Book Description

From the reviews of Numerical Solution of Partial Differential Equations in Science and Engineering:
* "The book by Lapidus and Pinder is a very comprehensive, even exhaustive, survey of the subject . . . [It] is unique in that it covers equally finite difference and finite element methods."-Burrelle's.
* "The authors have selected an elementary (but not simplistic) mode of presentation. Many different computational schemes are described in great detail . . . Numerous practical examples and applications are described from beginning to the end, often with calculated results given."-Mathematics of Computing.
* "This volume . . . devotes its considerable number of pages to lucid developments of the methods [for solving partial differential equations] . . . the writing is very polished and I found it a pleasure to read!"-Mathematics of Computation

Of related interest . . .NUMERICAL ANALYSIS FOR APPLIED SCIENCE Myron B. Allen and Eli L. Isaacson. A modern, practical look at numerical analysis, this book guides readers through a broad selection of numerical methods, implementation, and basic theoretical results, with an emphasis on methods used in scientific computation involving differential equations. 1997 (0-471-55266-6) 512 pp.

APPLIED MATHEMATICS Second Edition, J. David Logan. Presenting an easily accessible treatment of mathematical methods for scientists and engineers, this acclaimed work covers fluid mechanics and calculus of variations as well as more modern methods-dimensional analysis and scaling, nonlinear wave propagation, bifurcation, and singular perturbation. 1996 (0-471-16513-1) 496 pp. ... Read more

Reviews (1)

Este libro tiene muy buena base...
Me ayudó mucho en lo que necesitaba aprender y fue muy fácil entenderlo... ... Read more

Book Description

A graduate-level text introducing the use of exterior differential forms as a powerful tool in the analysis of a variety of mathematical problems in the physical and engineering sciences. Directed primarily to graduate-level engineers and physical scientists, it has also been used successfully to introduce modern differential geometry to graduate students in mathematics. 45 illustrations. Index.
... Read more

Reviews (4)

Concise
As a survey of the applications of differential forms to various aspects of the physical sciences, this book works quite well. However, for the non-mathematician, there are more intuitive texts available, although it is unlikely that they will have the same scope. The exposition is extremely concise, far more so than, for example, Lovelock and Rund, and at times this makes the text somewhat heavy going. Having said that though, the reader with a reasonable level of mathematical maturity will find the book very rewarding. A reader who is mainly focussed on differential geometry though would probably be better off with Lovelock and Rund.

Read on
Why is it that any book that contains the word "physics" or "physical" in its title feels free to be sloppy? Flander's book is sloppy. There are no shortcuts to understanding - if you want to know differential geometry you have to learn it right, not from Flander's book. For example: he devotes 3 pages to antisymmetrical tensors. The way he manages to do this is by avoiding precise definitions, or giving elaborate proofs. He does not even mention the word tensor! This is unacceptable for anyone who seeks to truly understand differential geometry. After all, if you intend to invest effort and time learning a subject, shouldn't you do it the right way?
I'm not saying Flander's book is without merit - especially the low price - which might make it worthwhile purchasing it is an additional source of information, but as a primary source it is, in my opinion, a very bad one.
There are many alternatives to Flander's book I suggest you check out before trying your luck with this one. The standard reference is Bishop & Goldberg's "Tensor analysis on manifolds". Another good book is "differential forms and connections" by Darling. A comprehensive book about diff. geometry is "Geometry and Physics" by Frankel. I'm sure others will have their personal favorites, but these are a good place to start.

Generalize vector calculus for general relativity
This book covers the basic math behind the "geometric approach" to tensor calculus. The math required is not heavy, but it requires some considerable mathematical maturity. If talk of "the boundary of a boundary is zero" or "exterior derivative" confuses you, this is a good book. An intuitive approach, not a sea of indices. If you want really heavy stuff then Bishop & Goldberg is good.

Most Dover math books are first rate. This one is.

