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Introduction to mathematical biology
Book
ference text for research workers in the field of nonlinear oscillations, periodic solutions and stability theory. This first book was not translated into English and, therefore, its influence did not extend to mathematics and science departments in the English-speaking countries. As a result of the wide scope of the second book covering the field of wave equations and the approximation of solutions by asymptotic methods its use will be somewhat different. Prerequisites for the study of this book are a basic knowledge of differential equations and complex function theory as usually presented in undergraduate courses. The first two chapters are devoted to the classical subjects of Fourier, Laplace and Mellin transforms and special functions (gamma functions and Bessel functions); Chap. 3 presents the basic results for the linear wave equation. The introduction to basic material is concluded in Chap. 4. in which, (for approximately 40 pages) asymptotic methods are discussed. All this material is used and applied in 200 pages where the following subjects are studied: scattering matrix theory; flow in open channel; asymptotic solutions of linear and nonlinear wave equations; seismic waves and waterwave theory. Nearly all the results in this book concern linear theory; the treatment both of the basic material in the first four chapters and in the applications is unusually detailed and mathematically sound. The concept of wave equation itself remains rather vague. It would have been helpful to the reader if a survey and classification of equations leading to wave phenomena had been presented. However, from the point of view of linear theory, Asymptotic Wave Theory will be very useful as a textbook for graduate students in applied mathematics and as a reference text for research workers in the field of wave equations. Only in a few instances, notably in Chap. 6 does the author touch upon nonlinear wave theory. No justice can be done to this rapidly expanding field in a few pages!
Journal of The Franklin
htitute
Reviews
and the reader is referred to the (unfortunately somewhat outdated) literature. Another minor criticism: there is little use for an index containing fewer key words than the table of contents; but this type of index is in an old French tradition. F. VERHULST Department of Mathematics, Rijksuniversiteit, Utrecht, The Netherlands
INTRODUCTIONTO MATHEMATKAL.BIOLOGY by S. I. Rubinow, 386pp., diagrams, illus., 6fx 9&, John Wiley, New York, 1975, Price $22.00. Catering as it does to two distinctly different readerships, this book on mathematical biology handles the task remarkably well. The author, in his preface, gives recognition to the fact that explanatory comments addressed to one group of students (of either biological or mathematical sciences) tend to ‘turn off the other group, but apologetically refuses to succumb to the obvious temptation of skipping such details. The book is divided into five chapters, this division being based more on biological and biophysical grounds rather than on underlying mathematical structures. However, it is claimed that the ordering of the five topics: cell growth, enzyme kinetics, physiological tracers, biological fluid dynamics and diffusion processes has been largely influenced by the desirability to present these topics in the order of increasing degree of mathematical sophistication. Perhaps, this has been achieved, although it may not be too obvious since there is very little mathematical sophistication per se in the treatment of any of the topics. The problem of cell growth, an immensely popular field of investigation in microbiology, is the subject of the fascinating analytical study presented in
141
