royal Statistical Society. Series A (General)*

**125**(1) (1962), 161-162.

This is another book from the prolific mind and pen (or electric typewriter) of Richard Bellman. It is written in an entertaining style with, at times , the use of a rich vocabulary. On the first page of the first chapter the word "gallimaufry" appears - a word which, as far as I can remember, has never occurred in the gallimaufry of literature to which I have been exposed. On nearly the last page he refers to the polynomials of a mathematician, Cebycev - a spelling perhaps correct, perhaps soon to be universally adopted, but not as familiar to me as five or six other attempts at the name. Between these flourishes is an account of the application of the methods of dynamic programming to several types of problem leading up to control processes. The author is careful to point out that just one of several possible approaches is discussed.

The book contains an introduction and eighteen chapters, each of which has a list of further references and comments. These forty pages or so of comments give a good indication of how much work has been going on and how great a proportion Bellman himself has had some part in. All the chapters are short. The first four show how feedback control processes lead to problems which seem to appear as examples in the classical calculus of variations but which present difficulties when attempted by those methods; some differential equations and variational problems are presented in a functional equation form which leads naturally to a treatment by dynamic programming methods. The "curse of dimensionality" is continually stressed and three of the chapters discuss the difficulties and possible methods of avoiding them when the number of state variables is as large as three. The book turns from the treatment of deterministic control processes to those with a stochastic element, first dealing with the case where the distributions concerned are fully known. Bellman shows that the functional equation technique yields a unified approach for both the deterministic and stochastic processes. There are some suggestions for the numerical solution of some non-linear, first-order differential equations in terms of an equation derived from a control process. Brief chapters on stochastic learning models and game theory are followed by those on adaptive processes in which there is incomplete knowledge about the characteristics of the stochastic elements. The last chapter "Successive Approximations" reviews applications of the classical approach by solving sequences of problems which converge to the desired one and contrasts with it the method of deriving a succession of approximate policies which converge to the desired optimum.

Throughout the book the emphasis is directed towards expressing in a suitable form the problems considered and finding methods of computing the solution. It seems to the reviewer a big disadvantage in a book of this type that there are no numerical examples at all. There are illustrations of how problems need to be formulated-although in nothing like the profusion of the author's Dynamic Programming-in order to be able to compute a solution, but there the problem is left. It is perhaps out of place to make this objection when I expect that it is amply met by the contents of a new book by S Dreyfus and the author on the Computational Aspects of Dynamic Programming, which I have not yet seen, but in the absence of such examples it is impossible to assess the success or importance of the techniques proposed. Apart from this criticism, the book is a useful addition to the literature on dynamic programming.

**12.9. Review by: C Derman.**

*Management Science*

**7**(4) (1961), 450.

This is a stimulating book originating out of a lecture series given by the author at the Hughes Aircraft Company. The title is slightly misleading in that the book is not really about adaptive control processes but about a mathematical approach to control processes, dynamic programming, a subject of some interest in itself and a useful tool in a variety of situations.

The main point of the book is that dynamic programming is a conceptually simple, often computationally feasible, almost universally applicable method for considering the problems of optimisation (under many constraints) arising in control processes. In relation to dynamic programming, topics such as theory of games, Markov processes, stochastic learning models, sequential machines, and successive approximations are discussed.

Coloured by the author's own scientific philosophy and personality, the book makes for interesting, educational, and entertaining reading. Should the reader wish to study further, ample references are given at the end of each chapter.

**12.10. Review by: A A Mullin.**

*Mathematical Reviews*MR0134403

**(24 #B456)**.

The author assumes that the reader of the book is quite familiar with the technical jargon of the field of control processes, e.g., nowhere does one find either a definition or an extensive classification of that which may be called a "process". "Invariant imbedding", mentioned at least four times in the book, will be somewhat of a mystery to those who have not followed the papers on mathematical physics by the author and others. Sometimes the sections are extremely sketchy, almost to the point of "name-dropping", e.g., section 8.10 entitled "Synthesis of logical systems" has fewer lines of text devoted to it than there are books on the subject. However such objections to the book are balanced, at least in part, by the ambitious and integrative nature of the far-reaching scope of the book.

**12.11. Review by: George W Morgenthaler.**

*Operations Research*

**10**(1) (1962) 143-146.

From hieroglyphics to Homer, with Hiawatha in between - what a guided tour! Everything about this book is different.

The striking jacket displays the hieroglyphic symbols for 'steersman' as emphasis for its adaptive-control-processes title. Just as the last chapter begins with a quotation from Homer's

*Iliad*, so every chapter is prefaced by a pithy saying or a profound quotation whose real significance can be discerned only after the main concept of the chapter has been understood (if then). The reference to Hiawatha is to the hilarious poem, "Hiawatha Designs an Experiment," by M G Kendall, found in the Bibliography and Comments part of Chapter IX. The Bibliography and Comments section of a given chapter may contain anything from a scholarly comment to a poem, from humour to philosophy, and it usually does.

This book does not fit neatly into any one of the usual categories of mathematical books. It is certainly not in the popular vein of Williams'

*The Compleat Strategist*or Huff's

*How to Lie with Statistics*. This is evident because it uses the integral symbol and partial derivatives. It is not a textbook, undergraduate or graduate, for it lacks the usual examples and problems built upon an assumed common background. It is not a reference book, for it provides no encyclopaedic source of theorems or formulas. Nor is it the usual mathematical monograph, for these are written for experts by experts and, while Dr Bellman is certainly an expert, he has taken pains to define and explain basic mathematical notions at several points in the book; hence, the audience need not be entirely expert.

...

The Introduction shows how a mathematical description of the behaviour of a physical system involving the system's present state and its past history may be obtained in terms of sets of vector differential and vector differential-integral equations. The concept of stability is introduced in passing, and the discussion leads to the inclusion of a control vector and the formulation of the control process in analytic terms. Control problems are found to be of three basic types: deterministic (a decision determines an unique outcome), stochastic (a decision determines a random output with known distribution), and adaptive (initial ignorance overcome by learn-as-you-go). While several approaches to these problems by others are called out, Bellman announces his intent to apply the technique of dynamic programming and to show the great versatility of this method.

...

The mathematical formulation of adaptive control processes which Bellman has devised is a valuable and courageous step forward. It may be that too much of the early part of the book reiterates the ideas of dynamic programming that have already been presented in his earlier book. The mathematician will squirm a bit about the 'formal' solutions that are not substantiated by suitably qualified existence theorems, and the numerical analyst will be awed by the computing tours de force sometimes implied. "Faith is the substance of things hoped for, the evidence of things not seen," is perhaps appropriate. But Bellman himself calls attention to these lapses. Occasionally he seems to forget control processes and wanders off into any convenient application of dynamic programming. To use another favourite quote of the reviewer, "He mounts his horse and rides off in all directions." The genuine sparkle of the book and its lack of frequent or obvious typographical error make one more than willing to look beyond these possible annoyances. The book is definitely of interest to the operations-research world.

**12.13. Review by: John E Gibson.**

*Quarterly of Applied Mathematics*

**20**(2) (1962), 195-196.

This is an edited and expanded version of a series of lectures on the modern concept of automatic control theory given by Bellman in 1959. Fortunately the editing has not meant the removal of the flavour and wit. This book should not be taken as a formal text or treatise on automatic control in general nor dynamic programming or adaptive control in particular. Rather it is a vigorous, philosophical presentation of a number of mathematical concepts which will be found useful in the study of adaptive control. The reader will find a very stimulating, informal discussion of the calculus of variations vs dynamic programming as applied to modern optimum control. The principle of optimality, which lies at the heart of Dynamic Programming, provides, in concept, a solution to the classical two-point boundary-value problem. It has been pointed out in recent years that the optimum control problem can be formulated in these terms, thus we see the reason for the interest of engineers in this subject Unfortunately the computational aspects of the problem as formulated in terms of Dynamic Programming are not trivial. In fact it is difficult if not impossible with the memory capacity of present day digital computers to solve the problem in systems of higher than third order without the imposition of artificial constraints.

There are a number of chapters expounding the philosophy of dynamic programming and sequential analysis. The reader will find discourses on game theory, communication theory and successive approximations. Some of the shorter chapters, like those on Stochastic Control Processes, Markovian Decision Processes, and Stochastic Learning Models for example are little more than a definition of terms and an annotated bibliography. The discussion of adaptive control itself occupies only about fifteen pages late in the text, and is essentially discursive in nature. We must wait until chapter 15 for a definition of adaptive control and then it is not given in one sentence nor is it operative and it is in a chapter with not a single equation. The list of topics in the table of contents is encyclopaedic thus the reader can expect only a page or two at most on each. This is obviously inadequate for the serious worker but it is ideal to an interested newcomer who wishes to be oriented on how the various topics are interrelated.

It is amusing for an engineer to hear a mathematician with an excellent classical training excoriating classical mathematics as only Bellman can. The danger lies with the fact that the naive engineer may be induced to give up what he never really learned. Though it may be true that Mathematics and mathematicians badly need new mathematics to solve new problems, we engineers still have much to learn and apply in the classical mathematics.