An intro to differential forms for physicists
This book introduces differential forms to physicists. It is suitable for those with no advanced training in math other than multivarible calculus and linear algebra. No prior knowledge of topology is needed, although it doesn't hurt. The book starts with a good intro explaining the motivation for forms and when they might be useful over tensors and vectors. The exterior algebra is then presented in chapter 2, followed by a chapter on the exterior derivative. The rest of the book prestents a good deal of applications of forms including the statement and proof of the Newton-Leibniz-Ostrogradksy-Gauss-Green-Stokes Theorem, usually just called Stoke's Theorem. I'd recommend this book to any grad student in Theoretical physics that plans on doing research on physics in the 21st century. ... Read more

Book Description

Integrable Hamiltonian systems have been of growing interest over the past 30 years and represent one of the most intriguing and mysterious classes of dynamical systems. This book explores the topology of integrable systems and the general theory underlying their qualitative properties, singularities, and topological invariants.The authors, both of whom have contributed significantly to the field, develop the classification theory for integrable systems with two degrees of freedom. This theory allows one to distinguish such systems up to two natural equivalence relations: the equivalence of the associated foliation into Liouville tori and the usual orbital equaivalence. The authors show that in both cases, one can find complete sets of invariants that give the solution of the classification problem. The first part of the book systematically presents the general construction of these invariants, including many examples and applications. In the second part, the authors apply the general methods of the classification theory to the classical integrable problems in rigid body dynamics and describe their topological portraits, bifurcations of Liouville tori, and local and global topological invariants. They show how the classification theory helps find hidden isomorphisms between integrable systems and present as an example their proof that two famous systems--the Euler case in rigid body dynamics and the Jacobi problem of geodesics on the ellipsoid--are orbitally equivalent.Integrable Hamiltonian Systems: Geometry, Topology, Classification offers a unique opportunity to explore important, previously unpublished results and acquire generally applicable techniques and tools that enable you to work with a broad class of integrable systems. ... Read more

Book Description

This text provides an introduction to the theory of partial differential equations. It introduces basic examples of partial differential equations, arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, including particularly Fourier analysis, distribution theory, and Sobolev spaces. These tools are applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations. Companion texts, which take the theory of partial differential equations further, are AMS volume 116, treating more advanced topics in linear PDE, and AMS volume 117, treating problems in nonlinear PDE. This book is addressed to graduate students in mathematics and to professional mathematicians, with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis ... Read more

Reviews (2)

Complete, accurate and well-written.
The author accomplished the goal of presenting this broad and many-faceted subject in a thorough and comprehensive manner. Beginning from the fundamentals of ODE theory to the most sophisticated methods for solving important PDE's of mathematical physics, this series of three volumes comprises all what a modern analyst must know about the topic and much more.

The contents of volume 1 are: Basic theory of ODE; Laplace and wave equations; Fourier analysis, distributions, and constant-coefficient linear PDE; Sobolev spaces; linear elliptic equations; linear evolution equations. Appendices: Outline of functional analysis; manifolds, vector bundles, and Lie groups.

Originally intended for graduate students and working mathematicians, most of the material is suitable for advanced undergraduate courses. Includes excercises for each section and extensive references.

ODE
Much material is included in this text, which is the first volume in a set of three and is suitable for graduate students and professionals. Chapter one details much material about ODE focusing on vector fields and much more including: Lie brackets, differential forms, geodesics, variable coefficient linear ODE, and of course, existence and uniqueness. Subsequent chapters cover Laplace's equation, Fourier Analysis and its relations to constant - coefficient linear PDE, linear Elliptic equations, Sobolev spaces, and linear evolution equations. Finally two important appendices are indcluded: One concerning functional analysis and another concerning differential geometry.

While these chapters are essential self contained the chapter on sobolev spaces is important for later volumes. Overall the writing is rigorous and demanding --- as most mathemtical texts of this level are. However, with the assistance of a good library and dilligent effort *much* can be obtained from this set of texts, which explains precisely what is currently known about the genernal theory of PDE. ... Read more

Book Description

The subject of partial differential equations has an unchanging core of material but is constantly expanding and evolving. Introduction to Partial Differential Equations with MATLAB presents the careful integration of traditional core topics with modem topics, taking full advantage of the computational power of MATLAB to enhance the learning experience.

This advanced text/reference is an introduction to partial differential equations covering the traditional topics within a modern context. To provide an up-to-date treatment, techniques of numerical computation have been included with carefully selected nonlinear topics, including nonlinear first order equations. Each equation studied is placed in the appropriate physical context. The analytical aspects of solutions are discussed in an integrated fashion with extensive examples and exercises, both analytical and computational. The book is excellent for classroom use and can be used for self-study purposes.

Topic and Features:• Nonlinear equations including nonlinear conservation laws;• Dispersive wave equations and the Schrodinger equation;• Numerical methods for each core equation including finite difference methods, finite element methods, and the fast Fourier transform;• Extensive use of MATLAB programs in exercise sets. MATLAB m files for numerical and graphics programs available by ftp from this web site (see top of page).

