Books by David Yang Gao
Theory,
Methods and Applications
By David Yang Gao
Kluwer Academic
Publishers,
- Hardbound, ISBN
0-7923-6145-8
December 1999, 472 pp.
NLG 375.00 / USD 199.00 / GBP 124.00 - Available at a
reduced price for course adoption.
Please contact Customer Services at (services@wkap.nl)
or the Editor John Martindale at (jmartindale@wkap.nl)
for further details.
Book Reviewed by G.A. Maugin:
…The present book, beautifully written by David Y. Gao and richly documented, present such a framework with the introduction of a wealth of fundamental notions and an illustration of these through many examples issued essentially from mechanics, although econometrics and nonlinear programming, and some fields of physics such as phase transitions certainly belong to the same chapter. The author, a mathematician, does his best to render an arid field as pleasant as possible while maintaining the required mathematical rigor. This is achieved thanks to the vast culture of the author who does not hesitate to quote many aphorisms of western and oriental origins at some critical points. …
… all mathematically-inclined mechanicians will find in this book some highly interesting solid substance, whether in the foundations or in the many applications. Selected chapters may be recommended for a graduate course. Applied mathematicians will find in it some inspiration via the numerous examples. Finally, literary critics and apprentice philosophers will get to know more about human nature through the numerous quotations and aphorisms.
Applied Mechanics Review, Vol. 53, no.
9, 2000, B96.
Book Reviewed by
Giulio Maier
Undergraduate students and, in general, beginners in
solid and structural mechanics are usually pleased to learn that the solutions of
their linear elasticity problems are characterized by two extremum
principles. Some celebrated professors of the past, like Arturo Danusso in
Similar manifestations of harmony and orderly architecture of concepts in mathematical modelling of observable events become more and more fascinating as one proceeds deeper and deeper in various scientific fields. This is one of the main messages arising from David Gao’s book on duality in broad sense. In view of the real difficulty of a clear account of its contents, I take the liberty to merely cite here the titles of the chapters.
Part I Symmetry in Convex Systems
1. Mono-duality in static systems
2. Bi-duality in dynamical systems
Part II Symmetry Breaking: Triality Theory in Nonconvex Systems
3. Tri-duality in nonconvex systems
4. Multi-duality and classifications of general systems
Part III Duality in Canonical Systems
5. Duality in geometrically linear systems
6. Duality in finite deformation systems
7. Applications, open problems and concluding remarks
Each chapter ends with a commentary section, primarily devoted to historical remarks and special applications.
The topics dealt with in three appendices are:
A- Duality in Linear Analysis
B- Linear Operators and Adjointness
C- Nonlinear Operators
As for the meaning and implications of bi-, tri- and multi-duality, it seems to me preferable, and more prudent, to refer to the book rather than to try explanations in this review.
The style adopted by the Author is consistent with the approach deliberately adopted: pure, abstract mathematics; rigour in statements and proofs; rich symbology introduced once for all; no indulgence in computational aspects or real-life problems.
As a consequence, the book turns out to be tough reading for engineering students and researchers. However, in my opinion, it is an excellent, impressive and appealing book, perhaps in a sense unique, as far as I know. In fact, the volume provides a unifying survey of little known, but fascinating and potentially useful, aspects and properties of mathematical models in diverse areas of sciences (primarily, but not exclusively, mechanical sciences); it contains several novelties and numerous very recent research results; finally, various passages give evidence that it was written with enthusiasm, emotional participation in advanced research and rare width and breadth of scientific and humanistic horizons.
“Through pure mathematical analysis the intrinsic inner beauty in physical natural can be revealed.” This is a quotation of a David Gao’s passage, but the book is enriched by many meaningful citations of thinkers in all times and continents, from mystic philosophers of ancient China (fourth century BC) to Einstein, Hilbert, Truesdell and other leaders of modern sciences.
The following quotation appeared in a recent issue of
this Journal: “The preface is the most important part of the book. Even
reviewers read a preface.” Both statements hold partly true for this
book. The latter might be related to the difficulty (but also the appeal) of
reading its chapters for an engineering-oriented reader like the present
reviewer. The former reflects the unusual depth and philosophical flavour of the thoughts expressed there (e.g.
“duality in nature is amazingly beautiful”, “simply
mysterious, for it is the way nature was created”) and the two pages of
acknowledgements (including the touching ones to the Author’s prematurely
lost wife and brother) which accompany the Preface.
Giulio Maier, Meccanica
Book Review by [J.Lovísek (Bratislava)] in Zentralblatt MATH - ZMATH
Zbl 0940.49001
Gao,
David Yang
Duality principles in nonconvex systems. Theory,
methods and applications. (English)
[B] Nonconvex Optimization and Its Applications. 39.
Duality allows us to associate a dual problem with a variational problem and to
study the relationship between the two problems. This is useful in mathematical
economics where the dual problem can be stated in terms of the price, in
mechanics where the primal and the dual problems are two well-known forms of
the conservation principles, characterizing the displacements and the
constraints, respectively, in numerics where the dual
problem may help us to solve the primal problem. In addition, the dual problem
enables us to define the generalized solution of a variational problem which
has no classical solution.
The present very important and useful book is of mathematics motivated by duality in natural phenomena, with particular emphasis on the mechanics rather than a book of mathematical analysis, proofs and applications. The book is therefore written in a dual way: by using the very simple examples, involving an elastostatic string and the dynamics of particle and then gives a rigorous analysis of the classical mono-duality of one-dimensional convex static equilibria, the nice bi-duality in dynamical systems, through the interesting two-duality in nonconvex problems to the complicated multi-duality in canonical systems. A potentially powerful sequential dual canonical transformation method is developed heuristically and illustrated by use of many interesting simple examples as well as extensive applications in a wide variety of nonlinear systems, including differential equations, variational problems and inequalities, constrained global optimization, multi-well phase transitions, nonsmooth post-bifurcation, large deformation mechanics, structural limit analysis, differential geometry and nonconvex dynamical systems.\par The book divides naturally into three closely interconnected parts with a total of seven chapters. Each chapter provides some motivation, both at the beginning and throughout, and concludes with substantial applications and commentaries.
