singular and degenerate cauchy problems.by r. w. carroll; r. e. showalter

3
Singular and Degenerate Cauchy Problems. by R. W. Carroll; R. E. Showalter Review by: Robert Hermann SIAM Review, Vol. 20, No. 3 (Jul., 1978), pp. 612-613 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/2030375 . Accessed: 17/06/2014 05:03 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extend access to SIAM Review. http://www.jstor.org This content downloaded from 62.122.79.69 on Tue, 17 Jun 2014 05:03:05 AM All use subject to JSTOR Terms and Conditions

Upload: review-by-robert-hermann

Post on 08-Jan-2017

212 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Singular and Degenerate Cauchy Problems.by R. W. Carroll; R. E. Showalter

Singular and Degenerate Cauchy Problems. by R. W. Carroll; R. E. ShowalterReview by: Robert HermannSIAM Review, Vol. 20, No. 3 (Jul., 1978), pp. 612-613Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2030375 .

Accessed: 17/06/2014 05:03

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to SIAM Review.

http://www.jstor.org

This content downloaded from 62.122.79.69 on Tue, 17 Jun 2014 05:03:05 AMAll use subject to JSTOR Terms and Conditions

Page 2: Singular and Degenerate Cauchy Problems.by R. W. Carroll; R. E. Showalter

612 BOOK REVIEWS

number of comeasurable lags lie entirely oin the left half-plane. The bibliography is quite extensive and after each chapter there is a commentary. This commentary gives an overview of the material in the particular chapter. It is relatively extensive after Chapters 1, 5, 11, 12 and 13. Not surprisingly these are the chapters which contain the majority of new material vis a' vis [4] and the areas where the most active research on functional differential equations is presently occurring. There are no exercises in the text, but many examples are given to illustrate the theory and suggestions are often made concerning problems which the author thinks warrant investigation.

REFERENCES

[1] R. BELLMAN AND K. COOKE, Differential Difference Equations, Academic Press, New York, 1963. [2] E. A. CODDINGTON AND N. LEVINSON, Theory of Ordinary Differential Equations, McGraw-Hill,

New York, 1955. [3] M. A. CRUZ AND J. K. HALE, Stability of functional differential equations of neutral type, J. Differential

Equations, 7 (1970), 334-355. [4] J. K. HALE, Functional Differential Equations, Appl. Math. Sci., vol. 3, Springer-Verlag, New York,

1971. [5] , Ordinary Differential Equations, John Wiley, New York, 1969. [6] P. H. HARTMAN, Ordinary Differential Equations, John Wiley, New York, 1964. [7] N. N. KRASOVSKII, Stability of Motion, Moscow, 1959; translation, Stanford University Press, 1963.

RICHARD F. DATKO Georgetown University

Singular and Degenerate Cauchy Problems. By R. W. CARROLL AND R. E. SHOWAL- TER. Academic Press, New York, 1976. viii+ 332 pp., $14.50. "Applied mathematics" should involve an interaction between "pure" mathe-

matics and the disciplines (e.g., engineering, physics, economics,...) which use mathematical ideas and concepts. Ideally, progress in "pure" directions is soon reflected in "applied," and conversely, there should be a flux of problems and concepts from "applied" to "pure." Unfortunately, this happens (under today's conditions) only spasmodically. Clearly, we need more systematic efforts to develop this process. Books such as this one-which present the current status of a field of mathematics with potential applications-are often an important catalyst.

The field where this process is the most clear-cut is that of partial differential equations. The most striking feature in "pure" work in PDE in the last twenty years has been its development in terms of the methodology of functional analysis together with a relatively modest infusion of new ideas from geometry and Lie theory. Now, all this new material is not necessarily readily digestible in terms of applications. Perhaps this is due to an over-concentration on the existence side-after all, when Newton wrote down F = MA it did not occur to him that the most important scientific question was the proof of the existence theorem for the underlying differential equation! It is in the direction of understanding qualitative properties of solutions of partial differential equations-especially those which are important for applications (e.g., asymptotic behavior, bifurcation on change of parameters, etc.)- that I belive the geometric and Lie-theoretic insights have the most to contribute.

