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In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;
methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and
methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.

As a result, the book represents a blend of new methods in general computational analysis,
and specific, but also generic, techniques for study of systems theory ant its particular
branches, such as optimal filtering and information compression.

- Best operator approximation,
- Non-Lagrange interpolation,
- Generic Karhunen-Loevetransform
- Generalised low-rank matrix approximation
- Optimal data compression
- Optimal nonlinear filtering ... Read more

Customer Reviews (1)

'Must read' for Theoretical Physicists
The Preface to a book can sometimes say a lot about the book itself...and this one took me by storm. I am one of the many who struggle to find Physical connections to mathematical equations..partly due to my lack of insight, and partly because I have been trained as a Physicist and I fail to understand the subtle aspects of Pure Mathematics vis a vis nature..until I found this book which instilled the idea that Mathematics is the mother of all sciences..I think I am in a better position to enjoy Physics now ! ... Read more

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Presents a conceptual underpinning for multivariable calculus that is as natural and intuitively simple as possible.This book focuses on modeling physical phenomena, especially from physics and engineering, and on developing geometric intuition. Geometric intuition is particularly stressed. The synthetic, coordinate-free geometries of 2- and 3-dimensional Euclidean spaces (E^2 and E^3) have a primary role. Wherever possible, coordinate-free definitions are used ... Read more

Customer Reviews (3)

Excellent book for a mathematical physicist.
This book offers a unique (in my experience) mix of visually oriented motivation, mathematical clarity & rigor, and examples from classical physics.It is well organized and (relatively) easy to understand, and it has a wealth of good exercises, many with answers.My only wish is for a solutions book.

A good book
I took Professor Rogers' class four years ago (based on a draft of the book). I still usethis book as a reference for my research.It is much better than othercalculus texts that I've consulted. By the way, pleasenote that'fred@mit.edu' does not exist. Maybe he dropped out afterfailing the class...

A good book from a great teacher
Perhaps I have a bias because the author was my professor for the course, but this is a high quality calculus text. The book clearly explains the subject matter with useful examples which are accompanied by appropriateproblems. BTW, I think the comment below is funny. You should ignore it... ... Read more

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This book provides a comprehensive discussion on the existence and regularity of minima of regular integrals in the calculus of variations and of solutions to elliptic partial differential equations and systems of the second order. While direct methods for the existence of solutions are well known and have been widely used in the last century, the regularity of the minima was always obtained by means of the Euler equation as a part of the general theory of partial differential equations. In this book, using the notion of the quasi-minimum introduced by Giaquinta and the author, the direct methods are extended to the regularity of the minima of functionals in the calculus of variations, and of solutions to partial differential equations. This unified treatment offers a substantial economy in the assumptions, and permits a deeper understanding of the nature of the regularity and singularities of the solutions. The book is essentially self-contained, and requires only a general knowledge of the elements of Lebesgue integration theory. ... Read more

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This book is divided into two parts. The first addresses the simpler variational problems in parametric and nonparametric form. The second covers extensions to optimal control theory. The author opens with the study of three classical problems whose solutions led to the theory of calculus of variations. They are the problem of geodesics, the brachistochrone, and the minimal surface of revolution. He gives a detailed discussion of the Hamilton-Jacobi theory, both in the parametric and nonparametric forms. This leads to the development of sufficiency theories describing properties of minimizing extremal arcs. Next, the author addresses existence theorems. He first develops Hilbert's basic existence theorem for parametric problems and studies some of its consequences. Finally, he develops the theory of generalized curves and ``automatic'' existence theorems. In the second part of the book, the author discusses optimal control problems. He notes that originally these problems were formulated as problems of Lagrange and Mayer in terms of differential constraints. In the control formulation, these constraints are expressed in a more convenient form in terms of control functions. After pointing out the new phenomenon that may arise, namely, the lack of controllability, the author develops the maximum principle and illustrates this principle by standard examples that show the switching phenomena that may occur. He extends the theory of geodesic coverings to optimal control problems. Finally, he extends the problem to generalized optimal control problems and obtains the corresponding existence theorems. ... Read more

Product Description

This book is developed for the study of vectorial problems in the calculus of variations. The subject is a very active one and almost half of the book consists of new material. This is a new edition of the earlier book published in 1989 and it is suitable for graduate students. The book has been updated with some new material and examples added. Applications are included.