This new text/reference is an excellent resources designed to introduce advanced students in mathematics, engineering and sciences to partial differential equations. It is also suitable as a self-study resource for professionals and practitioners. ... Read more

Reviews (2)

Not bad, but the 2nd edition will hopefully be better
This is a fairly challenging text. It's currently being used in my undergrad PDE course, and the flow of material is a bit out of the ordinary. The book starts off with the method of characteristics and weak solutions, which are slightly more sophisticated and unusual topics than some of the later material, making it difficult to get started. The focus on using Matlab is nice, and most things are fairly well-explained. Typographical errors are rampant, however, making it very poor for self-study (you need someone to point out the errors to you).

All in all, not bad -- the second edition will hopefully be much better, and if you have a decent grounding in multivariate calc and ODE's, you'll be OK.

An excellent rough draft of a textbook.
Cooper's book does some things quite well. While most courses on PDE's are reduced to repetitive applications of separation of variables and Fourier series, Cooper offers a new approach. He goes by equation rather than by technique, and introduces nonlinear equations and numerical methods very early. Separation of variables and Fourier series do not come up until Chapter 4. The first chapter, which reviews elements of analysis, is a very good reference. MATLAB is well incorporated throughout the text.

The reason I do not rate this book more highly is that the writing is unbelievably sloppy. There are at least five typos per chapter, usually more. Crucial things, like Green's first identity, are misprinted. There are dozens of typos in the answers in the back, which makes it very hard to check your work.

Judging just by the content, this is a very good book. However, the misprints in the math (to say nothing of those in the text) are so severe that I would not recommend buying it until a better-proofread edition comes out. ... Read more

Book Description

Algorithmic, or automatic, differentiation (AD) is concerned with the accurate and efficient evaluation of derivatives for functions defined by computer programs. No truncation errors are incurred, and the resulting numerical derivative values can be used for all scientific computations that are based on linear, quadratic, or even higher order approximations to nonlinear scalar or vector functions. In particular, AD has been applied to optimization, parameter identification, equation solving, the numerical integration of differential equations, and combinations thereof. Apart from quantifying sensitivities numerically, AD techniques can also provide structural information, e.g., sparsity pattern and generic rank of Jacobian matrices.

This first comprehensive treatment of AD describes all chainrule-based techniques for evaluating derivatives of composite functions with particular emphasis on the reverse, or adjoint, mode. The corresponding complexity analysis shows that gradients are always relatively cheap, while the cost of evaluating Jacobian and Hessian matrices is found to be strongly dependent on problem structure and its efficient exploitation. Attempts to minimize operations count and/or memory requirement lead to hard combinatorial optimization problems in the case of Jacobians and a well-defined trade-off curve between spatial and temporal complexity for gradient evaluations.

The book is divided into three parts: a stand-alone introduction to the fundamentals of AD and its software, a thorough treatment of methods for sparse problems, and final chapters on higher derivatives, nonsmooth problems, and program reversal schedules. Each of the chapters concludes with examples and exercises suitable for students with a basic understanding of differential calculus, procedural programming, and numerical linear algebra. ... Read more

Book Description

Reviews (4)

Concise and to the point
The author's discussion of the material is excellent.He presents good examples and tries to integrate proofs or theorems into what he has told you.

Unlike many engineering books, he actually puts solutions in the back of the text.

This is a refreshing change from publishers who tend to keep this information "restricted".

When you are working with a higher level subject such as Signals and Systems, you want to build practice with problems.

Another point I especially enjoyed was the clear discusion of state space analysis in both the laplace and the Z domains.

GREAT BOOK
Easy to read.Good balance of just enough theory and an ample amount of examples and problems to test one's knowledge.Can be used as either a main textbook or as a supplement for self-study.Perfect for those looking to use MATLAB as an integrated study tool.

An easy-to-learn-from text
Now in an expanded and updated second edition, Signals And Systems With MATLAB Applications by electrical engineer and educator Steven T. Karris is an invaluable textbook and resource especially written and organized for students of electrical engineering. An easy-to-learn-from text, with practical MATLAB applications to use and drive the theory home, Signals And Systems authoritatively covers a wide selection of mathematical and engineering topics in seasoned and technical detail, including delta functions, the laplace transformation, fourier series, analog and digital filters, and much more. Signals And Systems is a confidently recommended advanced instructional resource especially for dedicated students or veteran electrical engineers looking to brush up their basic skills.

