- Mathematics applied to deterministic problems in the natural sciences (1974), by C C Lin and L A Segel.
1.1. Review by: Editors.
Mathematical Reviews MR0368520 (51 #4761).There are many books on the methods of applied mathematics or mathematical physics. This is such a book and yet it is not: the methods are covered but they are covered in the service of concrete problems, i.e., this is a book on applied mathematics, with emphasis on "the construction, analysis and interpretation of mathematical models that shed light on significant problems in the natural sciences". The book is in three parts. Part A (An overview of the interaction of mathematics and natural science) contains chapters entitled: What is applied mathematics? Deterministic systems and ordinary differential equations; Random processes and partial differential equations; Superposition, heat flow, and Fourier analysis; Further developments in Fourier analysis. Part B (Some fundamental procedures illustrated on ordinary differential equations) contains chapters entitled: Simplification, dimensional analysis, and scaling; Regular perturbation theory; Illustration of techniques on a physiological flow problem; Introduction to singular perturbation theory; Singular perturbation theory applied to a problem in biochemical kinetics; Three techniques applied to the simple pendulum. Finally, Part C (Introduction to theories of continuous fields) contains chapters entitled: Longitudinal motion of a bar; The continuous medium; Field equations of continuous mechanics; Inviscid fluid flow; Potential theory.
- Mathematics applied to continuum mechanics (1977), by Lee A Segel.
2.1. From the Preface:
This text develops and uses mathematics to analyse continuum models of fluid flow and solid deformation. It is intended for upper level undergraduate and graduate students in applied mathematics, science, and engineering; all of the material has been tested in various courses for such students. There is an emphasis on the process of achieving understanding, even at the expense of limiting the topics covered. Although applied mathematics has flowered far from its original roots in physics, continuum mechanics remains a core course. One reason is that the concepts that theoreticians use to organise experience are developed most fully in classical subjects such as continuum mechanics. Subjects of comparable depth - electromagnetism and quantum physics, for example - do not share with continuum mechanics a concentration on relatively familiar natural phenomena and therefore are not so well suited to the rapid development of physical intuition. Moreover, because it deals with problems that are at once deep and widely understandable, continuum mechanics has been for two centuries a major source of significant new applied mathematics. Boundary layer theory provides a paradigm for the development of a concept from one invented for a particular class of problems in continuum mechanics to one presently used in a wide variety of applications. Another example is provided by the study of water waves, which continues to stimulate the generation of new techniques for analysing nonlinear partial differential equations.
2.2. Review by: John Howarth.
The Mathematical Gazette 62 (419) (1978), 67-68.The book is a sequel to the (in my opinion excellent) text of C C Lin and L A Segel, 'Mathematics applied to deterministic problems in the natural sciences', and there is no doubt that the earlier book should be to hand in order to obtain best results from this volume, for back-reference is frequently made. The style is informal and lucid; the author frequently states what he is about to do, and why, before he does it, at once motivating the reader and cultivating the desired attitude. It goes almost without saying that this makes the book much more accessible to the student reading it for himself. .... I only wish that it had been available when I was an undergraduate (despite the price). A 'must' for every university and college library, and strongly recommended to those undergraduates who can afford to buy it for themselves.
2.3. Review by: Harry Hochstadt.
SIAM Review 21 (3) (1979), 414-415.The book under review is a sequel to 'Mathematics applied to deterministic problems in the natural sciences' by C C Lin and L A Segel and leans heavily for prerequisite material on that volume. ... The strength of the book lies in [its] philosophical orientation ... All too often authors are interested in specific physical problems and will present ad hoc derivations of the differential equations associated with the problem under discussion. Such a method is efficient in terms of quickly setting up the problem, but doesn't explain how other problems can be set up. Segel's treatment is very sophisticated and begins with the most general equations. Then he gradually introduces additional assumptions, such as irrotationality, isotropy, reduction to lower dimensionality etc. that apply to special situations and produce more tractable equations. The implications of each assumption are discussed and in many instances the conclusions are correlated with experimental evidence. ... In summary the book has an excellent presentation of the derivation of the equations of continuum mechanics. The reader is shown in great detail how to proceed from the most general equations, through a succession of steps, to specific problems.
2.4. Review by: D Iesan.
Mathematical Reviews MR0502508 (58 #19516).This book is designed for upper level undergraduates and graduate students in applied mathematics and engineering. The book presents various mathematical methods used in continuum mechanics. Continuum models of fluid flow and solid deformation are studied.