This book should not be looked upon as a formal, technical effort and it should probably be kept out of the hands of that overly serious, pedantic creature, the Ph.D. candidate. For every one else concerned with modern mathematics and the "new automatic control"; in fact with any modern science it is enthusiastically recommended. It is guaranteed to be almost as much fun to read as it obviously was to write.

**13. Applied dynamic programming (1962), by Richard Bellman and Stuart E Dreyfus.**

**13.1. From the publisher.**

This comprehensive study of dynamic programming applied to numerical solution of optimisation problems. It will interest aerodynamic, control, and industrial engineers, numerical analysts, and computer specialists, applied mathematicians, economists, and operations and systems analysts.

**13.2. From the Preface.**

In the period following World War II, it began to be recognised that there were a large number of interesting and significant activities which could be classified as multistage decision processes. It was soon seen that the mathematical problems that arose in their study stretched the conventional confines of analysis, and required new methods for their successful treatment. The classical techniques of calculus and the calculus of variations were occasionally valuable and useful in these new areas, but were clearly limited in range and versatility, and were definitely lacking as far as furnishing numerical answers was concerned.

The recognition of these facts led to the creation of a number of novel mathematical theories and methods. Among these was the theory of dynamic programming, a new approach based on the use of functional equations and the principle of optimality, with one eye on the potentialities of the burgeoning field of digital computers.

The first task we set ourselves in the development of the theory was the examination of a variety of activities in the engineering, economic, industrial, and military domains with the aim of seeing which could be formulated in dynamic programming terms and with what level of complexity. This is not always a routine operation, since in the description of an optimisation process there is a good deal of leeway permitted in the choice of state variables and criteria. It often happens that one type of mathematical model is well suited to one type of analytic approach and not to another. Generally, the three principal parts of a mathematical model, the conceptual, analytic, and computational aspects, must be considered simultaneously and not separately. Consequently, a certain amount of effort is involved in translating verbal problems posed in such vague terms of efficiency, feasibility, cost, and so on, into precise analytic problems requiring the solutions of equations and the determination of extrema.

Once various translations had been affected, it was essential to study the relation between the solutions of the functional equations obtained and the optimal policies of the original decision process. This study of the existence and uniqueness of solutions is required before one can engage in any computational solution or structural analysis. Fortunately, in a number of significant cases, the demonstration of existence and uniqueness was readily carried out so that we could, with confidence, turn to the study of the nature of optimal policies and of the maximum return, using the basic functional equations.

**13.3. Review by: P B Coaker.**

*OR*

**15**(2) (1964), 155-156.

To many people, Bellman's Dynamic Programming was difficult reading, but most of them will find this book to be much easier. In fact, Bellman and Dreyfus have produced a book which is a good guide to Dynamic Programming for the operational research worker, the control engineer, the flight engineer and the mathematician. It is only a guide; much is left for the reader to follow up, but extensive bibliographies with reference to the section headings are given at the end of each chapter. The non-mathematical operational research worker will want to skip many chapters but the essential ideas of the technique can be readily acquired by selected reading. I suggest that he should concentrate on Chapters I, II, III, VII and XI. The more mathematical sections may well leave the reader with a similar sense of awe to that when he first heard of the use of Laplace transforms for the solution of differential equations.

The authors have used a novel method of presentation, which is introduced and developed in the first chapter. After a preliminary discussion in which one is convinced that such a technique is an essential tool, the complexities of the resultant calculations are admitted. Then all further problems are not solved in detail but a computer flow diagram is presented with examples of the typical results obtained. In the first problem the flow diagram is explained fully, subsequently the reader is left to follow it himself. In many instances the merits of different computational techniques are discussed, in particular the time factors involved are compared. In all cases the development of any formulae used is in sufficient detail to be readily understood, and the application of the principle of optimality becomes very familiar as one proceeds through the book. In fact one becomes amazed at the many applications which are indicated.

A word of warning should be given for those proceeding to Chapter III. A number of printer's errors were found at this stage, some being minor, but [others are more serious]. ...

One could blame these and other printer's errors on the proof readers, but mention of them should not discourage anyone from reading this excellent introduction to Dynamic Programming. One hopes, however, that there may be a third book with a monotonic decrease in complexity which might be called "Dynamic Programming for Management".

**13.4. Review by: Marshall Freimer.**

*Journal of the American Statistical Association*

**59**(305) (1964), 293.

This book takes up the story of dynamic programming where Bellman's

*Dynamic Programming*(1957) left off. In the earlier book the emphasis was on proving existence and uniqueness theorems for the functional equations and determining mathematical properties of the solutions. Here the emphasis is on developing the properties to the point of being able to obtain numerical solutions to the problems. Throughout the book the reader will find flow diagrams for computer routines and tables or graphs of actual solutions. In addition, new techniques such as Bellman's use of Lagrange multipliers, Howard's policy space iteration technique for Markovian decision processes, and Bryson's use of successive approximations in policy space are presented.

The chapter titles will give some idea of the scope of the book: I. One-dimensional allocation processes, II. Multi-dimensional allocation processes, III. One-dimensional smoothing and scheduling processes, IV. Optimal search techniques, V. Dynamic programming and the calculus of variations, VI. Optimal trajectories, VII. Multistage production processes utilising complexes of industries, VIII. Feedback control processes, IX. Computational results for feedback control processes, X. Linear equations and quadratic criteria, XI. Markovian decision processes, XII. Numerical analysis.

This book is highly readable, has numerous references to the literature, but totally lacks the exercises and research problems that distinguished Bellman's earlier book. Even so, it would serve as an excellent textbook.

**13.5. Review by: George R Desi.**

*Journal of Marketing Research*

**2**(1) (1965), 94-95.

This is a book for the mathematician, computer specialist, or the operations research analyst. The person moderately well-informed in classical mathematics will at least derive an appreciation for the subject. Dynamic programming is a systematic approach for the determination of optimal policies-where such policies can be mathematically formulated-applicable to a large class of problems. The classical (calculus) approach to finding maxima or minima (which mathematically usually defines an optimal policy) breaks down rapidly in the face of complex real-life problems. Some limitations of the calculus the authors point out are:

- Not all real-life problems are continuous.

- Calculus gives local maxima when the absolute maximum is required. For example, a function involving only two variables will give a three-dimensional structure having, except in the simplest cases, many peaks. Optimal policy generally requires the determination of the highest of these peaks. The methods of calculus can find the peak in some neighbourhood. In fact, if the relationships are linear, the optimal policy appears at the boundary of the domain of definitions-the calculus can only cope with functions in 'open intervals' which excludes any natural boundaries.

- Many useful functions are not single-valued. They are defined within a certain tolerance. The resulting ambiguity cannot be handled by calculus.

...

Some of the examples, despite their mathematical validity, will strike the reader as using a sledgehammer to kill an ant. For example, the caterer problem consists of deciding how napkins should be divided between two laundry services, one faster but more expensive than the other. Since napkins are small and hence present no real storage problem, it would seem an obvious solution to buy so many napkins that clean ones will always be available no matter how slow the cheap laundry is-unless indeed they lose them altogether

**13.6. Review by: J A Bather.**

*Journal of the Royal Statistical Society. Series A (General)*

**126**(4) (1963), 598-599.

This book is a sequel to Bellman's

*Dynamic Programming*and is concerned with the practical and computational aspects of the method. Problems of optimisation are formulated as multi-stage decision processes in such a way that suitable policies, which may depend upon a large number of variables, can be constructed from a succession of much simpler decisions. The functional equations which result from this approach are solved numerically with the aid of a digital computer, although, in a few cases, explicit solutions are presented.

The general method is first described in connection with the problem of allocating a given quantity of an economic resource between a number of activities, each providing a different return. Much more complicated situations arise if there are several types of resource, but it is shown that a simplification can sometimes be achieved by the use of Lagrange multipliers. After a discussion of the Hitchcock-Koopmans transportation model, there is a chapter on optimal search techniques for locating the maximum of a unimodal function and the zeros of a convex function. Next, the authors derive some of the classical equations of the calculus of variations in a simple and natural manner. The discussion of supersonic flight which follows is obscured because of an assumption that the reader is already familiar with the special notation. The later chapters contain a number of stochastic applications which include the control of a simple autoregression and some of the work done by R Howard on Markovian decision processes.

The book contains many interesting examples from widely different backgrounds and its presentation is for the most part elementary. Some of the special topics are treated superficially; in particular, those involving stochastic models, although comprehensive references are given at the end of each chapter. A similar criticism applies to the discussion of particular methods for simplifying the computations. For example, there is no warning that an indiscriminate use of the Lagrange multiplier technique might lead to greater complication. Nevertheless, the dynamic programming approach is undoubtedly powerful and the book should encourage further applications.

**13.7. Review by: W Prager.**

*Quarterly of Applied Mathematics*

**21**(2) (1963), 174-175.

This is a most valuable supplement to the senior author's

*Dynamic Programming*(by R Bellman, 1957). While there the emphasis has been on the general theory much of the present volume is concerned with the computational feasibility of dynamic programming.