Main result: Much of the book contains material that is new,
both in its manner of presentation and in its research development. With especially
coherent and lucid exposition, the work fills a big gap between the
mathematical and engineering sciences. It shows how to use formal language and
duality methods to model natural phenomena, to construct intrinsic framework in
different fields and to provide ideas, concepts and powerful methods for
solving nonconvex nonsmooth problems arising naturally in engineering and
science.
[J.Lovísek (Bratislava)]
MSC 2000:
*49-02
Research monographs (calculus of variations)
90-02
Research monographs (optimization)
49N15
Duality theory (optimization)
90C46
Optimality conditions, duality
Keywords: classical mono-duality of convex static equilibria;
dual problem; variational problem; bi-duality in dynamical systems; constrained
global optimization; multi-well phase transitions; nonsmooth post-bifurcation;
large deformation mechanics; structural limit analysis; differential geometry;
nonconvex dynamical systems
Cited in: Zbl 1104.90038 Zbl 1119.90034 Zbl 1081.74037 Zbl 1075.90074 Zbl 1040.49036
PDF
XML ASCII DVI
PS
BibTeX Online
Ordering
Book Review by Mathematics Review:
At the start of Chapter two, the following quote by Einstein is given: “It is a splendid feeling to realize the unity of a complex of phenomena that by physical perception appear to be completely separated”. Readers of this book will no doubt appreciate the wisdom of this quote and enjoy the experience of a “splendid feeling” as reflected in the way duality is presented by the author David Gao. In applied mathematics departments all around the world, unnecessary demarcation, and sometimes rivalry, seems to exist between subgroups of specialists in continuum/fluid mechanics, dynamical systems, optimization/operations research, mathematical physics, numerical analysis, etc. Likewise, Engineering departments in mainstream universities traditionally compartmentalize the major areas of mechanical, civil, electrical and chemical Engineering, and little effort has been made to allow these related disciplines to be studied as a unified subject. In 1986 Gilbert Strang made a fabulous attempt to break down this demarcation in his book “Introduction to Applied Mathematics”. It is clear that Strang’s book has a profound influence on the author where he continues to drive the philosophy of “unity of a complex of phenomena” to a greater depth.
Duality is indeed an important concept that drives the unity that Einstein is alluding to. While there is no shortage of books that address duality from a theoretical point of view, hitherto few, if any, have stressed the unity of a remarkably wide range of superficially unrelated disciplines as lucidly as David Gao has.
The book contains a significant number of examples and illustrations
By 
David Y. Gao, Ray W. Ogden and G.E. Stavroulakis
Kluwer Academic Publishers, Dordrecht/Boston
- Hardbound, ISBN
0-7923-6786-3
February 2001, 516 pp.
EUR 145.00 / USD 157.00 / GBP 99.00
The Table of Contents
Buy this book from
publisher
Buy
this book from Amazon
Book Review by Giulio Maier
Nonsmooth /
Nonconvex Mechanics: Modeling, Analysis and
Numerical Methods
D. Y. Gao, R. W. Ogden and G. E. Stavroulakis,
Kluwer Academic Publishers,
The untimely death of Professor Panagiotis Panagiotopoulos (Panos for His many friends) unexpectedly occurred in August 1998. He was 48 years old, extremely active in research, already internationally acknowledged as a leading researcher in new areas of theoretical and engineering mechanics and applied mathematics. This review in “Meccanica” gives an opportunity to pay once again homage to this unforgettable outstanding Colleague and to remember that among Panos’many links with scientific communities there was a warm and fruitful relationship with the Italian one (incidentally, our National Academy of Lincei conferred upon Him one of its most prestigious prices in 1995).
This book is dedicated to the memory of Professor Paniagiotopoulos:
a well deserved and a very appropriate and suitable homage. In fact, its title
and subjects concern the fascinating area to which Panos
devoted most of His pioneering works. Almost all 22 Chapters due to 37 authors
contain contributions to this field, contributions which are consistent with Panagiotopoulos’ scientific legacy and are more or less
directly related to His lasting research results and seminal ideas.
Several chapters are intensely mathematical and the assessment of
their novelty and importance is, unfortunately, beyond this reviewer’s
competence. From the standpoint of mechanics, both theoretical and, to a lesser
extent, computational, this book contains a number of meaningful contributions.
They consist of either presentations or critical and systematic surveys of
novel or recent results which are at the present forefront of research in the
fast growing area indicated by the title.
Within the unifying framework provided by this area and its mathematical
peculiarities, numerous and diverse mechanical issues are dealt with in the
book, often very effectively. They include: quasi-static evolution
stability; inverse problems of optimum design and parameter
identification and relevant sensitivity analysis; unilateral frictional
contact; interface modelling and debonding in composites and laminates; phase
transitions; nonlinear and chaotic dynamics and wave propagation;
variational principles and duality and triality theories.
The variety of the subjects considered by the numerous Authors entails
unavoidable heterogeneity in style, originality and approach and makes it difficult to outline and evaluate in a
journal review the contents of the individual contributions.
However, I would like to express a positive overall evaluation of this
book: in my opinion it is interesting and useful to all researchers in
theoretical and applied mechanics, since it represents a rich source of
information on recent developments concerning mathematical models which at
present arise more and more frequently in various technological and engineering
sectors.