Book Description

The Darboux transformation approach is one of the most effective methods for constructing explicit solutions of partial differential equations which are called integrable systems and play important roles in mechanics, physics and differential geometry.

This book presents the Darboux transformations in matrix form and provides purely algebraic algorithms for constructing the explicit solutions. A basis for using symbolic computations to obtain the explicit exact solutions for many integrable systems is established. Moreover, the behavior of simple and multi-solutions, even in multi-dimensional cases, can be elucidated clearly. The method covers a series of important equations such as various kinds of AKNS systems in R¹⁺ⁿ, harmonic maps from 2-dimensional manifolds, self-dual Yang-Mills fields and the generalizations to higher dimensional case, theory of line congruences in three dimensions or higher dimensional space etc. All these cases are explained in detail. This book contains many results that were obtained by the authors in the past few years.

Audience: The book has been written for specialists, teachers and graduate students (or undergraduate students of higher grade) in mathematics and physics.

Book Description

This invaluable book contains selected papers of Prof Chuan-Chih Hsiung, renowned mathematician in differential geometry and founder and editor-in-chief of a unique international journal in this field, the Journal of Differential Geometry.

During the period of 1935-1943, Prof Hsiung was in China working on projective differential geometry under Prof Buchin Su. In 1946, he went to the United States, where he gradually shifted to global problems. Altogether Prof Hsiung has published about 100 research papers, from which he has selected 64 (in chronological order) for this volume. ... Read more

Book Description

Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research. The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme.

This methodically designed book contains a rich collection of exercises, examples, and illustrations within each individual chapter, concluding with an extensive bibliography of monographs, research papers, and a thorough index. Seeking to capture the imagination of advanced undergraduate and graduate students with a basic background in complex analysis—and also to spark the interest of seasoned workers in the field—the book imparts a solid education both in complex analysis and in how modern mathematics works.

Book Description

The focus in this book is on the theory of regulators and secondary invariants, with articles written and refereed by experts in their respective fields. A short historical and mathematical overview of the theory of regulators from its number theoretic origins, and its connections to analysis, topology, differential geometry, and algebra, is presented by the editors in the introduction, with key topics noted as follows: hyperbolic volume and the Borel regulator, the Chern-Simons invariant, the Bloch-Beilinson regulator, polylogarithms (classical and elliptic), p-adic analalogues, and analytic torsion.

Contributors include:D. Blasius, S. Bloch, C. Deninger, H. Esnault, H. Gangl, A. Goncharov, K. Khler, K. Knnemann, A. Levin, J. Lott, V. Maillot, J. Rogawski, and J. Wildeshaus.

This work is an outgrowth of a conference held at the Hebrew University in Jerusalem on "Regulators in Analysis, Geometry and Number Theory", and should appeal to a broad audience of graduate students and research mathematicians.

Series: Progress in Mathematics, Vol. 171 ... Read more

Book Description

This well-organized and coherent collection of papers leads the reader to the frontiers of  present research in the theory of nonlinear partial differential equations and the calculus of variations and offers insight into some exciting developments. In addition, most articles also provide an excellent introduction to their background, describing extensively as they do the history of those problems presented, as well as the state of the art and offer a well-chosen guide to the literature. Part I contains the contributions of geometric nature: From spectral theory on regular and singular spaces to regularity theory of solutions of variational problems.Part II consists of articles on partial differential equations which originate from problems in physics, biology and stochastics. They cover elliptic, hyperbolic and parabolic cases. ... Read more

Book Description

The papers in this book originate from lectures which were held at the „Vienna Workshop on Nonlinear Models and Analysis“ – May 20–24, 2002. They represent a cross-section of the research field Applied Nonlinear Analysis with emphasis on free boundaries, fully nonlinear partial differential equations, variational methods, quasilinear partial differential equations and nonlinear kinetic models. ... Read more

Book Description

This book is a compendium of survey lectures presented at aconference on Riemannian Geometry sponsored by The FieldsInstitute for Research in Mathematical Sciences (Waterloo,Canada) in August 1993. Attended by over 80 participants, theaim of the conference was to promote research activity inRiemannian geometry. A select group of internationallyestablished researchers in the field were invited to discuss andpresent current developments in a selection of contemporarytopics in Riemannian geometry. This volume contains four of thefive survey lectures presented at the conference.

Features:

Basic notions of volume and entropy and the difficult and deeprelations of these invariants to curvature.

$LP$ cohomology, in which the methods combine various areas ofmathematics going beyond Riemannian geometry.

Curvature inequalities from a general point of view, leading tothe study of general spaces. ... Read more

Reviews (5)

A must for mathematicians interested in cosmology
Besides providing in clear-cut fashion the mathematics essential to research in cosmology, the authors simplify many concepts the physicists make opaque.