Chapter 1 (One-dimensional allocation problems) deals with processes that lead to sequences of functions of a single variable and introduces the reader to the basic ideas of both, dynamic programming and the computational techniques that are to be developed in the remainder of the book. Chapter 2 (Multidimensional allocation processes) extends these ideas to problems involving sequences of functions of two variables, stressing the use of Lagrange's multipliers and successive approximations in policy space. Whereas these chapters treat static allocation processes, smoothing and scheduling: processes, which naturally present themselves in dynamic form, are considered in Chapter 3 (One-dimensional smoothing and scheduling processes). Problems of dynamic programming that arise in the numerical solutions of dynamic-programming problems are discussed in Chapter IV (Optimal search techniques). In Chapter V (Dynamic programming and calculus of variations) dynamic programming is used to derive a nonlinear partial differential equation from which the principal results of the calculus of variations are readily obtained. Chapters 6 (Optimal trajectories), 7 (Multistage production processes utilising complexes of industries), and 8 (Feedback control processes) exploit this approach. Numerical solutions of typical feedback control processes are discussed in Chapter 9 (Computational results for feedback control processes). Chapter 10 (Linear equations and quadratic criteria) is concerned with computational procedures that apply when the equations describing a process are linear but the criterion functions are quadratic. Other ways of taking advantage of the analytical structure of a specific problem to improve the computational procedure are discussed in Chapter 11 (Markovian decision processes). Chapter 12, finally, presents preliminary results concerning questions of accuracy, reduction of computing time, and reduction of dimensionality. Five appendices contain supplementary results (I. On a transcendental curve, by O Gross - II. A new approach to the duality theory of mathematical programming, by S Dreyfus and M Freimer - III. A computational technique based on successive approximations in policy space, by S Dreyfus - IV. A new functional transform in analysis: the maximum transform, by R Bellman and S Dreyfus) and a description of the computer used by the authors - (V. The Rand Johnniac computer, by S Dreyfus).

As far as possible, the authors have tried to make this stimulating work independent of the volume mentioned at the beginning of this review.

**13.8. Review by: A Charnes.**

*SIAM Review*

**6**(2) (1964), 184-185.

This volume is a repository of further results, conjectures, illuminations, and ingenious approaches to the formulation or solution of an extremely wide swath of problems by means of the functional equations techniques which have come to be identified with Richard Bellman. As in previous books by the senior author, the procedure appears to be to rough out ideas by induction from striking examples and generally to leave detailed systematic developments to papers or other items cited in the bibliography. While this technique is conducive to rapid presentation of insights, the serious reader is hereby warned to heed and read citations to avoid misinterpretation or confusion as to the results already achieved either by the authors or other contributors to the field.

For example, Y Schreider in a recent Russian paper rederives a special case of the general formulation and proofs of existence and uniqueness of solutions that Samuel Karlin accomplished ten years ago. Schreider's proof even employs Karlin's technique of the Tychonov compactness theorem. Yet Schreider does cite Bellman's 1957 book,

*Dynamic Programming*in which Karlin's work is referenced! One can surmise that Schreider was confused by Bellman's usage there of contraction operators. Actually Bellman had himself presented more general results in his 1953 Rand Corporation book. The point would seem to be that since Karlin had already presented the general results in 1954, Bellman preferred to present a new approach in 1957, even though it meant sacrificing generality. As the authors put it, "Too often in applications of mathematics to the physical world particular methods are taken as Gospel and it is often forgotten that others exist."

...

... whether the reader is interested in fly-away kits, difficult crossing problems, multistage rockets, the $n$th shortest path, the "bang-bang" control problem nuclear reactor shutdown, adaptive feedback control, stochastic control processes, or the Kuhn-Tucker equations, he will find unusual insightful approaches together with guides to the literature in this book.

**13.9. Review by: B Gluss.**

*Mathematical Reviews*MR0140369

**(25 #3791)**.

This volume gives a basic introduction to the field that develops the theory clearly and systematically, while at the same time presents detailed indications of its practical application to a wealth of different problem areas. Whenever the complexity of the computational solutions merits them, there are well-presented flow diagrams that computer programmers will find most helpful.

The chapter headings, which give an idea of the structure of the book, are (1) One-dimensional allocation processes, (2) Multi-dimensional allocation processes, (3) One-dimensional smoothing and scheduling processes, (4) Optimal search techniques, (5) Dynamic programming and the calculus of variations, (6) Optimal trajectories, (7) Multistage production processes utilising complexes of industries, (8) Feedback control processes, (9) Computational results for feedback control processes, (10) Linear equations and quadratic criteria, (11) Markovian decision processes, and (12) Numerical analysis. Also, hidden away at the back are some appendices of considerable interest, especially one on the maximum transform, an important new mathematical concept of which more will undoubtedly be heard.

Along with the consistently thorough model-building and attack throughout the book, there are frequent detailed comparative discussions of methods of approach other than dynamic programming.

Parenthetically, there are a couple of points that might perhaps have been stressed more: first, that the principle of optimality depends in general on the decision processes being Markovian; and second, that one of the most important features of dynamic programming is that bounds on the state vector computationally reduce the problem, while classical methods often fail when such bounds are present.

The volume is well organised, is written in a lucid style typical of its authors, and is well qualified to be the definitive introduction to dynamic programming and its applications.

**13.10. Review by: George W Morgenthaler.**

*Operations Research*

**10**(6) (1962), 916-918.

A major aim of the present book is to "present a detailed account of the application of the theory of dynamic programming to the numerical solution of optimisation problems.". The computational results presented were developed on the RAND Johnniac computer, whose vital statistics are given in Appendix V, together with the note that the corresponding computer times on more recent computers are estimated to take about 10 as long as those shown for the Johnniac.

...

The book contains very little new theory or new application. It does contain the latest thinking on earlier ideas, and the comments and bibliography at the end of each chapter are particularly informative and of service to the reader. Pedagogically, the introduction to the concepts of dynamic programming and the detailed description of how to compute answers seems much improved over the 1957 book.

...

In summary, only the comments on computability will be new to the dynamic- programming expert. The case for dynamic programming seems completely numerically established for discrete problems, but somewhat less so for the continuous case. The book is strongly recommended for the OR practitioner who wants a simpler demonstrated introduction to dynamic programming with emphasis on how to arrive at answers. In fact, having read all three books cited in this review, the reviewer is inclined to recommend to the newcomer that for best results the books be read in the reverse of their publication order if resulting biblio- graphic and commentary anachronisms can be overlooked.

**14. Invariant imbedding and radiative transfer in slabs of finite thickness (1962), by R E Bellman, R E Kalaba and M C Prestrud.**

**14.1. From the Publisher.**

One of a projected series of Rand publications on the application of the theory of invariant imbedding to certain computational problems in radiative transfer. The method stems from an extension of the ideas of Ambarzumian and Chandrasekhar. After presenting the fundamental equations of classical transport theory and of invariant imbedding, the authors explain the computational methods and explore certain properties of the solutions. Tables are given, for the ideal case of spherical or isotropic elementary scattering and for various values of the absorption coefficient, of the reflected intensity in various directions when an incident beam falls at various angles on a slab of finite thickness.

**14.2. Review by: S Ueno.**

*Mathematical Reviews*MR0143618

**(26 #1171)**.

In this monograph the theory of invariant imbedding is applied to the diffuse reflection problem of parallel rays by a finite, homogeneous and isotropically scattering slab, putting emphasis on the numerical computation.

It consists of four chapters. The first chapter is devoted to the explanation of the physical background of the problem and the formulation of the fundamental equations of classical transport theory and of invariant imbedding. In the second chapter, with the aid of the Gaussian quadrature method, the procedure of numerical solutions of the equation is elucidated. The third chapter contains several analytical discussions of the solution in terms of the thickness of the finite slab for the existence, uniqueness and others, allowing for the limiting behaviour of the reflected flux as the thickness becomes semi-infinite. In the last chapter the computational results are presented in the tables of values of the reflection function as a function of the angle of incidence, the angle of reflection, the thickness of the slab, and the albedo for a single scattering.

It is of interest to mention that, by means of invariant imbedding technique, a study of the time-dependent diffuse reflection problem by a finite flat layer is later made by Bellman, Kalaba and Ueno, and that the numerical computation is now in progress in the RAND Corporation.

**14.3. Review by: G M Wing.**

*Mathematical Reviews*MR0153436

**(27 #3403)**.

The method of invariant imbedding, with special emphasis on its applications to transport theory, has been under extensive investigation for several years. In this book the authors discuss the problem of radiative transfer in a finite homogeneous slab of material, with albedo not greater than unity, and with isotropic scattering. The invariant imbedding equations for the reflection function are developed and approximations to them derived, using Gaussian quadrature. A description then follows of the use of these equations in computation, including a presentation of the actual FORTRAN program employed.

About 90% of the book consists of tables of the reflection function reproduced from the computing sheets by photo-offset. Calculations have been done for seven ingoing and seven outgoing angles, using albedos from 0.1 to 1.0 in intervals of 0.1. Slabs of terminal thicknesses ranging from 3.0 to 20.0 are computed, depending upon the albedo. Results have been checked when possible against the relatively few calculations previously available and are found to be completely satisfactory. In addition the effects of decreasing or increasing the number of angles has been studied, and other self-consistency checks have been made.