Book Reviews Chap. 1. While those with mathematical backgrounds are exposed to the jargon of microbiologists, biology majors are introduced to the mathematical concepts of differential equations needed to describe exponential growth, growth rates and other such physically understood aspects of cell growth phenomenon. Nutrient uptake by cells, growth of a microbial colony, interacting predator-prey systems (a long studied problem of interest to investigators from a variety of inter-disciplinary fields) are some of the problems tackled here to illustrate the application of mathematical techniques to the field of microbiology. Enzyme Kinetics, the topic of Chap. 2 is treated at greater length. The reader is first introduced to enzymes terminology followed by a detailed presentation of Michaelis-Menten theory and its applications to enzymatically controlled reaction problems. Graph theory and its relevance in the context of steady state ‘enzyme kinetics equations and their solutions, is another important topic covered. Chapter 3, on tracers in physiological systems, starts off with an introduction to the concept of compartment systems, and progresses step-by-step from an analysis of a one-compartment system through two and three-compartment catenary systems to n-compartment systems, illustrating each with examples. Biological fluid dynamics is dealt with in detail in Chap. 4. As is to be expected, the presentation begins with an exposure to the inevitable Navier-Stokes equations and the attendant mathematical sophistications in terms of partial differential equations and their solutions. Properties of blood followed by an analysis of blood flow through arteries and veins are covered in succeeding sections. The swimming of microorganisms, such as spermatozoa, parasitic worms and other such self -propelling organisms, is considered. The final chapter entitled “Diffusion in Biology,” begins with a discussion of Fick’s Laws of Diffusion. The phenome-
142
non of Olfactory Communication in animals is cited as an example of a diffusion process in the realm of biological studies. Next, membrane transport as a diffusion process (through a slab) is studied in depth. The Gaussian function encountered in the study of a convective transport problem: ionic flow in an axon, is explained by introducing the reader to some basic probability theory concepts like random variable, probability density function moments. and Ultracentrifugation, sedimentation velocity method, solution to the Lamm equation, sedimentation equilibrium method and transcapillary exchange are some of the other topics covered. A commendable feature of the book is the set of interesting problems offered at the end of each chapter. While offering students of biology an opportunity to test their understanding of the mathematical techniques presented, these problems expose the mathematically oriented reader to additional problem areas in the field of biology and related sciences where their tools can be meaningfully employed. There is a section on solutions to the problems. This facilitates readers, who are trying to wade through the book on their own. Of course, availability of solutions to the problems within the book itself makes it rather inconvenient, (in terms of setting up home assignments), to a professor intending to use this book for a regular credit course. But the advantage to the uninitiated reader far outweighs this inconvenience. More than 175 references listed at the end significantly add to the value of the book. Appendices A and B help the mathematically undeveloped by reviewing some elementary notions such as variables, functions, integration, differentiation, exponential and logarithm function, Taylor series, determinants, vectors and matrices. Introduction to In conclusion, Mathematical Biology, having set for itself the difficult task of catering to the needs
Journal of The
Franklin
Institute
Book of those with vastly differing backgrounds, carries it out quite satisfactorily. The only significant omission is the vast body of literature commonly referred to as biostatistics. This is pointed out in the preface by the author himself, but he justifies it by restricting his definition of the term mathematical biology accordingly. However, in this reviewer’s opinion, it would have been worth the effort if a brief chapter or two had been included to give the reader an awareness of this area of mathematical biology also. B. V. DASARATHV M & S Computing Inc. Huntsville, Alabama U.S.A.
INTRODU
Vol. 304, No. Z/3, August/September 1977
Reviews
The book would be very useful as a reference for additional reading rather than as a required text for a course which has bond graphs as the vehicle for system modelling. A. ANDRV Department of Mechanical Engineering, Michigan State University, East Lansing, Michigan U.S.A.