Good text for self study
I found this text to be very interesting.It contains material that I have not seen in a single book.All theorems and definitions are well and simply expressed and illustrated with practical examples.I've found few but insignificant references to equations and figures, and I've informed the author. It is an ideal text to learn both the material and MATLAB.Appendix A
is a very good introduction to MATLAB and it helped me very much.
M. Phillips ... Read more

Book Description

UNDERSTANDING NONLINEAR DYNAMICS is based on an undergraduate course taught for many years to students in the biological sciences. The text provides a clear and accessible development of many concepts from contemporary dynamics, including stability and multistability, cellular automata and excitable media, fractals, cycles, and chaos. A chapter on time-series analysis builds on this foundation to provide an introduction to techniques for extracting information about dynamics from data. The text will be useful for courses offered in the life sciences or other applied science programs, or as a supplement to emphasize the application of subjects presented in mathematics or physics courses. Extensive examples are derived from the experimental literature, and numerous exercise sets can be used in teaching basic mathematical concepts and their applications. Concrete applications of the mathematics are illustrated in such areas as biochemistry, neurophysiology, cardiology, and ecology. The text also provides an entry point for researchers not familiar with mathematics but interested in applications of nonlinear dynamics to the life sciences. ... Read more

Reviews (3)

A good introduction to the mechanical side...
It is funny that there is so much hype about Wolfram's "A New Kind of Science" when this book neatly pointed out his big 'discoveries' back in 1995. That is, all of Wolfram's opus can be found in chapter 2 of this book.

The book provides a good (and relatively gentle) introduction to the mechanistic view of non-linear dynamics. Many different applications of non-linear dynamics, from physics to biology, are overviewed and many detailed examples are provided. In all, this is a great reference.

The only shortcomings are the lack of discussions about frameworks, particularly mechanistic systems, and how they influence the applicability of the models. That is, some understanding of systems theory (von Bertalanffy in particular and the work of cybernetics as well) should have been included to point out the fact that the linear causality implied by many models immediately limits their applicability.

Gentle introduction to nonlinear dynamics
This is a gentle introdution to nonlinear dynamics, maths are well explained, illustrations are abundant, the overview is very good overall, even if subjects are only touched. It could also serve as an introduction to finite-difference and differential equations for non-maths students.

Undergrads looking for something smooth on nonlinear dynamics should get this one...but for advanced readers it is probably a waste of time.

Simply an Iluminating path to Understand Nonlinear Dynamics
It happens that in any field there is a book that iluminates. This is the one in Nonlinear Dynamics. With it's clear and simple but mathematically and conceptually rich presentation this text opens the mind towards the mathemathics that you learn in your two first University years, as aplicated to innumerable Life Science examples, but at the same time interesting for any other field's applications (Physics, Maths, Chemists,..). This book deals with Dynamics in a Superb Way. Contents: 1-Finite Difference Equations. 2-Boolean Networks and Celular Automata. 3-Self Similarity and Fractal Geometry. 4-One Dimensional Differential Equations. 5-Two Dimensional Differential Equations. 6-Time Series Analysis. Apendix-A: A Multifunctional Appendix. Appendix-B: A Note on Computer Notation. Buy it, you might learn how much you can do with you mathematics knowledge! ... Read more

Book Description

This book combines mathematics (geometry and topology), computer science (algorithms), and engineering (mesh generation) in order to solve the conceptual and technical problems in the combining of elements of combinatorial and numerical algorithms.The book develops methods from areas that are amenable to combination and explains recent breakthrough solutions to meshing that fit into this category. It should be an ideal graduate text for courses on mesh generation.The specific material is selected giving preference to topics that are elementary, attractive, lend themselves to teaching, are useful, and interesting. ... Read more

Book Description

Under the auspices of the Istituto Nazionale di Alta Matematica, a conference was held in October 1992 in Cortona, Italy, to study partial differential equations of elliptic type. These equations arise from many real systems and have been studied in depth for many years. Here special emphasis is placed on the geometric aspects of the subject, giving this volume a unique flavour. Many of the world's leading figures in this subject area attended the meeting, and this volume collects the best papers, covering the latest advances and shedding new light on old problems. As an account of the present state of the subject, these papers are unparalleled, and all workers on partial differential equations will find that this book will be of lasting value. ... Read more