These tables should be of considerable value to astrophysicists, neutron physicists, and others interested in transport phenomena.

**14.4. Review by: I P Grant.**

*The Mathematical Gazette*

**50**(371) (1966), 93-94.

This book is the first of a series dedicated to Modern Analytic and Computational Methods in Science and Mathematics which will be edited by the first two authors. According to the dust jacket the book will be of interest to 'research workers in the areas of pure and applied mathematics, physics, astrophysics, and engineering, with particular significance for those concerned with transport processes.' It is intended to provide a self-contained account of the application of invariant imbedding to the calculation of the diffuse reflection of radiation from a plane stratified inhomogeneous slab. This reviewer is not convinced that such an attempt to be all things to all men will satisfy everyone concerned.

The bulk of the text, over 300 pages, consists of tables of the diffuse reflection coefficient for homogeneous plane stratified slabs of various thicknesses for various values of the albedo for single scattering. These tables, which for many will provide the justification for buying this book, have a number of uses. In the first place, this particular problem has been extensively studied analytically, so that the results of different analytical and computational methods can be compared and the validity of the numerical procedure assessed. And secondly, once this has been done, the tables provide a standard against which one can measure the changes in diffuse reflection due to polarisation, to anisotropic scattering and to variation of the albedo for single scattering with position.

This is clearly a valuable piece of work, and it is therefore disappointing that the authors could not have given a more comprehensive treatment of the mathematical and computational aspects of their method in a book of this length. ...

**15. Differential-difference equations (1963), by Richard Bellman and Kenneth L Cooke.**

**15.1. From the Publisher.**

A basic text in differential-difference and functional-differential equations used by mathematicians and physicists in attacking problems involving the description and prediction of the behaviour of physical systems. The subjects covered include the use of the Laplace transform to derive a contour-integral expression permitting a study of asymptotic behaviour, the stability by solutions of linear and nonlinear differential-difference equations, and applications of these techniques to problems encountered in contemporary science and engineering.

**15.2. Review by: Jack K Hale.**

*Mathematical Reviews*MR0147745

**(26 #5259)**.

A differential-difference equation (or, more properly, a functional-differential equation) is an equation in which the derivative of a function is expressed in terms of its past, present and future values. Although many particular equations of this general class have appeared in the mathematical literature over the last hundred years, only within the last few years have they been intensively and extensively cultivated. Up to the present time, there has not been, in any language, a very systematic account of the properties of such equations. This book is the first serious attempt to fulfil this need and will be a welcome addition to the library of students as well as research scientists and mathematicians. Overall, it is well-written, proceeding from the simpler problems to the more complex, with ample exercises for the student. In addition, interspersed in the exercises are research problems with appropriate references to the journal in which the problem has been solved. A careful reading of the research problems is necessary for a good understanding of the current status of the present research in this field.

...

In spite of its shortcomings, the book is a significant contribution to the field and will be very useful.

**15.3. Review by: G Temple.**

*The Mathematical Gazette*

**51**(377) (1967), 276.

This splendid volume offers the first comprehensive account of differential-difference equations "in any language," according to the well justified claim of the publishers. It covers a really immense range of topics. The subject of linear differential-difference equations and of systems of such equations is examined in detail with special reference to the distinctions between the retarded, neutral and advanced types, and detailed application to the "renewal equation" ...

The theory is developed to deal with existence theorems, series expansions, asymptotic behaviour and stability, and covers a number of auxiliary techniques such as the use of the Laplace Transform and the asymptotic location of the zeros of exponential polynomials.

The method of exposition is admirably adapted to the needs of the student and the teacher, and proceeds by well motivated induction from early and familiar examples to the careful construction of deep and far reaching theorems. There is a wealth of interesting and illustrative examples, many of which are well described as "research problems". Each chapter is also provided with an extensive bibliography enriched with illuminating comments on the scope and character of the references.

Apart from its intrinsic mathematical interest this volume is also of the greatest value to physicists, engineers and biologists by reason of the wide variety of applications which can be made of the theory.

**16. Perturbation techniques in mathematics, physics, and engineering (1964), by Richard E Bellman.**

**16.1. From the Publisher.**

Graduate students receive a stimulating introduction to analytical approximation techniques for solving differential equations in this text, which introduces a series of interesting and scientifically significant problems, indicates useful solutions, and supplies a guide to further reading. Intermediate calculus and a basic grasp of ordinary differential equations are prerequisites.

**16.2. Review by: Stephen P Diliberto.**

*Mathematical Reviews*MR0161003

**(28 #4212)**.

For some years now there has been an ever-increasing need for books and/or monographs which attempted surveys of perturbation techniques in ordinary differential equations. The present work, despite its many faults, will prove extremely useful. This utility arises from two counts: The book contains more information about perturbation techniques for ordinary differential equations than any recent text in the field; and secondly, because of the informal or casual presentation, the basic ideas will be more accessible to the engineers and applied scientists.

The informal approach used in this book is one that makes the subject essentially non-mathematical; the author simply outlines a manipulation via a specific example and seldom indicates whether there is a theorem justifying the procedure.

...

The elementary nature of much of the material makes a standard text on ordinary differential equations a more suitable reference source than "research" papers. Since the book includes material not completable to known (mathematical) results, it would have been convenient (for the uninitiated) to have labelled "standard" techniques as such and "proposed" techniques as such. Unless the reader is widely read, he will be unable to determine from the text what techniques are mathematically justified.

**16.3. Review by: M L Cartwright.**

*The Mathematical Gazette*

**57**(40) (1973), 143-144.

This is a very readable booklet; in the words of the author, "What we have attempted to do is to introduce the reader to a plethora of problems, all interesting and all of scientific significance, to indicate a number of useful methods, some old, some new and to furnish a guide to further reading". The enthusiasm which makes the book so readable engenders an outlook which makes it desirable to read somewhat critically. Owing to the restricted space, he says very naturally "we shall focus our attention almost exclusively on methods and soft pedal proofs". The references are numerous and notes on them helpful and there are many exercises, but it seemed to me that, although he pointed out various problems of convergence, in the earlier pages he almost encouraged the reader to ignore the problems of whether, and to what extent, the remainder after a few terms of power series can be neglected. On the other hand there is a discussion of this point in connection with asymptotic series in the last chapter.

...

He assumes a course of intermediate calculus and the rudiments of ordinary differential equations and says that "starting from this level the book is self-consistent". It is not clear to me how much of the Sturm-Liouville theory is supposed to belong to the "rudiments" but definitions of words like "adjoint" and "orthogonal" might have helped many readers who could manage most of the text (with recourse to the references).

**16.4. Review by: D S Carter.**

*SIAM Review*

**7**(3) (1965), 433.

This compact monograph presents a profusion of perturbation methods for the approximate solution of various types of functional equations, especially ordinary differential equations. Part 1, which occupies the first half of the book, discusses Classical perturbation techniques, ranging from the fundamental Lagrange expansion formula, through the Poincaré-Lyapunov Theorem, to a brief discussion of singular perturbation theory. Special techniques for handling periodic solutions of nonlinear differential equations are presented in Part 2. Part 3 deals with questions of asymptotic behaviour and stability, as illustrated by the equation $u" + a^{2}(t)u = 0$. The Liouville transformation of this equation, the WKB approximation, and a short introduction to the theory of asymptotic expansions are emphasised in these final pages. In addition to the methods mentioned above, many useful but less familiar results from the research literature are brought forward. Several methods are related to works of the author and his colleagues in dynamic programming, stability theory, and invariant imbedding.

The book is clearly not intended as a complete or mathematically rigorous treatise on perturbation theory. Rather, it is a kind of illustrated guide through the outer labyrinths of this broad subject. The material is divided into short sections averaging less than two pages each. A typical section begins by describing a particular technique, either in formal outline or in terms of a simple example. This description is followed by a list of illustrative exercises, and the section concludes with several comments and references for further reading. The level of difficulty is not quite uniform, at least between the text and some of the more demanding exercises. To the competent reader, however, this book will appeal as an elementary, thought provoking, and highly readable survey.

**17. Invariant imbedding and time-dependent transport processes (1964), by R E Bellman, H H Kagiwada, R E Kalaba and M C Prestrud.**

**17.1. From the Publisher.**

An examination of time dependence approached through the theory of invariant imbedding with the aid of the technique of the numerical inversion of Laplace transforms. The principal objective is to demonstrate that a simple and direct application of invariant imbedding, combined with a relatively unsophisticated use of the modern digital computer, will yield a number of significant results in radiative transfer. These have immediate application to the study of planetary atmospheres and other important physical problems. The authors present FORTRAN programs and various useful tables and graphs using the numerical inversion procedure.

An examination of time dependence approached through the theory of invariant imbedding with the aid of the technique of the numerical inversion of Laplace transforms. The principal objective is to demonstrate that a simple and direct application of invariant imbedding, combined with a relatively unsophisticated use of the modern digital computer, will yield a number of significant results in radiative transfer. These have immediate application to the study of planetary atmospheres and other important physical problems. The authors present FORTRAN programs and various useful tables and graphs using the numerical inversion procedure.