MICROPROCESSORS AND MICROCOMPUTERS by B. Soucck, 607 pp., diagrams, 6; x 9f in., John Wiley-Interscience, New York, 1976, Price $23.00. Some people like to pick books by their weight. If so, they may be naturally drawn to this entry in the new crop of microcomputer books. However, they may be disappointed. This book gets much of its volume from light rewrites of the author’s previous book on minicomputers (Minicomputers in Data Processing and Simulation, Wiley-Interscience, 1972). It gets the rest from lightly edited material supplied by microcomputer manufactures. The book is intended as a text for students and as a reference for practicing engineers and scientists. It assumes a minimal background in such topics as programming, computer organization, and electronics. It includes many problems and hence could be used as a text assuming one would want to after reading this book and comparing it with similar ones. The book is divided into three parts. Part I; Microprocessor Programming and Interfacing Techniques consists of five chapters of traditional computer organization and programming, retitled but hardly rewritten. Chapter 1, Number Systems and Digital Codes, presents the obligatory material on decimal and binary number systems, binary arithmetic, and octal, BCD, and Gray codes. On page 17, the author presents the formats for fixed and floating-point representations for 36-bit words. The reader may conclude that he is
143
ference text for research workers in the field of nonlinear oscillations, periodic solutions and stability theory. This first book was not translated into English and, therefore, its influence did not extend to mathematics and science departments in the English-speaking countries. As a result of the wide scope of the second book covering the field of wave equations and the approximation of solutions by asymptotic methods its use will be somewhat different. Prerequisites for the study of this book are a basic knowledge of differential equations and complex function theory as usually presented in undergraduate courses. The first two chapters are devoted to the classical subjects of Fourier, Laplace and Mellin transforms and special functions (gamma functions and Bessel functions); Chap. 3 presents the basic results for the linear wave equation. The introduction to basic material is concluded in Chap. 4. in which, (for approximately 40 pages) asymptotic methods are discussed. All this material is used and applied in 200 pages where the following subjects are studied: scattering matrix theory; flow in open channel; asymptotic solutions of linear and nonlinear wave equations; seismic waves and waterwave theory. Nearly all the results in this book concern linear theory; the treatment both of the basic material in the first four chapters and in the applications is unusually detailed and mathematically sound. The concept of wave equation itself remains rather vague. It would have been helpful to the reader if a survey and classification of equations leading to wave phenomena had been presented. However, from the point of view of linear theory, Asymptotic Wave Theory will be very useful as a textbook for graduate students in applied mathematics and as a reference text for research workers in the field of wave equations. Only in a few instances, notably in Chap. 6 does the author touch upon nonlinear wave theory. No justice can be done to this rapidly expanding field in a few pages!
Journal of The Franklin
htitute
Reviews
and the reader is referred to the (unfortunately somewhat outdated) literature. Another minor criticism: there is little use for an index containing fewer key words than the table of contents; but this type of index is in an old French tradition. F. VERHULST Department of Mathematics, Rijksuniversiteit, Utrecht, The Netherlands
INTRODUCTIONTO MATHEMATKAL.BIOLOGY by S. I. Rubinow, 386pp., diagrams, illus., 6fx 9&, John Wiley, New York, 1975, Price $22.00. Catering as it does to two distinctly different readerships, this book on mathematical biology handles the task remarkably well. The author, in his preface, gives recognition to the fact that explanatory comments addressed to one group of students (of either biological or mathematical sciences) tend to ‘turn off the other group, but apologetically refuses to succumb to the obvious temptation of skipping such details. The book is divided into five chapters, this division being based more on biological and biophysical grounds rather than on underlying mathematical structures. However, it is claimed that the ordering of the five topics: cell growth, enzyme kinetics, physiological tracers, biological fluid dynamics and diffusion processes has been largely influenced by the desirability to present these topics in the order of increasing degree of mathematical sophistication. Perhaps, this has been achieved, although it may not be too obvious since there is very little mathematical sophistication per se in the treatment of any of the topics. The problem of cell growth, an immensely popular field of investigation in microbiology, is the subject of the fascinating analytical study presented in
141