Book Description

This is the second of three volumes on partial differential equations. It builds upon the basic theory of linear PDE given in Volume 1, and pursues some more advanced topics in linear PDE. Analytical tools introduced in Volume 2 for these studies include pseudodifferential operators, the functional analysis of self-adjoint operators, and Wiener measure. There is also a development of basic differential geometrical concepts, centered about curvature. Topics covered include spectral theory of elliptic differential operators, the theory of scattering of waves by obstacles, index theory for Dirac operators, and Brownian motion and diffusion. The book is addressed to graduate students in mathematics and to professional mathematicians, with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis. ... Read more

Reviews (2)

Modern approach to important topics of mathematical physics.
This second volume focuses on several interesting problems arising from mathematical physiscs, making a perfect connection between the purely theoretical digression and the sources from which it originated.

The contents are: Pseudodifferential operators; spectral theory; scattering by obstacles; Dirac operators and index theory; Brownian motion and potential theory; the "D"-bar Neumann problem. Appendix: Connections and curvature.

Useful for mathematics and physics advanced undergraduates, graduate students, and researchers. Plenty of excercises and references.

Pseudo differential operators, dirac operators, & diffusion
In this second volume, of a set of three, the author takes up linear equations such as Dirac operators and those found in Brownian / diffusive processes. The first chapter presents fundamental results concerning pseudo-differential operators including Egorov's theorem and Weyl Calculus, while the second chapter presents spectral theory with applications to the laplace operator on n spheres & noncompact manifolds (such has hyperbolae and cones). A chapter on scattering from obstacles includes general discussion of wave operators, scattering from a sphere and the inverse scattering problem. Index theory, clifford algebra, spin manifolds, Chern -Gauss -Bonnet, and the Riemann -Roch theorem are covered in the chapter on dirac operators. The chapter on Brownian motion includes general discussion of Ito integrals and SDE as well as the Feynman-Kac solution of the heat eqaution. Finally, the Neumann boundary problem is the subject of the final chapter.

As with the first volume of this set, the writing is dense and challenging. A reader should be a professional or advanced graduate student with access to a large library and plenty of time. However, if one wishes to have a definitive text that covers the modern understanding of PDE in general these three texts are it. ... Read more

Book Description

This text is a clearly written and complete introductory study of differential equations and their applications. Suitable for a one- or two-semester course on the subject, the material is fully comprehensible to anyone who has studied college-level calculus. This book distinguishes itslef from other differential equation texts through its application of the subject matter to fascinating events and its incorporation of relatively recent developments in the field. Some applications include:* a proof that the painting, "Disciples of Emmaus," bought by the Rembrandt Society of Belgium was a modern forgery;* a mathematical explanation of the Tacoma Bridge disaster;* a model of the blood glucose regulatory system which leads to a criterion for the diagnosis of diabetes;*an explanation of why the predator portion (sharks, skates, rays, etc. ) of all fish caught in the port of Flume, Italy, rose dramatically during the years of World War I;* a mathematical verification of Darwin's law that "the more similar two species are, the fiercer is the struggle for existence between them. "An introduction to bifurcation theory, computer programs in C, Pascal, and Fortran, and many original and challenging exercises contribute to the quality of this text. ... Read more

Reviews (3)

Nice try, but...
This is the typical mathematics book intended for non-mathematicians. The author made a great effort to compile the most diverse examples of problems (better said MODELS of problems) that can be solved employing mostly ODE and some PDE techniques. However, I think this kind of books fail to show what mathematics is, and how mathematics should be used to solve problems coming from other fields of knowledge.

First, the book does not cover thoroughly the theory for the linear case, which could be carried out without any effort up to the n-th order. Second, it does not tell explicitly the great difference between the linear and the non-linear cases. Third, eventhough it includes some material on systems of differential equations, it doesn't exploit the fact that a n-th order equation can be viewed as a n-th order system of of 1st order equations. Fourth, everything is mixed up: in the same chapter you find elementary concepts of linear algebra (that the student should be supposed to master), some miscelaneous applications, and, if you are lucky, some theory on differential equations. Fifth, it gives a wrong idea of the meaning of solving an ACTUAL problem. The fact that we are able to give a nice solution to a highly simplified model we have built does not mean that we have found a practical solution to the problem we actually face up in the real world, and moreover it does not mean that we will always succeed to give a useful answer by employing these tools. When you try to solve a problem that you don't know thoroughly you take the risk of building castles in the air. I think that if you are interested in applying mathematics you should start first by learning all the necessary theory and then study carefully the problem you are interested in. And don't forget the most important of all: the theoretical solution to an actual problem must be confirmed by experimentation, something this book does not point out.