**17.2. Review by: G M Wing.**

*Mathematical Reviews*MR0162581

**(28 #5779)**.

In the first book of this series the problem of a time-independent plane parallel flux of particles impinging at a specified angle on a finite slab, assuming absorption and isotropic scattering, was investigated [R E Kalaba, R E Bellman, M C Prestrud, Invariant imbedding and radiative transfer in slabs of finite thickness, 1962. In the present volume the steady beam is assumed to impinge at $t = 0$ and time-dependence of the reflected flux is accounted for.

The authors reduce the invariant imbedding equations for the reflected flux to the time-independent case by means of the Laplace transform, then apply a scheme for the numerical inversion of this transform to obtain the desired information. The first part of the book contains a description of the imbedding method and develops the inversion scheme in some detail. Tables of the reflection function for various times, angles, etc., occupy the next hundred or so pages. Results agree for large $t$ with those obtained in the first volume. Details of the FORTRAN program, etc., are also given.

**17.3. Review by: I P Grant.**

*The Mathematical Gazette*

**50**(372) (1966), 233-235.

The authors of this short monograph discuss a numerical approach for solving the equation describing time-dependent reflection of monochromatic radiation (or of neutrons) by finite slabs. The twin obstacles barring the way to a simple numerical difference approximation to the equation are the presence of a time- as well as a space-derivative and of a convolution integral over time. In addition to the difficulty of assessing the accuracy and stability of any approximate method, the evaluation of the convolution integral presents an awkward storage problem. However, by taking Laplace transforms, the equation reduces to a form, closely resembling the equation for the time-independent problem, which can be solved by simple methods such as those discussed in Volume One of the series. The central problem of this book, occupying Chapter 1, is therefore to assess the possibility of numerical inversion of Laplace transformations.

...

Because of its wide range of potential application, the chapter on numerical inversion of Laplace transforms may be regarded as the most important of this book. It is questionable, however, whether the tentative and incomplete nature of the investigation makes the work suitable for publication in book form. The authors implicitly recognise the incompleteness of their work by including a section on alternative numerical techniques, but do not try to compare the methods with one another. Some attempt to define the class of inverse transforms to which the methods can be applied and to estimate error bounds would have been most valuable. For a book claiming to be self-contained it is remarkably reticent on these topics.

The data in the later appendices will clearly be useful to anyone who wishes to find numerically the inverse Laplace transform of a function with suitable characteristics. The justification for printing the extensive Appendix One, which makes up nearly half the book, is much less obvious. We are told in the introduction that the "results have immediate application ... to planetary atmospheres..." but there is no indication of the nature of the problems that the authors have in mind, nor of the way in which their results are to be used. It is this kind of thing that makes an otherwise exciting piece of work a dis- appointment to read.

**17.4. Review by: E H Bareiss.**

*SIAM Review*

**7**(1) (1965), 150-151.

The authors determine the time and angular dependence of the intensity of the diffusely reflected radiation due to incident parallel rays of radiation on a slab, initiated at time zero, and continuing indefinitely.

**17.5. Review by: Stephen Prager.**

*Quarterly of Applied Mathematics*

**24**(3) (1966), 274-275.

This volume extends the results of the authors' earlier book

*Invariant imbedding and radiative transfer in slabs of finite thickness*(1962) on the invariant imbedding technique to certain time-dependent problems, in particular one-dimensional neutron multiplication in a rod and diffuse reflection of incident radiation from a slab. The basic approach is to apply a Laplace transformation with respect to the time variable, which has the effect of replacing time derivatives by absorption terms, so that the transformed equations can formally be thought of as representing a steady state situation.

The small amount of textual material (there are only about fifty pages) is devoted mainly to the numerical inversion of Laplace transforms, with special reference to computer methods. The remainder of the book consists of appendices which tabulate results for the diffuse reflection problem and also various quantities arising in connection with the Gaussian quadrature techniques used to invert Laplace transforms. Fortran programs for a number of calculations are also given.

On the whole, this is a highly specialised book, and will be of use mainly to readers interested in the particular physical situations discussed by the authors.

**18. Quasilinearization and nonlinear boundary-value problems (1965), by Richard E Bellman and Robert E Kalaba.**

**18.1 From the Publisher.**

An introduction to quasilinearisation for both those solely interested in the analysis and those primarily concerned with applications. The Report contains chapters on: (1) the Riccati Equation; (2) two-point boundary-value problems for second-order differential equations; (3) monotone behaviour and differential inequalities; (4) systems of differential equations, storage, and differential approximation; (5) partial differential equations; (6) applications in physics, engineering, and biology; and (7) dynamic programming and quasilinearisation. The authors believe that, in addition to other uses, the theory of quasilinearisation will aid in obtaining a uniform approach to the study of the existence and uniqueness of the solutions of ordinary and partial differential equations subject to initial and boundary-value conditions.

**18.2. Review by: L Collatz.**

*Mathematical Reviews*MR0178571

**(31 #2828)**.

Since initial value problems can now be solved numerically on computer systems even for very large systems of ordinary nonlinear differential equations (with hundreds of unknowns), new aspects also arise for boundary value problems and various types of functional equations. The aim of the book is to show, prepare and stimulate further research for direct treatment in a reasonable time for tasks of a very complex type on computing systems.

**18.3. Review by: L E Payne.**

*The American Mathematical Monthly*

**74**(9) (1967), 1157.

This book is a well written and well motivated introduction to quasilinearisation and its applications to nonlinear boundary-value problems. A number of analytical concepts are introduced and illustrated through the use of simple nontrivial examples. The book begins with a discussion of the application of quasilinearisation to the study of the Riccati equation. Here quasi linearisation results in an explicit analytic representation for the solution in terms of quadratures and a maximum operation. This representation yields simple upper and lower bounds for the solution. In nonlinear two point boundary value problems the authors then show how quasilinearisation may lead to a sequence of approximations with quadratic convergence. Chapter 3 examines conditions under which the sequence of approximations converges monotonically. Here the Green's function is encountered and its variation diminishing property emphasised. Chapter 5 deals with higher order equations and systems and develops techniques for using successive approximations without excessive reliance on rapid-access storage. Various methods for overcoming storage problems are discussed. Some nonlinear partial differential equations are dealt with in Chapter 5, and a number of examples are given in Chapter 6 which illustrate ways in which quasilinearisation may be applied to treat various descriptive and variational processes. The final chapter illustrates how quasilinearisation and dynamic programming may be effectively combined in the treatment of multi-dimensional variational problems, and identification problems. Computer programs for a number of problems are contained in the various appendices.

Each chapter begins with an introductory paragraph which motivates and describes the content of the chapter. The exposition is clear throughout, the references are numerous, and the misprints are few.

**18.4. Review by: L B Rall.**

*SIAM Review*

**8**(3) (1966), 401-402.

This book deals with a number of interesting and important nonlinear problems, mostly boundary-value problems for ordinary differential equations. The methods proposed for treating these problems on a computational basis are collectively described as "quasilinearisation," but actually fall into at least three distinct classes: (1) the transformation of nonlinear equations into initial-value problems for ordinary differential equations, which is the method proposed for nonlinear functional equations by D F Davidenko; (2) replacement of a nonlinear equation by a linear equation with a solution which is a majorant of the solution of the original equation, a technique associated with the name of S A Caplygin; and (3) Newton's method in the generality obtained by L V Kantorovic. ...

The above methods have usually one or more of the properties of computational simplicity, monotonicity, and quadratic convergence. A number of examples are given in which these properties are exploited, but, except for some very simple cases, nothing is proved which would give an investigator an indication as to when to expect these desirable features to appear, except on a "try it to see" basis.

The writing and editing of the book shows certain effects of the division of chores: equation numbers cited in the text do not correspond at times with the actual ones, and some section numbers in the references are incorrect or omitted. While $u'$ is used for $\large\frac{du}{dt}\normalsize$ in most of the text, $\dot {u}$ makes an abrupt appearance on p. 99. There are other minor gaps, omissions, and misstatements. Throughout, the elementary aspects of the problems considered are explained clearly and in detail, and motivate the methodology developed extremely well. The discussion of more advanced aspects is sketchy, somewhat jargonised, and some sections serve only to point out the existence of reference material, so that the utility of the book is reduced greatly if one does not have access to the literature cited, most of it due to the authors and their colleagues.

The book concludes with about forty pages of printout of computer programs used for solving example problems in the text. These pages could have been better used to convey additional theoretical information, or omitted to reduce the price of the book, at least presumably.

Here is the list of chapters: 1. The Riccati Equation, 2. Two-Point Boundary- Value Problems for Second-Order Differential Equations, 3. Monotone Behavior and Differential Inequalities, 4. Systems of Differential Equations, Storage and Differential Approximations, 5. Partial Differential Equations, 6. Applications in Physics, Engineering, and Biology, 7.

**19. Inequalities (2nd edition) (1965), by Edwin Beckenbach and Richard Bellman.**

**19.1. Review by: Editors.**

*Mathematical Reviews*MR0192009

**(33 #236)**.

A reprinting of the 1961 book with correction of some minor errors.