2.5. Review by: Albert Schild.
SIAM Review 22 (3) (1980), 384-385.This volume is a sequel to C C Lin and L A Segel's 'Mathematics applied to deterministic problems in the natural sciences'. While knowledge of the material of the Lin-Segel book is certainly useful, it is not a prerequisite for this book. ... The book could be used as a text for good upper level undergraduate and graduate students in applied mathematics and engineering. It would be nice if such a course were made a requirement for students in "pure" mathematics, so that the student would get a feel for the variety of ways in which mathematics is applied to interesting and significant real world problems.
2.6. Review by: J R Ockendon.
J. Fluid Mech. 113 (1981), 533-535.This book describes various aspects of applying mathematics to problems in the natural sciences, beginning with physical descriptions of the problems and ending with physical interpretations of the mathematical results. About nine such problems, many of which have not appeared in a text book before, are treated in the first two parts of the book, while the third part comprises an introduction to the theory of continuum mechanics with some associated problems which will be very familiar to Journal of Fluid Mechanics readers. This final part is a good introduction to the book's successor, which is a more conventional volume and is described later. All the material, both physical and mathematical, is introduced at freshman level and this is the principal reason why the book is so long. However, the authors are ambitious enough to have sprinkled many lists and footnotes throughout the text indicating important areas of current research, but many of these will be lost on beginning students. The general approach is typified by the large number of exercises with carefully thought-out answers and the plethora of footnotes containing pieces of advice about applied mathematical philosophy. ... I am sure the most difficult part of the task they have set themselves is the description of how a model is set up and for many beginners it is almost impossible to be too painstaking over this. Despite the joyless way in which some parts of the book may appear to read, the thorough descriptions which the authors have given of examples from biology, chemistry, physics and engineering will surely prove a great boon to future generations of students. Many of these students would otherwise be resourceless or at best have to learn modelling by far more haphazard routes.
- Mathematical Models in Molecular and Cellular Biology (1980), edited by Lee A Segel.
3.1. Review by: Stephen Childress.
SIAM Review 25 (1) (1983), 138-139.In the spring of 1978 a course on "Mathematical models in biology" was offered at the Weizmann Institute of Science, with support from the Institute as well as the European Molecular Biology association. L A Segel has performed a singular service to all interested scientists in the preparation of the present volume, which records the contributions of the twenty-one authors of the course notes. The result is, however, much more than a "proceedings"; careful editing and presumably some elaboration of the material have produced a valuable reference work with (excluding one or two topics) a remarkably balanced organisation and scope. The course was aimed at the experimental biologist with a background in basic calculus, and the various sections can be divided into roughly three categories which reflect that intent: First there are introductory sections designed to develop basic tools, including "Mathematical Topics" collected into an appendix (calculus and linear algebra refresher, ordinary differential equations, dimensional analysis). Second, "case studies" of specific systems are taken up, sometimes at a more descriptive level and with a generous dose of biological background and data. Finally, there are the more technical sections, which require a good grasp of applied mathematics, particularly differential equations. It is perhaps the collaborative nature of the work which makes it appealing on so many levels both to applied mathematicians as well as to mathematically inclined biologists.
3.2. Review by: John J Tyson.
Mathematical Reviews MR0603811 (82c:92001).This book stems from a course on mathematical models in biology given in 1978 at the Weizmann Institute of Science. The purpose of the book is identical to that of the course: to demonstrate to experimental biologists with a minimum of mathematical background the possible usefulness of mathematical models in cellular and molecular biology. Secondarily, the book is intended to serve applied mathematicians as a source of mathematical applications, predominantly using differential equations, to this area of biology.
- Modeling dynamic phenomena in molecular and cellular biology (1984), by Lee A Segel.
4.1. Review by: Leon Glass.
SIAM Review 28 (2) (1986), 259-260.The mathematical education of biology students in most institutions of higher learning is abysmal. A typical background is a year of calculus, some statistics and occasionally a bit of differential equations. Examples are frequently drawn from the physical sciences, and the many ways in which theoretical analysis using mathematics has been applied to biology are often ignored. This book is based on a semester course, required for graduate students in the biological sciences, which has been taught at the Weizmann Institute for the past ten years ... The book's main theoretical line is to analyse bifurcations (i.e. changes in qualitative dynamics) in mathematical models of biological systems. ... Most biologists will agree that mathematics continues to play an important role in many fields such as the structure of proteins and DNA, population genetics, biochemical kinetics, tomography, membrane biophysics. Yet there remains a curious reluctance of biologists to require that their students acquire even a rudimentary knowledge of mathematics. Segel's book provides a superb introduction to dynamic phenomena in biology. I am using it in a course I teach, and urge others to do likewise.