To tell something positive about it let me say that you will find a lot of challenging excercises, some of its "application" problems are interesting, and if you are a teacher it will serve well as a companion text for real treatises on differential equations like Coddington's or Kaplan's.

Please check my other reviews at my member page (just click on my name above).

Learning D.E.'s Can Actually be Fun
I have used this to teach DE's in a one to one tutoring context for a couple of years now, since I first picked it off a library shelf and felt literally like jumping for joy over how good it was. Not that I dislike Boyce / DiPrima, but suddenly I had a text that was really fun! Now, this is definitely a "what DE's are good for" kind of text, so if you feel that your students should suffer teething pains on a dry theoretical tome, this is NOT the book for you. Nevertheless, having had students chew up all their available time on this book, because they loved it, makes me recommend it highly.

One of the best in the field
This book covers its material thoroughly and clearly with the most interesting set of applications I've seen in a high-level text. It also contains all the linear algebra needed to understand the material. This book was based on a course the author taught. Indeed, Braun's claim that students taking linear algebra would refer to his course for a refresher is well-deserved. Exercises are generally difficult but the examples are easy to follow. Buy this if you want to truly understand the subject matter. ... Read more

Book Description

Multigrid presents both an elementary introduction to multigrid methods for solving partial differential equations and a contemporary survey of advanced multigrid techniques and real-life applications.

Multigrid methods are invaluable to researchers in scientific disciplines including physics, chemistry, meteorology, fluid and continuum mechanics, geology, biology, and all engineering disciplines. They are also becoming increasingly important in economics and financial mathematics.

Readers are presented with an invaluable summary covering 25 years of practical experience acquired by the multigrid research group at the Germany National Research Center for Information Technology. The book presents both practical and theoretical points of view.

* Covers the whole field of multigrid methods from its elements up to the most advanced applications
* Style is essentially elementary but mathematically rigorous
* No other book is so comprehensive and written for both practitioners and students ... Read more

Book Description

The book contains seven survey papers about ordinary differential equations.

The common feature of all papers consists in the fact that nonlinear equations are focused on. This reflects the situation in modern mathematical modelling - nonlinear mathematical models are more realistic and describe the real world problems more accurately. The implications are that new methods and approaches have to be looked for, developed and adopted in order to understand and solve nonlinear ordinary differential equations.

The purpose of this volume is to inform the mathematical community and also other scientists interested in and using the mathematical apparatus of ordinary differential equations, about some of these methods and possible applications. ... Read more

Book Description

After Karl Jöreskog's first presentation in 1970, Structural Equation Modelling or SEM has become a main statistical tool in many fields of science. It is the standard approach of factor analytic and causal modelling in such diverse fields as sociology, education, psychology, economics, management and medical sciences. In addition to an extension of its application area, Structural Equation Modelling also features a continual renewal and extension of its theoretical background. The sixteen contributions to this book, written by experts from many countries, present important new developments and interesting applications in Structural Equation Modelling. The book addresses methodologists and statisticians professionally dealing with Structural Equation Modelling to enhance their knowledge of the type of models covered and the technical problems involved in their formulation. In addition, the book offers applied researchers new ideas about the use of Structural Equation Modeling in solving their problems. Finally, methodologists, mathematicians and applied researchers alike are addressed, who simply want to update their knowledge of recent approaches in data analysis and mathematical modelling. ... Read more

Book Description

This book is a comprehensive description of the current theoretical and numerical aspects of inverse problems in partial differential equations. Applications include recovery of inclusions from anomalies of their gravity fields, reconstruction of the interior of the human body from exterior electrical, ultrasonic, and magnetic measurement. The reconstruction of the interior structural parameters of machines and of the underground are also discussed together with further scientific and engineering applications. By presenting this data in a readable and informative manner, the book introduces both scientific and engineering researchers and graduate students to significant work done in this area in recent years, relating it to broader themes in mathematical analysis. ... Read more

Book Description

This book provides a self-contained comprehensive exposition of the theory of dynamical systems. The book begins with a discussion of several elementary but crucial examples.These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods.The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth.The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate and up. ... Read more

Reviews (4)

Great, advanced intro to dynamical systems
This is really one of the very best books on dynamical systems available today. Nearly every topic in modern dynamical systems is treated in detail. The authors have provided many important comments and historical notes on the material presented in the main text. The writing is clear and the many topics discussed are given appropriate motivation and background.