**20. Dynamic programming and modern control theory (1965), by Richard E Bellman and Robert E Kalaba.**

**20.1. Review by: S E Dreyfus.**

*Mathematical Reviews*MR0204191

**(34 #4037)**.

This little paperback book is an introduction to the dynamic programming treatment of sequential decision processes. It includes philosophical discussions and elementary results; it is a self-contained and very readable introduction to the subject, with many problems and references to the literature of dynamic programming.

The book is primarily concerned with presenting the concepts and procedures of dynamic programming and not necessarily with the efficient solution of the particular problems chosen for illustrative purposes. Hence, little or no note is taken of non-dynamic programming treatments of such problems as shortest path determination or of gradient or other procedures for solving deterministic decision problems.

Chapter titles, which are self-explanatory and, if understood to refer exclusively to dynamic programming, indicate accurately the contents of the book, are as follows: (I) Multistage processes, (II) Multistage decision processes, (III) Computational aspects, (IV) Analytical results in control and communication theory, and (V) Adaptive control processes.

**20.2. Review by: David G Luenberger.**

*American Scientist*

**54**(4) (1966), 482A.

This little book contains a clear and pleasantly readable introduction to the theory of multistage processes. The title is somewhat misleading, however, since modern control theory is the subject of less than 20 per cent of the book, and its treatment is light and specialised.

The titles of the first three chapters are: "Multistage Processes," "Multistage Decision Processes," and "Computation Aspects." These chapters, which comprise 72 of the 105 pages of text, constitute an excellent brief introduction to multistage processes and dynamic programming. Although not intended as a comprehensive treatment, these sections effectively pinpoint the essential, elements of the theory, the major practical limitations together with some techniques for alleviating them, and a scant yet fairly convincing set of examples which demonstrate the power and generality of the approach.

The fourth and fifth chapters are: "Analytic Results in Control and Communications/' and "Adaptive Control Processes." These final chapters serve further to illustrate the utility and breadth of application of dynamic programming, both as a computational technique and as a tool for analysis. Examples are given of the role of dynamic programming as a new approach to classical problems and as a possible approach to research problems. These chapters cannot, however, be regarded as a fair introduction to control or communication theory.

The book contains an ample supply of well-chosen exercises in each chapter, and, in general, the book is to be recommended as an enjoyable short treatment of dynamic programming suitable as an introduction for almost anyone having a scientific or technical background.

**21. Numerical inversion of the Laplace transform: Applications to biology, economics, engineering and physics (1966), by Richard E Bellman, Robert E Kalaba and Jo Ann Lockett.**

**21.1. Review by: P L Butzer.**

*Mathematical Reviews*MR0205454

**(34 #5282)**.

The authors' aim is to solve ordinary and partial differential equations and more bizarre types of functional equations, such as differential-difference equations and integral equations of the Volterra type, occurring in mathematical physics, biology and economics, by means of the Laplace transform and then apply a scheme for the numerical inversion of this transform to obtain the solution.

...

The first chapter is devoted to a presentation of the elementary properties of the Laplace transform, the second to the numerical inversion technique, the third to the computational solution of sample linear functional equations such as the scalar renewal equation and equations of chemotherapy. The last two chapters contain a discussion of non-linear equations and dynamic programming to obtain accurate solutions of ill-conditioned systems of linear algebraic equations. FORTRAN programs and various tables necessary for the numerical inversion are given.

**21.2. Review by: W L Miranker.**

*Science, New Series*

**157**(3793) (1967), 1163-1164.

The authors' stated purpose in writing this book is to advance the point of view that many problems which take the form of equations can be solved numerically by means of the intermediary use of the Laplace transform. The authors point out that the transform "reduces the order of transcendence" of many types of equations, notably linear equations and equations involving convolutions. They recommend solving these simpler problems numerically and then numerically inverting the transform to obtain a numerical answer to the original problem. They discuss a variety of approximate linearising methods of an iterative sort, such as Picard iterations and the method of quasi-linearisation, as means of extending their solution methods to certain nonlinear equations.

As for the numerical inversion of the Laplace transform, the authors suggest a procedure which amounts to replacing the relation defining the transform with a system of linear equations which are obtained by evaluating the integral by a method of numerical quadrature of Gaussian type. The authors also include a section on dynamic programming. Numerous examples are included in the text.

The authors suggest that this book is well suited to the purposes of certain nonmathematicians, such as biologists, who want to get answers to equations arising in their work. These methods, they claim, will enable such investigators to relegate the onerous task of computation to assistants. In my opinion the authors have succeeded in their aims, which were to convey a superficial understanding. I strongly recommend this book to those whose needs will be satisfied by superficiality. To those who want something else, the book is a poor investment of time.

**21.3. Review by: Y L L.**

*Mathematics of Computation*

**22**(101) (1968), 215-218.

In numerous applied problems, characterised by ordinary differential equations, difference-differential equations, partial differential equations or other functional equations, the Laplace transform is often a powerful tool for obtaining a solution. When the Laplace transform approach is applicable, getting the Laplace transform of the solution is relatively easy. The major problem is inverting the transform. It is often the case that closed-form representations in terms of tabulated functions for the inverse are not known, and so one must resort to numerical methods.

A comprehensive volume on the numerical inversion of Laplace transforms replete with examples would fill an important gap in the literature. Although the volume under review has much which is commendable, it is not comprehensive in its coverage, as the authors seem completely unaware of important segments of the literature. We return to this point later, but first we explore the contents of the volume and present a generalisation of the basic tool used by the authors. ...

**22. Introduction to the mathematical theory of control processes. Vol. 1: Linear equations and quadratic criteria (1967), by Richard E Bellman.**

**22.1. From the Publisher.**

This work discusses the theory of control processes. The extremely rapid growth of the theory, associated intimately with the continuing trend toward automation, makes it imperative that the courses of this nature rest upon a broad basis. The work discusses the fundamentals of the calculus of variations, dynamic programming, discrete control processes, use of the digital computer, and functional analysis. Introductory courses in control theory are essential for training the modern graduate student in pure and applied mathematics, engineering, mathematical physics, economics, biology, operations research, and related fields. The work also describes the dual approaches of the calculus of variations and dynamic programming in the scalar case and illustrates ways to tackle the multidimensional optimisation problems.

**22.2. Review by: David W Miller.**

*Management Science*

**15**(8) (1969), B450-B451.

This remarkably fine book is intended to be the first of three volumes on the mathematical theory of control processes. This first volume deals with deterministic control processes which can be formulated in terms of quadratic functional and linear differential equations. The second volume will treat deterministic problems with more general functionals and nonlinear differential equations. The third volume will deal with stochastic control processes.

The present book is intended for students with good calculus, differential equations, and some matrix theory. It is difficult to imagine a more felicitous presentation of this material. A short introductory chapter describes control theory. The second chapter gives the necessary material on second-order linear differential and difference equations. The third chapter uses a simple pendulum model to provide a crystal-clear statement of the ideas of stability and control. Chapters 4 and 5 deal with the scalar case. Chapter 4 uses calculus of variations to treat continuous variational processes and Chapter 5 uses dynamic programming for the same problem and also for discrete control processes. Chapter 6 reviews the applications of matrix theory to multidimensional linear differential equations. Chapters 7 and 8 then generalise, respectively, Chapters 4 and 5 to multidimensional optimisation problems, with some discussion of the numerical problems in these cases. Chapter 9 uses functional analysis in a re- consideration of the problems of the earlier chapters. This last chapter assumes some minimal knowledge of Hilbert space theory. There are numerous problems, including many fairly difficult ones.

The exposition is of almost unequalled clarity. I can recall only a handful of mathematical books which have achieved such transparency in their presentation. How does Professor Bellman accomplish this? I do not know but I notice at least three contributing factors. First, in the author's words: "We try as carefully as we can to avoid a confusion of purely analytic difficulties due to the nature of the equation with conceptual complexities inherent in a theory of control." Second, the organisation of the material is in terms of the logic of the problem, not the logic of the mathematical apparatus. Third, the style of writing is very pleasant and creates the illusion that the reader is in complicity with the author in surmounting the difficulties en route. These factors support, but do not suffice to explain, the excellence of this book. Highly recommended to anyone who wants to know something about control theory.

**22.3. Review by: G M Ewing.**

*Mathematical Reviews*MR0224382

**(36 #7426)**.

This book is directed at readers who have had an undergraduate training in calculus and differential equations and a course in matrix theory. It includes a descriptive introduction to systems, stability and control; discussion of a succession of particular Lagrange problems with quadratic integrands and linear side conditions; the usual heuristic derivation of the partial differential equation of dynamic programming and its application to examples; a brief treatment of $L^{2}$ with application to quadratic functionals. There are occasional sections on computational aspects. Extensive lists of exercises and bibliographies are at the ends of chapters. A second volume on more general deterministic problems and a third on problems with stochastic ingredients are promised.