4.2. Review by: Howard C Berg.
The Quarterly Review of Biology 61 (1) (1986), 78-79.This compact textbook summarises a course taught to first-year graduate students in the biological sciences at the Weizmann Institute (Rehovoth, Israel). Its intent is cross-cultural: to impress on biologists the usefulness of mathematical modelling, chiefly in molecular and cellular biology, and to expose mathematicians to applications in biological dynamics. ... The strength of this book is that it provides examples of problems in a sufficiently broad range of biological scenarios, with enough mathematical rigour and depth that a biologist can judge for himself whether the effort required to acquire the technology, or to collaborate effectively with one who is more deeply versed, is likely to prove rewarding.
4.3. Review by: H C Tuckwell.
Mathematical Reviews MR0762637 (86j:92003).This is a most interesting book. It is well written and well edited. ... The author specifies that his book is aimed primarily at biology graduate students. Since the reviewer found that the reader would certainly profit from a reasonably extensive biological background, he agrees. However, the biology is well explained in general and so the book should be useful for teaching applied mathematics modelling, or courses in biomathematics. The students would need to have a genuine interest to provide the motivation for persevering with some fairly elaborate models. However, the biology is, after all, interesting and should provide the impetus.
4.4. Review by: Thomas A Cole.
Journal of College Science Teaching 16 (4) (1987), 391.According to the for preface, this book is designed for "students of biology who have studied calculus for one year." Although many undergraduate biology majors have completed a calculus course, most would find that a course using this text would be quite challenging in both its biological and mathematical foundations. Of the 300 pages, some 63 are devoted to mathematical and analytical appendices. There are nine chapters on topics related to optimal strategies for metabolism of storage materials, population dynamics, population genetics, enzyme kinetics, the chemostat, cAMP signalling in slime moulds, diffusion, pattern formation and morphogenesis. T he last two chapters (on pattern formation and morphogenesis) address different aspects of developmental processes: chemical signalling and mechanical signalling. ... The undergraduate student who masters the material of this volume will be recognized as having a bright future in the study of biophysical, modelling, and theoretical approaches to biological problems.
- Biograph: A Graphical Computer Simulation Package with Exercises to Accompany Lee A Segel's Modeling Dynamic Phenomena in Molecular Cellular Biology (1987), by Garrett M Odell and Lee A Segel.
5.1. Review by: John G Milton.
SIAM Review 31 (1) (1989), 151-153.The dynamical behaviour of a nonlinear system can change qualitatively as a consequence of quantitative variations of its parameters. This important concept in nonlinear dynamics is often one of the most difficult to teach in an introductory course in biological or applied mathematics. Here the authors present an inexpensive graphical computer simulation package designed to show at a glance what it often takes pages of a book to describe. However, BIOGRAPH "is not just another software manual. It is the completion of an integrated educational tool". The other arm of this tool is the text 'Modelling Dynamic Phenomena in Molecular and Cellular Biology' written previously by one of the authors. The integration is so comprehensive that a short section in the BIOGRAPH manual lists minor corrections and additions to the original 'Modelling Dynamic Phenomena in Molecular and Cellular Biology' text. ... In summary, the authors have developed a welcome tool for teaching introductory level nonlinear dynamics. This package is inexpensive, easy to use, and fun. Moreover, the package should help motivate biologists to use computers for exploring biological phenomena.
- Mathematics applied to deterministic problems in the natural sciences. 2nd edition (1988), by C C Lin and L A Segel.
6.1. From the Publisher:
Addresses the construction, analysis, and interpretation of mathematical models that shed light on significant problems in the physical sciences. The authors' case studies approach leads to excitement in teaching realistic problems. The many problems and exercises reinforce, test and extend the reader's understanding. This reprint volume may be used as an upper level undergraduate or graduate textbook as well as a reference for researchers working on fluid mechanics, elasticity, perturbation methods, dimensional analysis, numerical analysis, continuum mechanics and differential equations.