Product Description
This monograph (in two volumes) deals with non scalar variational problems arising in geometry, as harmonic mappings between Riemannian manifolds and minimal graphs, and in physics, as stable equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and accessible to non specialists. Topics are treated as far as possible in an elementary way, illustrating results with simple examples; in principle, chapters and even sections are readable independently of the general context, so that parts can be easily used for graduate courses. Open questions are often mentioned and the final section of each chapter discusses references to the literature and sometimes supplementary results. Finally, a detailed Table of Contents and an extensive Index are of help to consult this monograph ... Read more

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Originally published in 1891.This volume from the Cornell University Library's print collections was scanned on an APT BookScan and converted to JPG 2000 format by Kirtas Technologies.All titles scanned cover to cover and pages may include marks notations and other marginalia present in the original volume. ... Read more

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This volume is produced from digital images created through the University of Michigan University Library's large-scale digitization efforts. The Library seeks to preserve the intellectual content of items in a manner that facilitates and promotes a variety of uses. The digital reformatting process results in an electronic version of the original text that can be both accessed online and used to create new print copies. The Library also understands and values the usefulness of print and makes reprints available to the public whenever possible. This book and hundreds of thousands of others can be found in the HathiTrust, an archive of the digitized collections of many great research libraries. For access to the University of Michigan Library's digital collections, please see http://www.lib.umich.edu and for information about the HathiTrust, please visit http://www.hathitrust.org ... Read more

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This monograph is unique in its treatment of the application of methods of nonstandard analysis to the theory of curves in the calculus of variations. It will be of particular value to researchers in the calculus of variations and optimal control theory. ... Read more

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A major problem in current applied mathematics is the lack ofefficient and accurate techniques to solve optimization problems inthe calculus of variations and optimal control theory. This issurprising since problems occur throughout many areas of appliedmathematics, engineering, physical sciences, economics, andbiomedicine. For instance, these techniques are used to solve rockettrajectory problems, current flow problems in electronicsmanufacturing, and financial risk problems in investing. The authorshave written a unique book to remedy this problem. The first half ofthe book contains classical material in the field, the second halfunique theoretical and numerical methods for constrained problems. ... Read more

Customer Reviews (1)

An excellent textbook for students
The book covers ordinary optimization problems (constrained and unconstrained) and nonstochastic continuous-time dynamic problems. Its chapter on numerical theory and its appendix on related books areparticular useful. The book presents optimization problems in a veryunderstandable way. With many examples and graphic illustrations, studentswould be able to understand the materials easily. I would use it as atextbook for students in economics. ... Read more

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Morse theory is a study of deep connections between analysis and topology. In its classical form, it provides a relationship between the critical points of certain smooth functions on a manifold and the topology of the manifold. It has been used by geometers, topologists, physicists, and others as a remarkably effective tool to study manifolds. In the 1980s and 1990s, Morse theory was extended to infinite dimensions with great success. This book is Morse's own exposition of his ideas. It has been called one of the most important and influential mathematical works of the twentieth century. Calculus of Variations in the Large is certainly one of the essential references on Morse theory. ... Read more

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This volume is produced from digital images from the Cornell University Library Historical Mathematics Monographs collection. ... Read more

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This monograph explores various aspects of the inverse problem of the calculus of variations for systems of ordinary differential equations. The main problem centers on determining the existence and degree of generality of Lagrangians whose system of Euler-Lagrange equations coincides with a given system of ordinary differential equations. The authors rederive the basic necessary and sufficient conditions of Douglas for second order equations and extend them to equations of higher order using methods of the variational bicomplex of Tulcyjew, Vinogradov, and Tsujishita. What emerges is a fundamental dichotomy between second and higher order systems: the most general Lagrangian for any higher order system can depend only upon finitely many constants. The authors present an algorithm, based upon exterior differential systems techniques, for solving the inverse problem for second order equations. A number of new examples illustrate the effectiveness of this approach. The monograph also contains a study of the inverse problem for a pair of geodesic equations arising from a two dimensional symmetric affine connection. The various possible solutions to the inverse problem for these equations are distinguished by geometric properties of the Ricci tensor. ... Read more

Product Description
This volume is produced from digital images created through the University of Michigan University Library's large-scale digitization efforts. The Library seeks to preserve the intellectual content of items in a manner that facilitates and promotes a variety of uses. The digital reformatting process results in an electronic version of the original text that can be both accessed online and used to create new print copies. The Library also understands and values the usefulness of print and makes reprints available to the public whenever possible. This book and hundreds of thousands of others can be found in the HathiTrust, an archive of the digitized collections of many great research libraries. For access to the University of Michigan Library's digital collections, please see http://www.lib.umich.edu and for information about the HathiTrust, please visit http://www.hathitrust.org ... Read more

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This is an EXACT reproduction of a book published before 1923. This IS NOT an OCR'd book with strange characters, introduced typographical errors, and jumbled words.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
This is a reproduction of a book published before 1923.This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process.We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide.We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ... Read more

Product Description
Publisher: Cambridge [Eng.], Printed by J. SmithPublication date: 1810Subjects: Calculus of variationsNotes: This is an OCR reprint. There may be typos or missing text. There are no illustrations or indexes.When you buy the General Books edition of this book you get free trial access to Million-Books.com where you can select from more than a million books for free. You can also preview the book there. ... Read more