This interesting book is mainly a progress report on a broad research program directed towards PDE's which have features in common with the classical Euler- Poisson-Darboux equations. (It is quite appropriately dedicated to Alexander Wein- stein.) Heavy functional analysis predominates; in mitigation there is an interesting

This content downloaded from 62.122.79.69 on Tue, 17 Jun 2014 05:03:05 AMAll use subject to JSTOR Terms and Conditions

Page 3: Singular and Degenerate Cauchy Problems.by R. W. Carroll; R. E. Showalter

BOOK REVIEWS 613

chapter developing the relation with modern work on Lie group harmonic analysis. (It seems to me that much more extensive work in this direction is possible and would be very fruitful.) Since it contains nothing about the background or motivation ("pure" or "applied") to these problems, it will be mainly of interest to the "expert"-however, as such it may be considered as a first attempt to put together in a book what is obviously an important and useful body of research.

ROBERT HERMANN Harvard University

Biochemical Systems Analysis. A Study of Function and Design in Molecular Biology. By MICHAEL A. SAVAGEAU. Addison-Wesley, Reading, MA, 1976. xvii + 379 pp., $16.50. In 1956 H. E. Umbarger provided evidence for a negative feedback mechanism

in the biosynthesis of isoleucine. He found that the first enzyme involved in the conversion of threonine to isoleucine was controlled by the level of isoleucine itself. R. A. Yates and A. B. Pardee simultaneously observed a similar effect in pyrimidine biosynthesis. These events had a profound impact on the subsequent direction of biochemical investigation. A major change in perspective occurred within the discipline. Up until this time, modification of enzyme activity was viewed as something done by the investigator to probe the mechanism of enzyme action. Following these discoveries, modification of enzyme activity was considered within the realm of normal cellular function. It marked the beginning of the study of metabolic regulation at the molecular level. It marked the end of classical enzyme kinetics developed by Michaelis and Menton in 1913 which, as described by Dr. Savageau in his book, became a 50 year paradigm for the rigorous description of enzyme-catalyzed chemical change.

Metabolic regulation, enzyme and gene control systems, is the substance of Savageau's book. This is, of course, a vast area of biochemistry. Savageau focuses on one aspect of molecular regulation. He returns to the revolutionary observations of Umbarger, Yates and Pardee and asks, "Why is feedback inhibition in a biosynthetic pathway predominantly exerted through the first enzyme in the pathway?" The question is charmingly simple in its statement. The question would have pleased the late Nicholas Rashevsky, who was fond of asking, "Why does a given mechanism exist and not some other?" The question is childlike in its flavor, yet a suitable reply could shed some much needed light on the adaptive strategies of biological systems.

A further characteristic of Savageau's question is its abstractness. That is, one would like to compare the properties of feedback inhibition of the first enzyme in a pathway with feedback inhibition of, say, the second enzyme. However, the latter does not appear to exist in nature. Abstract, simple, well-posed questions lend themselves to mathematical investigations. Thus much of Savageau's book is bor- rowed from mathematical control theory. The mathematics is relatively unsophisti- cated requiring little more than some facility in calculus and linear equations. This, coupled with Savageau's thorough, detailed presentation makes for a very readable, self-contained text.

There are a number of possible answers to Savageau's basic question and he considers all of the plausible ones. What emerges is the demonstration that feedback inhibition of the first enzyme in a biochemical pathway minimizes sensitivity to environmental perturbations of the pathway structure. Structural stability appears to be the adaptive strategy of the organism. Savageau's approach is convincing and the

This content downloaded from 62.122.79.69 on Tue, 17 Jun 2014 05:03:05 AMAll use subject to JSTOR Terms and Conditions