6.2. From the Foreword:
This volume is a classic in the pedagogy of applied mathematics. Its three authors are internationally known leaders in their respective research communities, which span mathematical approaches in astronomy, biology, and continuum mechanics. They are also naturally gifted teachers who have been long concerned about curricula and other issues of education and they have been remarkably successful in showing how mathematics can be applied in the physical and biological sciences. Students and other readers will sense the authors' enthusiasm and conviction and will learn that relatively elementary mathematics can often provide substantial understanding of natural phenomena.
6.3. Contents:
Part A: An Overview of the Interaction of Mathematics and Natural Science;
Chapter 1: What is Applied Mathematics; On the nature of applied mathematics; Introduction to the analysis of galactic structure; Aggregation of slime mould amebae;
Chapter 2: Deterministic Systems and Ordinary Differential Equations; Planetary orbits; Elements of perturbation theory, including Poincare's method for periodic orbits; A system of ordinary differential equations;
Chapter 3: Random Processes and Partial Differential Equations; Random walk in one dimension; Langevin's equation; Asymptotic series, Laplace's method, gamma function, Stirling's formula; A difference equation and its limit; Further considerations pertinent to the relationship between probability and partial differential equations;
Chapter 4: Superposition, Heat Flow, and Fourier Analysis; Conduction of heat; Fourier's theorem; On the nature of Fourier series;
Chapter 5: Further Developments in Fourier Analysis; Other aspects of heat conduction; Sturn-Liouville systems; Brief introduction to Fourier transform; Generalized harmonic analysis;
Part B: Some Fundamental Procedures Illustrated on Ordinary Differential Equations;
Chapter 6: Simplification, Dimensional Analysis, and Scaling; The basic simplification procedure; Dimensional analysis; Scaling;
Chapter 7: Regular Perturbation Theory; The series method applied to the simple pendulum; Projectile problem solved by perturbation theory;
Chapter 8: Illustration of Techniques on a Physiological Flow Problem; Physical formulation and dimensional analysis of a model for ""standing gradient" osmotically driven flow; A mathematical model and its dimensional analysis; Obtaining the final scaled dimensionless form of the mathematical model; Solution and interpretation;
Chapter 9: Introduction to Singular Perturbation Theory; Roots of polynomial equations; Boundary value problems for ordinary differential equations;
Chapter 10: Singular Perturbation Theory Applied to a Problem in Biochemical Kinetics; Formulation of an initial value problem for a one enzyme-one substrate chemical reaction; Approximate solution by singular perturbation methods;
Chapter 11: Three Techniques Applied to the Simple Pendulum; Stability of normal and inverted equilibrium of the pendulum; A multiple scale expansion; The phase plane;
Part C: Introduction to Theories of Continuous Fields;
Chapter 12: Longitudinal Motion of a Bar; Derivation of the governing equations; One-dimensional elastic wave propagation; Discontinuous solutions; Work, energy, and vibrations;
Chapter 13: The Continuous Medium; The continuum model; Kinematics of deformable media; the material derivative; The Jacobian and its material derivative;
Chapter 14: Field Equations of Continuum Mechanics; Conservation of mass; Balance of linear momentum; Balance of angular momentum; Energy and entropy; On constitutive equations, covariance; and the continuum model;
Chapter 15: Inviscid Fluid Flow; Stress in motionless and inviscid fluids; Stability of a stratified fluid; Compression waves in gases; Uniform flow past a circular cylinder;
Chapter 16: Potential Theory; Equations of Laplace and Poisson; Green's functions; Diffraction of acoustic waves by a hole.6.4. Review by: John Adam.
Mathematical Reviews MR0982711 (91a:00013).This is a beautiful book. It is a pleasure to review, not least because it was very influential in the reviewer's own development as an applied mathematician. ... The book differs from the standard "techniques applied to classical problems'' approach, and, while one should not belittle this approach, the difference is one of the strengths of the present volume. The approach used is more of the mathematical modelling/case study kind, stressing the importance and difficulty of formulating "real world'' problems. There is a nice balance between rigorous, formal and heuristic reasoning which reflects well the nature of research in applied mathematics.
There are three largely independent components to the book:
Part A. An overview of the interaction of mathematics and natural science.
Part B. Some fundamental procedures illustrated on ordinary differential equations.
Part C. Introduction to theories of continuous fields.
... In summary, the book is a highly effective ambassador for applied mathematics. It illustrates the beauty and richness of the subject, but not only this. The reader is also introduced to the philosophical fabric behind the activity itself. It is a much-valued addition to my library. It encourages the reader to refrain from getting so deeply immersed in the practice of applied mathematics that its broader implications are forgotten - what are we actually doing and why does it work? - Design Principles for the Immune System and Other Distributed Autonomous Systems (2001), by Lee A Segel and Irun R Cohen.