There are only two potential drawbacks. First, the prerequisites for this book are quite high. The read should be familiar with real and functional analysis, differential geometry, topology, and measure theory, for starters. Fortunately a well-organized appendix collects the key results of each of the branches of math for the reader's reference. Second, many dynamical systems of interest to applied mathematicians, scientists, and engineers arise from differential equations. This book does not discuss in much detail the connection between ODEs and continuous dynamical systems. Other books (e.g. Perko) treat this connection more thoroughly.

For completeness, clarity, and rigor, Katok and Hasselblatt is without equal. If you work in dynamical systems, you should definitely have this excellent text on your bookshelf. Highly recommended.

Great book with lots of detail
This book is a comprehensive overview of modern dynamical systems that covers the major areas. The authors begin with an overview of the main areas of dynamics: ergodic theory, where the emphasis is on measure and information theory; topological dynamics, where the phase space is a topological space and the "flows" are continuous transformations on these spaces; differentiable dynamics where the phase space is a smooth manifold and the flows are one-parameter groups of diffeomorphisms; and Hamiltonian dynamics, which is the most physical and generalizes classical mechanics. Noticeably missing in the list of references for individuals contributing to these areas are Churchill, Pecelli, and Rod, who have done interesting work in the area of both topological and Hamiltonian mechanics. No doubt size and time constraints forced the authors to make major omissions in an already sizable book.

Some elementary examples of dynamical systems are given in the first chapter, including definitions of the more important concepts such as topological transitivity and gradient flows. The authors are careful to distinguish between topologically mixing and topological transitivity. This (subtle) difference is sometimes not clear in other books. Symbolic dynamics, so important in the study of dynamical systems, is also treated in detail.

The classification of dynamical systems is begun in Chapter 2, with equivalence under conjugacy and semi-conjugacy defined and characterized. The very important Smale horseshoe map and the construction of Markov partitions are discussed. The authors are careful to distinguish the orbit structure of flows from the case in discrete-time systems.

Chapter 3 moves on to the characterization of the asymptotic behavior of smooth dynamical systems. This is done with a detailed introduction to the zeta-function and topological entropy. In symbolic dynamics, the topological entropy is known to be uncomputable for some dynamical systems (such as cellular automata), but this is not discussed here. The discussion of the algebraic entropy of the fundamental group is particularly illuminating.

Measure and ergodic theory are introduced in the following chapter. Detailed proofs are given of most of the results, and it is good to see that the authors have chosen to include a discussion of Hamiltonian systems, so important to physical applications.

The existence of invariant measures for smooth dynamical systems follows in the next chapter with a good introduction to Lagrangian mechanics.

Part 2 of the book is a rigorous overview of hyperbolicity with a very insightful discussion of stable and unstable manifolds. Homoclinicity and the horseshoe map are also discussed, and even though these constructions are not useful in practical applications, an in-depth understanding of them is important for gaining insight as to the behavior of chaotic dynamical systems. Also, a very good discussion of Morse theory is given in this part in the context of the variational theory of dynamics.

The third part of the book covers the important area of low dimensional dynamics. The authors motivate the subject well, explaining the need for using low dimensional dynamics to gain an intuition in higher dimensions. The examples given are helpful to those who might be interested in the quantization of dynamical systems, as the number-theoretic constructions employed by the author are similar to those used in "quantum chaos" studies. Knot theorists will appreciate the discussion on kneading theory.

The authors return to the subject of hyperbolic dynamical systems in the last part of the book. The discussion is very rigorous and very well-written, especially the sections on shadowing and equilibrium states. The shadowing results have been misused in the literature, with many false statements about their applicability. The shadowing theorem is proved along with the structural stability theorem.

The authors give a supplement to the book on Pesin theory. The details of Pesin theory are usually time-consuming to get through, but the authors do a good job of explaining the main ideas. The multiplicative ergodic theorem is proved, and this is nice since the proof in the literature is difficult.

Excellent rigorous introduction to chaotic dynamical system
This remarkable book is by far the best rigorous introduction to many facets of the modern theory of (chaotic) dynamical systems. It introduces and rigorously develops the central concepts and methods in dynamical systems in a hands-on and highly insightful fashion. The authors are world experts in smooth dynamical systems and have played a major role in the development of the modern theory and this shows througout the book.