**22.4. Review by: E B Lee and K S P Kumar.**

*The American Mathematical Monthly*

**81**(1) (1974), 100-103.

Because of a large and growing human population it is becoming ever more necessary to utilise efficiently the limited resources of our planet. Control theory provides a scientific approach to the decision making involved in the allocation of resources. In the two books authored by Bellman, control theory is defined as the care and feeding of systems - control theory being the technique whereby the best feeding plan is established. The four books under review are each devoted to the topic of control theory.

The control theory approach is to assume that the controlled system (be it an airplane, a human, or an economy) is fixed, usually unalterable, and equivalent from an input-output point of view to its mathematical model. The inputs can be thought of as the influence variables (the quantities which are subject to control) and the output can be thought of as the sensed variables. Control theory provides the connection between the outputs and the inputs so that the controlled system behaves in a satisfactory (or perhaps even in a most efficient) manner. This connection between output and input is the concept of feedback. Models commonly used for systems are difference equations, differential equations (or the equivalent transfer functions), functional differential equations, partial differential equations or combinations of these.

The mathematical model for a system is obtained in a number of ways. In engineering it is common to build the model based on the known structure of the system; for example, it may be composed of electrical devices such as resistors and capacitors, which have a well established relationship between their inputs and outputs, and rules (Kirchhoff's current and voltage laws) for interconnecting which determine the model of the composed system. Mechanical and chemical systems can be modelled in a similar fashion. Another approach to modelling is to postulate a relationship between the input and output variables with unknown parameters which are determined by a comparison with experimental data from the controlled system.

To evaluate the performance of the controlled system a comparison is usually made between the performance of the actual controlled system and some ideal system in certain critical situations. In control engineering this comparison may be in terms of such quantities as the rise time of the output for a step type input, the peak overshoot, phase margin, gain margin, etc. The modern control approach makes this comparison by considering as the performance index the mean square error between the actual system response and the ideal system response (or some other mathematical expression of efficiency). Frequently the goals of the system are not clear and one chooses an index of performance for mathematical simplicity to carry out the controller synthesis and afterwards compares the controlled system with other norms.

Another part of the problem to consider is the constraints and restraints that the system may be subject to. For example, the controlled inputs may be restricted as to size or shape or to the total number available. Such constraints are frequently stated in mathematical terms by putting a limit on the absolute value of a control variable or by a limitation on the integral of the control variable squared.

The mathematical control problem is then to select the best relationship (in the sense of the performance criteria) between the outputs and the inputs subject to the mathematical model and constraints. This mathematical control problem is the central topic of the four books under review.

In the two books authored by Bellman, the mathematical control problem for systems modelled by ordinary differential equations is considered from two points of view: a classical variational one and the dynamic programming (feedback) approach. The control of processes with difference equations and partial differential equation models are briefly discussed.

Volume I begins with motivational material in the form of nine very short sections concerning mathematical modelling, uncertainty, questions of observing systems, the concept of feedback, and the like. Chapter 2 is devoted to second order linear differential and difference equations, while in Chapter 3 the concepts of stability and control are discussed and the mathematical control problem (linear model and quadratic performance index) is formulated. The remaining 6 chapters elucidate techniques for solving mathematical control problems when various constraints and generalisations are admitted. Such questions as the existence of an optimising control function, its uniqueness, and necessary and sufficient conditions that the optimising control function must satisfy are considered. Chapter 9 outlines briefly how the theory of functional analysis can be applied to the quadratic optimal control problem.

**23. Modern elementary differential equations (1968), by Richard E Bellman and Kenneth Cooke.**

**23.1. From the Preface.**

We feel that the following are minimum objectives for the basic training of a student: 1. An understanding of how to use DEs in order to describe a variety of physical processes. By this we mean that the reader should understand how to convert plausible scientific assumptions into various kinds of DEs. A part of this understanding is an appreciation of when DEs alone do not suffice. 2. An ability to deal easily with linear DEs with constant coefficients, and an awareness of how many important physical phenomena, such as resonance, criticality, and so on, can be understood both qualitatively and quantitatively in terms of the behaviour of the solutions of DEs as functions of both time and system parameters. 3. An ability to use power-series expansions to obtain analytic and computational behaviour of the solutions of linear and nonlinear DEs. In addition, the reader should learn to appreciate the use of perturbation expansions in terms of various parameters in the equation. 4. An understanding of the basic methods used to obtain numerical solutions of DEs using both desk and digital computers. We assume that he has learned the rudiments of FORTRAN programming prior to taking this course, or simultaneously. However, this ability is required only in one part of the text, Chapter 4. The remainder of the book can be read by the computerless student. 5. An appreciation for the use of successive approximations as an all-purpose tool for obtaining analytic and numerical solutions of DEs, and as a technique for establishing existence theorems.

**23.2. Review by: Judith M Elkins.**

*The American Mathematical Monthly*

**76**(1) (1969), 102-103.

A glance at this book will tempt many to adopt it as a text at the sophomore level for engineers and scientists. It departs from the traditional approach of a compilation of special techniques and introduces the reader to modern applications and those techniques most useful for scientists: linear differential equations, power series solutions, perturbation techniques, numerical solutions, and successive approximations.

This book was used by the reviewer in a one-semester course for sophomore engineers and scientists who had completed three semesters of calculus. The students were unable to grasp the point of many sections. Many important ideas were hidden in the problems; for example, the solution of the general first order linear equation and the method of finding a particular solution by variation of parameters. Neither the index nor the section headings would allow this book to be used for reference. The problems were quite interesting, but too involved for the students. It was necessary to supply extra problems at every stage. The solution of partial differential equations by separation of variables was omitted entirely.

There are many unusual and good features to this book. Applications from diverse fields serve as good motivation for the reader. There is a short but valuable section on the direct use of the differential equation to obtain information about the solution. The behaviour of the solutions of the second order linear equation is examined to gain an understanding of the related physical phenomena for electrical circuits. The chapter on power series is quite well done on the whole. Also, the inclusion of the chapter on numerical solutions has long been needed.

This book could be used as supplementary material or to introduce those with limited knowledge to modern techniques, but I do not recommend its use as an introductory text.

**23.3. Review by: C V R.**

*Current Science*

**37**(15) (1968), 448.

The book is intended to provide students, as well as practising engineers and scientists, with an understanding of the basic methods used to obtain numerical solution of differential equations using both desk and digital computers. Prior or concurrent study of the rudiments of FORTRAN programming is required for an understanding of the chapter discussing these methods. The balance of the book, however, assumes only an elementary course in calculus, containing the rudiments of the theory of power series.

The subject-matter in this volume is dealt with in six chapters, viz., The Origins of Differential Equations, Second-Order Linear Differential Equations with Constant Coefficients. Power-Series Solutions, The Numerical Solution of Differential Equations, Linear Systems and Nth-Order Differential Equations and Existence and Uniqueness Theorems.

**24. Some vistas of modern mathematics. Dynamic programming, invariant imbedding, and the mathematical biosciences (1968), by Richard E Bellman.**

**24.1. From the Publisher.**

Rapid advances in the physical and biological sciences and in related technologies have brought about equally far reaching changes in mathematical research. Focusing on control theory, invariant imbedding, dynamic programming, and quasilinearisation, Mr. Bellman explores with ease and clarity the mathematical research problems arising from scientific questions in engineering, physics, biology, and medicine. Special attention is paid in these essays to the use of the digital computer in obtaining the numerical solution of numerical problems, its influence in the formulation of new and old scientific problems in new terms, and to some of the effects of the computer revolution on educational and social systems. The new opportunities for mathematical research presage, Bellman concludes, a renaissance of mathematics in human affairs by involving it closely in the problems of society.

**24.2. From the Preface.**

This age is one of the great ages of mankind.

As such, it is a time of ferment. All over the earth, we witness the birth pangs of new societies and new cultures. In our own country, it is a period of emancipation. Some are escaping from the tyranny of centuries-old prejudice, others from the blight of inherited poverty. All of us are gradually emancipating ourselves from cheerless puritanism and are looking forward to freedom from the dread diseases and disabilities that have plagued humanity from the beginning.

Science by itself plays no major part in shaping the philosophical attitudes that make human beings desire a better world for themselves and others. Once, however, these attitudes have been formed, science plays a fundamental role in making the wishes come true. To grow the food to feed the hungry, to introduce effective birth-control measures, to build cities and to make them habitable, to staff and supply the hospitals, to perform a thousand other tasks in a complex society, we urgently need the scientist's help. Inasmuch as mathematics is the language of science, we can readily conclude that mathematics itself must occupy a major position in the world of the future.

It follows that we must pay careful attention to the kind of mathematical training given in the colleges and universities. This is the reasonable view of a society that needs mathematics for its feasible operation. It is an important view because so much of the financial support of the schools comes from the state and federal government. It is hard to see how we can avoid the conclusion that the taxpayer has a right to know how his tax dollar is spent. In all conscience, we cannot cry "academic freedom" or "research" or some other convenient slogan and claim immunity from a critical gaze.

In addition to being a tool of science, mathematics is also an art form. It possesses its own structures and its own aesthetic goals. How well are we turning out young artists? We know from a study of the contemporary scene, or from the histories of the other art forms, that the fields of painting, sculpture, literature, and the drama all react strongly to periods of turmoil in culture and technology. Professional mathematicians, dedicated both to research and to passing on the torch, are very much concerned with the sluggish reaction of the universities to the new developments within mathematics itself, to the new interactions between mathematics and science, and to the significance of the computer. No art form can remain static without becoming sterile. Mathematics is no exception to this universal rule.