7.1. Review by: Alan S Perelson.
SIAM Review 44 (4) (2002), 740.This book, which is a volume in the Santa Fe Institute Studies in the Sciences of Complexity series, had its origins in a workshop of the same title held at the Santa Fe Institute in July 1999. The idea behind the book is to examine features of systems that can be described as a set of interacting entities or agents, which act autonomously or, as the editors put it, "without a boss." Here distributed could mean distributed in space but the authors use the term to describe systems composed of large numbers of entities, such as the trillions of cells of the immune system. How such systems function successfully without a boss is the central question unifying the book. ... The book is exciting for those of us dealing with theoretical immunology. However, it should have broader appeal. Using the immune system as a model from which to deduce biological and engineering principles is an idea that is gaining credence and which has spawned a subfield called artificial immune systems. Here one might be interested in seeing whether the pattern recognition algorithms of the immune system can be generalized or whether the algorithms that the immune system uses to detect when a person is infected can be applied to detecting a computer virus or a network intrusion.
- Mathematics applied to continuum mechanics. Reprint of the 1977 original (2007), by Lee A Segel.
8.1. From the Publisher:
This book focuses on the fundamental ideas of continuum mechanics by analysing models of fluid flow and solid deformation and examining problems in elasticity, water waves, and extremum principles. Mathematics Applied to Continuum Mechanics gives an excellent overview of the subject, with an emphasis on clarity, explanation, and motivation. Extensive exercises and a valuable section containing hints and answers make this an excellent text for both classroom use and independent study.
8.2. From the Preface:
In the 1960s, continuum mechanics was undergoing a revolution from a 'feudal' science - with fiefs of fluid mechanics, elasticity, and esoteric combinations of those sciences with electromagnetics - to a 'cosmopolitan' science consisting of a theory of everything continuum. A E H Love's A treatise on the mathematical theory of elasticity and H Lamb's Hydrodynamics had brought rigor to these areas, and Truesdell's two 'Handbuch der Physik' articles rigorised the more abstract 'rational mechanics' approach. To further complicate matters, science was starting to become computerized, with codes to approximate solutions to increasingly complex problems.
Against this backdrop, Lee Segel set out to write a book for a course in applied mathematics at Rensselaer. The text for the first semester, written with C C Lin, was Mathematics applied to deterministic problems in the natural sciences, also published by SIAM. The present volume was the text for the second semester of this course.
As texts for such a course, the first enjoyed more success than the second. This is partly because mathematics was starting to be applied to a wide spectrum of disciplines in the physical and social sciences and partly because continuum mechanics took on less of a role as a source for interesting and challenging mathematical problems.
Still, in the decades since this book's first publication, continuum approaches were attempted for the description of a multitude of mechanics problems, including mixtures, reacting fluids, heterogeneous solids, multiphase flows, and structure-fluid interactions. Successful treatment of each added complications that required an understanding of the fundamentals of continuum mechanics that went beyond the manipulations required for describing Navier-Stokes fluids or linear elasticity.
This text possesses a quality that makes it an important scientific tool that can be recommended to anyone interested in understanding the vagaries of continuum mechanics. It is aimed at explaining the science of continuum mechanics with clarity winning out over rigor, with explanation and motivation emphasized over manipulation.
This is a book from which it is possible to learn continuum mechanics. As such, it shares a niche with many others, both modern and classical. But the student who uses this book to add understanding to a rigorous text will reach a transcendent appreciation of the science of continuum mechanics.
- A primer on mathematical models in biology (2013), by Lee A Segel and Leah Edelstein-Keshet.
9.1. From the Publisher:
This textbook introduces differential equations, biological applications, and simulations and emphasises molecular events (biochemistry and enzyme kinetics), excitable systems (neural signals), and small protein and genetic circuits. A primer on mathematical models in biology will appeal to readers because it
(i) represents the unique perspective developed by the popular and highly respected applied mathematician Lee Segel in a course he taught at the Weizmann Institute of Science;
(ii) combines clear and useful mathematical methods with applications that illustrate the power of such tools; and
(iii) includes many exercises in reasoning, modelling, and simulations.
This book is intended for upper-level undergraduates in mathematics, graduate students in biology, and lower-level graduate students in mathematics who would like exposure to biological applications.