Excellent, But Has Flawed Editing
I found this book much more comprehensible than the Bishop & Goldberg text (perhaps because the specifics of my particular graduate program made this approach more accessible). The only problem is that in some important areas of the text there are typographical errors; this text needs a new corrected edition--otherwise I have no complaints. The only recommendation I would make to a prospective reader is to obtain a copy of the excellent, old, out-of-print (but still readily avilable used, on the web) Vector and Tensor Analysis by Louis Brand and master the tensor chapter in it. This will prepare the reader to gain an idea of where the authors are heading with their modern, abstract functional analysis approach.
The reader will be greatly assisted by a solid understanding of linear algebra and a preparatory course in functional analysis wouldn't hurt, either. This material is challenging even for a math graduate student; any high school student who could master this book would have to be gifted, indeed.

Good introduction
I agree with previous reviewers, and only wish to add a few comments: 1. This book assumes very little on the part of the reader, which makes it ideal for beginners, as long as they're mature readers. 2. Like many books out there, everything in this book is real and finite dimensional, which is a bit disappointing. 3. It's not as advanced as the writers or reviewers would like to think. For instance, no differential forms, no killing vectors, and although there's a chapter on lie groups it treats only their geometrical aspects and not the algebraic ones.4. However, it contains two (extensive) chapters on SR and GR which are pure gold, I say! Everything is done from the geometricalpoint of view, and only AFTER all of the math has been introduced, so the discussion is mature and elegant.In short, this is a good book to read for the geometrical intuition but don't count on it to explain everything about differential geometry. Enjoy!

Serious math for mathematical readers, but excellent
I found learning GR very frustrating, as I am a mathematician by training and found the woolly "picture the vector" approach of say Graviatation (Misner et al) very loose. it drove me crazy. When I found this I was pleased, and found thye rigourous footing I wanted for the subject. Bundles! After this I could read Gravitation with some comfort.

NOT repeat NOT for anyone but mathematically trained readers.

A great, usefull book that is easy to understand
Considering the fact that I am a high school student and I had no problem understanding the majority of this text, I would call this book wonderful! It is a book that no student of general relativity or differential geometrycan do without. It develops insightful geometric premises early on so thatthe whole picture of tensors can be absorbed, not just those"definitive" transformation equations. The book is not dry likemost math books either. It contains almost witty sections as well asenlightening mathematical ideas. I would recommend this book to anyonestudying tensors on their own, or as a supplementary text book for a class. ... Read more

Book Description

Appliies variational methods and critical point theory on infinite dimenstional manifolds to some problems in Lorentzian geometry which have a variational nature, such as existence and multiplicity results on geodesics and relations between such geodesics and the topology of the manifold. ... Read more

Book Description

This is the first account in book form of the theory of harmonic morphisms between Riemannian manifolds. Harmonic morphisms are maps which preserve Laplace's equation. They can be characterized as harmonic maps which satisfy an additional first order condition. Examples include harmonic functions, conformal mappings in the plane, and holomorphic functions with values in a Riemann surface. There are connections with many conepts in differential geometry, for example, Killing fields, geodesics, foliations, Clifford systems, twistor spaces, Hermitian structures, iso-parametric mappings, and Einstein metrics and also the Brownain pathpreserving maps of probability theory. Giving a complete account of the fundamental aspects of the subject, this book is self-contained, assuming only a basic knowledge of differential geometry. ... Read more

Book Description

The geometrical theory of nonlinear differential equations originates from classical works by S. Lie and A. Backlund. It obtained a new impulse in the sixties when the complete integrability of the Korteweg--de Vries equation was found and it became clear that some basic and quite general geometrical and algebraic structures govern this property of integrability.Nowadays the geometrical and algebraic approach to partial differential equations constitutes a special branch of modern mathematics.In 1993, a workshop on algebra and geometry of differential equations took place at the University of Twente (The Netherlands), where the state-of-the-art of the main problems was fixed. This book contains a collection of invited lectures presented at this workshop. The material presented is of interest to those who work in pure and applied mathematics and especially in mathematical physics. ... Read more

Book Description

In 1920, Pierre Fatou expressed the conjecture that--except for special cases--all critical points of a rational map of the Riemann sphere tend to periodic orbits under iteration. This conjecture remains the main open problem in the dynamics of iterated maps. For the logistic family x- ax(1-x), it can be interpreted to mean that for a dense set of parameters "a," an attracting periodic orbit exists. The same question appears naturally in science, where the logistic family is used to construct models in physics, ecology, and economics.

In this book, Jacek Graczyk and Grzegorz Swiatek provide a rigorous proof of the Real Fatou Conjecture. In spite of the apparently elementary nature of the problem, its solution requires advanced tools of complex analysis. The authors have written a self-contained and complete version of the argument, accessible to someone with no knowledge of complex dynamics and only basic familiarity with interval maps. The book will thus be useful to specialists in real dynamics as well as to graduate students. ... Read more