Between the formless entity called "society" and the ethereal concept of artistic conscience is something quite real and tangible, the individual. Just as we are not dedicated solely to creating an efficient state, so we are not merely high priests at the sacred shrine of mathematics. We are very much concerned with the happiness of the individual, with his growth, fruition, and emotional satisfaction. We know that his university training, particularly his graduate training, will be a determining factor in his life.

If we take this responsibility seriously, as the overwhelming majority of university professors do, it is essential to discuss openly the many different choices that exist as far as both training and research are concerned. Furthermore, we must attempt to point out the possibilities for success and happiness in these directions. It is imperative to let the student know which fields are promising and which ones have been well worked out for twenty-five years. He may wish to make a deliberate choice to work on the Riemann hypothesis or the four-colour problem. But he should be aware of the probabilities of success in these endeavours as compared to a number of other activities.

Much of current university education is woefully wrong because it is narrow and provincial, providing no broad view of the exciting activities of the outside world. It calls to mind the following paragraph from Gulliver's Travels:

There was a Man born blind, who had several Apprentices in his own Condition; Their Employment was to mix Colours for Painters, which their Master taught them to distinguish by feeling and smelling. It was indeed my Misfortune to find them at that Time not very perfect in their Lessons; and the Professor himself happened to be generally mistaken: This Artist is much encouraged and esteemed by the whole Fraternity.Discussions of matters like this are frustratingly difficult because we cannot pursue them in the same manner in which we present a mathematical proof. We cannot expect to establish any principles in a rigorous fashion. Nor can we pursue the discussion purely rationally. Much is subjective; much is a matter of judgment. But this perhaps is the major point. We must bring this fact clearly to the student's attention so he is forced to exercise his own judgment and common sense rather than accept unquestioningly the traditional material. If we can only get him to ask, "Why?" instead of just "What?" we will have made an essential step in his education. We are not interested so much in determining his precise path as in forcing him to make his own decision from a wide range of worthwhile choices. This procedure is difficult. We must enthusiastically promote certain ideas and theories because we believe they merit study and research, yet we must constantly point out the existence of still other points of view and other endeavours. The task is not easy for a human being. As Mark Twain might have said, it wouldn't be a cinch for an angel.

The purpose of this book, originally delivered as lectures at the University of Kentucky during Thanksgiving week of 1966, is to describe some of the ways that the problems of the modem world provide interesting mathematical questions and open up entirely new domains of mathematics. Rather than concentrating solely on "What," I have indeed tried to answer "Why," even at the risk of some verbosity here and there. In particular, I have constantly emphasised the fact that there are many alternate formulations of physical processes in mathematical terms.

I hope that this book will be of interest to a number of different groups of readers. To the student, I hope it will provide some discussion of why certain mathematical theories are created and where and how certain types of problems arise. To the practicing mathematician in another field and to the scientist, I hope it will satisfy his curiosity as to what is going on in other parts of the forest. To the chairman of a department, or to a dean, I hope it will be useful in planning the new curriculum for the department of mathematics or in explaining the many different kinds of interaction that can exist between a computer installation and the Department of Mathematics. I have tried to preserve some of the informal flavour of a lecture in the presentation.

**24.3. Review by: David W Miller.**

*Management Science*

**15**(10) (1969), B575-B576.

Here is another superb book by Professor Bellman. The three chapters in this book were originally given as lectures at the University of Kentucky in late 1966. In the preface, the author tells us that "we must pay careful attention to the kind of mathematical training given in the colleges and universities." In this context he states that professional mathematicians "are very much concerned with the sluggish reaction of the universities to the new developments within mathematics itself, to the new interaction between mathematics and science, and to the significance of the computer." And he tells us that the purpose of this book "is to describe some of the ways that the problems of the modern world provide interesting mathematical questions and open up entirely new domains of mathematics."

In summary, then, Professor Bellman has directed this book to students, practicing mathematicians, scientists, chairmen of departments, deans-anyone who needs to know "where it is really at" in mathematics. He has little patience with some of the main trends in contemporary pure mathematics. For example, "the Bourbaki caused an unfortunate regression into medievalism." Generalising with regard to medievalism in modern mathematics, he adds: "An interesting relic of medievalism is this idea that only the difficult can be distinguished, that only the arcane can be erudite, that what is readily understood cannot be profound." And again: "This goal is a blow at medievalism, in which knowledge is power and the object is to impress, not to inform. Recall that Newton informed Leibniz of his newfound ability to integrate differential equations using power series by means of an anagram. Remarkably, this practice has continued to the present in many scientific journals and books." The author's sentiments are as apt as his expression of them is felicitous.

As alternatives the author offers three of the relatively recent developments in mathematics. The first one is "Dynamic Programming and Modern Control Theory." Starting with the use of differential equations to describe the behaviour of systems, Professor Bellman discusses approximation and stability, control theory and the calculus of variations, feedback control, stochastic pursuit processes, and dynamic programming. The second topic is "Invariant Imbedding and Mathematical Physics". The author exemplifies the imbedding approach with the problem of determining the maximum height reached by a stone thrown straight up. Here is the author's summary of the classical imbedding technique: "The first step in the solution process consists of an imbedding within a family of problems. The second step consists of finding relations between various members of this family. In this fashion, we hope to pass by simple steps from the known to the unknown." This procedure is illustrated by a one-dimensional neutron transport process, the construction of a shield for a nuclear reactor, and the propagation of light through the atmosphere. The latter problem leads to invariant imbedding and the discussion of a variety of resulting problems. The third topic is "The Challenge of the Mathematical Biosciences." This is a more elementary survey than the two preceding chapters: the uses of computers for data storage and retrieval, medical diagnosis, pattern recognition, teaching of doctors and similar topics.

The exposition is very fine throughout. At one point Professor Bellman contrasts Poincare's style with "the constipated, crabbed style of the Bourbaki." Certainly, part of the reason for Poincare's extraordinary clarity was his immersion in the great French mathematical tradition, which includes Lagrange, Cauchy, Hadamard, and many others. It seems to me that these men generally started their efforts with a real-world problem and ended them with a major attempt to communicate their conclusions. Professor Bellman is surely in the same tradition. Every serious quantitative analyst will want to read this book.

**24.4. Review by: H Taylor.**

*Mathematical Reviews*MR0232610

**(38 #935)**.

The stated purpose of this book, originally delivered as lectures at the University of Kentucky during Thanks-giving week of 1966, is "to describe some of the ways that the problems of the modern world provide interesting mathematical questions and open up entirely new domains of mathematics". The belief that "the growth of vital mathematics depends crucially on continuing interaction with the real world" is emphasised repeatedly.

The first two chapters survey some of the author's experiences in the fields of dynamic programming, control theory, invariant imbedding and quasilinearisation. The aim is to show by example how the quest for numerical answers to numerical questions leads to new and interesting mathematical ideas and problems. A number of ways in which the digital computer is influencing mathematics are presented. The last chapter surveys several new areas of research that have opened up in the mathematical biosciences.

The presentation is informal. The book contains no new results.

**24.5. Review by: R Kalaba.**

*The American Mathematical Monthly*

**76**(10) (1969), 1159.

This is a remarkable book, written in the tradition of Poincaré

*Science and Method*, Courant and Robbins

*What is Mathematics*, and Khinchin

*The Teaching of Mathematics*(Telegraphic Review June 1969). Dr Bellman lays before us three personal and penetrating essays devoted to modern control theory, invariant imbedding, and mathematical biosciences. Each of these areas has, of course, undergone pronounced developments at his hands.

The aim throughout is to lead the reader to the frontiers of current mathematical research. This is done by stressing ideas and the role of the modern computing machine, rather than trivial details. The first two essays emphasise the transformation of two-point boundary-value problems into initial-value problems. Semi-group notions come to the fore.

The last essay deals with cancer chemotherapy, pattern recognition, and electrocardiography. In a book brimming with quotable sentences, Dr Bellman offers this prescription for curing discontent in mathematics departments: "This involvement has been seriously lacking over the last twenty years and has been responsible for much discontent on college campuses. The rise of the mathematical biosciences will change much of this situation by providing the intellectual with the proper combination of challenge and service."

**25. Stability theory in differential equations (reprint) (1969), by Richard E Bellman.**

**25.1. Review by: Editors.**

*Mathematical Reviews*MR0247201

**(40 #470)**.

This is an unabridged and unaltered republication of the 1953 first edition.

**26. Methods of nonlinear analysis. Vol. 1 (1970), by Richard E Bellman.**

**26.1. From the Publisher.**

In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation; methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.

As a result, the book represents a blend of new methods in general computational analysis, and specific, but also generic, techniques for study of systems theory ant its particular branches, such as optimal filtering and information compression.

- Best operator approximation,

- Non-Lagrange interpolation,

- Generic Karhunen-Loeve transform

- Generalised low-rank matrix approximation

Last Updated August 2024