Book Reviews Chap. 1. While those with mathematical backgrounds are exposed to the jargon of microbiologists, biology majors are introduced to the mathematical concepts of differential equations needed to describe exponential growth, growth rates and other such physically understood aspects of cell growth phenomenon. Nutrient uptake by cells, growth of a microbial colony, interacting predator-prey systems (a long studied problem of interest to investigators from a variety of inter-disciplinary fields) are some of the problems tackled here to illustrate the application of mathematical techniques to the field of microbiology. Enzyme Kinetics, the topic of Chap. 2 is treated at greater length. The reader is first introduced to enzymes terminology followed by a detailed presentation of Michaelis-Menten theory and its applications to enzymatically controlled reaction problems. Graph theory and its relevance in the context of steady state ‘enzyme kinetics equations and their solutions, is another important topic covered. Chapter 3, on tracers in physiological systems, starts off with an introduction to the concept of compartment systems, and progresses step-by-step from an analysis of a one-compartment system through two and three-compartment catenary systems to n-compartment systems, illustrating each with examples. Biological fluid dynamics is dealt with in detail in Chap. 4. As is to be expected, the presentation begins with an exposure to the inevitable Navier-Stokes equations and the attendant mathematical sophistications in terms of partial differential equations and their solutions. Properties of blood followed by an analysis of blood flow through arteries and veins are covered in succeeding sections. The swimming of microorganisms, such as spermatozoa, parasitic worms and other such self -propelling organisms, is considered. The final chapter entitled “Diffusion in Biology,” begins with a discussion of Fick’s Laws of Diffusion. The phenome-
142
non of Olfactory Communication in animals is cited as an example of a diffusion process in the realm of biological studies. Next, membrane transport as a diffusion process (through a slab) is studied in depth. The Gaussian function encountered in the study of a convective transport problem: ionic flow in an axon, is explained by introducing the reader to some basic probability theory concepts like random variable, probability density function moments. and Ultracentrifugation, sedimentation velocity method, solution to the Lamm equation, sedimentation equilibrium method and transcapillary exchange are some of the other topics covered. A commendable feature of the book is the set of interesting problems offered at the end of each chapter. While offering students of biology an opportunity to test their understanding of the mathematical techniques presented, these problems expose the mathematically oriented reader to additional problem areas in the field of biology and related sciences where their tools can be meaningfully employed. There is a section on solutions to the problems. This facilitates readers, who are trying to wade through the book on their own. Of course, availability of solutions to the problems within the book itself makes it rather inconvenient, (in terms of setting up home assignments), to a professor intending to use this book for a regular credit course. But the advantage to the uninitiated reader far outweighs this inconvenience. More than 175 references listed at the end significantly add to the value of the book. Appendices A and B help the mathematically undeveloped by reviewing some elementary notions such as variables, functions, integration, differentiation, exponential and logarithm function, Taylor series, determinants, vectors and matrices. Introduction to In conclusion, Mathematical Biology, having set for itself the difficult task of catering to the needs
Journal of The
Franklin
Institute
Book of those with vastly differing backgrounds, carries it out quite satisfactorily. The only significant omission is the vast body of literature commonly referred to as biostatistics. This is pointed out in the preface by the author himself, but he justifies it by restricting his definition of the term mathematical biology accordingly. However, in this reviewer’s opinion, it would have been worth the effort if a brief chapter or two had been included to give the reader an awareness of this area of mathematical biology also. B. V. DASARATHV M & S Computing Inc. Huntsville, Alabama U.S.A.
INTRODU
Vol. 304, No. Z/3, August/September 1977
Reviews
The book would be very useful as a reference for additional reading rather than as a required text for a course which has bond graphs as the vehicle for system modelling. A. ANDRV Department of Mechanical Engineering, Michigan State University, East Lansing, Michigan U.S.A.
MICROPROCESSORS AND MICROCOMPUTERS by B. Soucck, 607 pp., diagrams, 6; x 9f in., John Wiley-Interscience, New York, 1976, Price $23.00. Some people like to pick books by their weight. If so, they may be naturally drawn to this entry in the new crop of microcomputer books. However, they may be disappointed. This book gets much of its volume from light rewrites of the author’s previous book on minicomputers (Minicomputers in Data Processing and Simulation, Wiley-Interscience, 1972). It gets the rest from lightly edited material supplied by microcomputer manufactures. The book is intended as a text for students and as a reference for practicing engineers and scientists. It assumes a minimal background in such topics as programming, computer organization, and electronics. It includes many problems and hence could be used as a text assuming one would want to after reading this book and comparing it with similar ones. The book is divided into three parts. Part I; Microprocessor Programming and Interfacing Techniques consists of five chapters of traditional computer organization and programming, retitled but hardly rewritten. Chapter 1, Number Systems and Digital Codes, presents the obligatory material on decimal and binary number systems, binary arithmetic, and octal, BCD, and Gray codes. On page 17, the author presents the formats for fixed and floating-point representations for 36-bit words. The reader may conclude that he is
143