The book starts with a comprehensive discussion of a series of elementary but fundamental examples. These examples are used to formulate the general program of the study of asymptotic properties as well as to introduce the principal notions (differentiable and topological equivalence, moduli, asymptotic orbit growth, entropies, ergodicity, etc.) and, in a simplified way, a number of important methods (fixed point methods, coding, KAM-type Newton method, local normal forms, etc.). This chapter alone is worth the price of the book.

The main theme of the second part is the interplay between local analysis near individual (e.g., periodic) orbits and the global complexity of the orbit structure. This is achieved by exploring hyperbolicity, transversality, global topological invariants, and variational methods. The methods include study of stable and unstable manifolds, bifurcations, index and degree, and construction of orbits as minima and minimaxes of action functionals.

In the third and fourth part the general program is carried out for low-dimensional and hyperbolic dynamical systems which are particularly amenable to such analysis. In addition these systems have interesting particular properties. For hyperbolic systems there are structural stability, theory of equilibrium (Gibbs) measures, and asymptotic distribution of periodic orbits, in low-dimensional dynamical systems classical Poincare-Denjoy theory, and Poincare-Bendixson theories are presented as well as more recent developments, including the theory of twist maps, interval exchange transformations and noninvertible interval maps.

This book should be on the desk (not bookshelf!) of any serious student of dynamical systems or any mathematically sophisticated scientist or engineer interested in using tools and paradigms of dynamical systems to model or study nonlinear systems.

Excellent rigorous introduction to chaotic dynamical systems
This remarkable book is by far the best rigorous introduction to many facets of the modern theory of (chaotic) dynamical systems. It introduces and rigorously develops the central concepts and methods in dynamical systems in a hands-on and highly insightful fashion. The authors are world experts in smooth dynamical systems and have played a major role in the development of the modern theory and this shows througout the book.

The book starts with a comprehensive discussion of a series of elementary but fundamental examples. These examples are used to formulate the general program of the study of asymptotic properties as well as to introduce the principal notions (differentiable and topological equivalence, moduli, asymptotic orbit growth, entropies, ergodicity, etc.) and, in a simplified way, a number of important methods (fixed point methods, coding, KAM-type Newton method, local normal forms, etc.). This chapter alone is worth the price of the book.

The main theme of the second part is the interplay between local analysis near individual (e.g., periodic) orbits and the global complexity of the orbit structure. This is achieved by exploring hyperbolicity, transversality, global topological invariants, and variational methods. The methods include study of stable and unstable manifolds, bifurcations, index and degree, and construction of orbits as minima and minimaxes of action functionals.

In the third and fourth part the general program is carried out for low-dimensional and hyperbolic dynamical systems which are particularly amenable to such analysis. In addition these systems have interesting particular properties. For hyperbolic systems there are structural stability, theory of equilibrium (Gibbs) measures, and asymptotic distribution of periodic orbits, in low-dimensional dynamical systems classical Poincare-Denjoy theory, and Poincare-Bendixson theories are presented as well as more recent developments, including the theory of twist maps, interval exchange transformations and noninvertible interval maps.

This book should be on the desk (not bookshelf!) of any serious student of dynamical systems or any mathematically sophisticated scientist or engineer interested in using tools and paradigms of dynamical systems to model or study nonlinear systems. ... Read more

Book Description

This book provides an introduction to the use of geometric partial differential equations in image processing and computer vision. It brings a number of new concepts into the field, providing a very fundamental and formal approach to image processing. State-of-the-art practical results in a large number of real problems are achieved with the techniques described. Applications covered include image segmentation, shape analysis, image enhancement, and tracking. The volume provides information for people investigating new solutions to image processing problems as well as for people searching for existent advanced solutions. ... Read more

Reviews (2)

Sapiro is smart!
Too bad Minnesota is so cold. Maybe we can buy him some gloves
and get another good book out him...

Not really a good book
well, I bought this book to learn image analysis using pde and
this book is not what I expected and unfortuanetly I ended up paying lots of money for it. The only use it has had so far is for me to know a topic's name and look for the mathematical definitions and implementation methods online. Most of the times, it is not clear what it's talking about, does not include real clarifying examples, it jumps from one kind of notation to another kind in the middle of explanation and the math at times is not clear and well written. Definitely not what you would expect from a book which invloves many fomulas/theories and proofs. ... Read more