Book Description

The monograph presents some of the authors' recent and originalresults concerning boundedness and compactness problems in Banachfunction spaces both for classical operators and integral transformsdefined, generally speaking, on nonhomogeneous spaces. It focuses onintegral operators naturally arising in boundary value problems forPDE, the spectral theory of differential operators, continuum andquantum mechanics, stochastic processes, etc. The book may beconsidered as a systematic and detailed analysis of a large class ofspecific integral operators from the boundedness and compactness pointof view. A characteristic feature of the monograph is that most of thestatements proved here have the form of criteria. We provide a list ofproblems which were open at the time of completion of the book.Audience: The book is aimed at a rather wide audience, rangingfrom researchers in functional and harmonic analysis to experts inapplied mathematics and prospective students. ... Read more

Book Description

Here is a genuine introduction to the differential geometry of plane curves for undergraduates in mathematics, or postgraduates and researchers in the engineering and physical sciences. This well-illustrated text contains several hundred worked examples and exercises, making it suitable for adoption as a course text. Key concepts are illustrated by named curves, of historical and scientific significance, leading to the central idea of curvature. The author introduces the core material of classical kinematics, developing the geometry of trajectories via the ideas of roulettes and centrodes, and culminating in the inflexion circle and cubic of stationary curvature. ... Read more

Book Description

In topology the three basic concepts of metrics, topologies and uniformities have been treated so far as separate entities by means of different methods and terminology. This is the first book to treat all three as a special case of the concept of approach spaces. ... Read more

Book Description

Elliptic cohomology is an extremely beautiful theory with bothgeometric and arithmetic aspects. The former is explained by the factthat the theory is a quotient of oriented cobordism localised awayfrom 2, the latter by the fact that the coefficients coincide with aring of modular forms. The aim of the book is to construct thiscohomology theory, and evaluate it on classifying spaces BG of finitegroups G. This class of spaces is important, since (using ideasborrowed from `Monstrous Moonshine') it is possible to give abundle-theoretic definition of EU-(BG). Concluding chapters alsodiscuss variants, generalisations and potential applications. ... Read more

Book Description

Self-contained, and collating for the first time material that has until now only been published in journals - often in Russian - this book will be of interest to functional analysts, especially those with interests in topological vector spaces, and to algebraists concerned with category theory. The closed graph theorem is one of the corner stones of functional analysis, both as a tool for applications and as an object for research. However, some of the spaces which arise in applications and for which one wants closed graph theorems are not of the type covered by the classical closed graph theorem of Banach or its immediate extensions. To remedy this, mathematicians such as Schwartz and De Wilde (in the West) and Rajkov (in the East) have introduced new ideas which have allowed them to establish closed graph theorems suitable for some of the desired applications. In this book, Professor Smirnov uses category theory to provide a very general framework, including the situations discussed by De Wilde, Rajkov and others. General properties of the spaces involved are discussed and applications are provided in measure theory, global analysis and differential equations. ... Read more

Book Description

Book Description

Professor Atiyah is one of the greatest living mathematicians and is well known throughout the mathematical world.He is a recipient of the Fields Medal, the mathematical equivalent of the Nobel Prize, and is still at the peak of his career.His huge number of published papers, focusing on the areas of algebraic geometry and topology, have here been collected into six volumes, divided thematically for easy reference by individuals interested in a particular subject. This six volume set of the collected works of professor Sir Michael Atiyah includes: Collected Works: Volume 1: Early Papers; General papers Collected Works: Volume 2: K-Theory Collected Works: Volume 3: Index Theory: 1 Collected Works: Volume 4: Index Theory: 2 Collected Works: Volume 5: Gauge Theories New for 2004-Collected Works: Volume 6: Publications since 1987, including his work on skyrmions, "Atiyah's axioms" for topological quantum field theories, monopolies, knots, K-Theory, equivalent problems, point particles, and M-Theory. ... Read more

Book Description

This volume is the first one to introduce a unified and homogeneous exposition of the theory of almost automorphic functions and its applications to the fast growing field of differential equations in abstract spaces (Banach and Hilbert spaces). It is based essentially on the work of M. Zaki, S. Zaidman, and the author during the last three decades. It is also the first one to present the theory of almost periodic functions in general topological vector spaces. ... Read more

Reviews (1)

A good introduction
This book is a short and elementary introduction to the Seiberg-Witten equations, which created quite a stir back in 1994 when they were first proposed. The book is elementary enough that it could be read by someone without a background in the intricacies of the geometry and topology of 4-dimensional topological and smooth manifolds, but the results can be better appreciated if one already has such a background. A background in quantum field theory, specifically the guage theory of the strong interaction, called quantum chromodynamics, will also help in the appreciation of the book. A lot of work has been done in elucidating the properties of the Seiberg-Witten equations since this book was written, but the book could still serve as an introduction to these developments.

The author gives a brief introduction to the use of Seiberg-Witten equations in chapter 1, along with a review of the background needed from the theory of vector bundles, differential geometry, and algebraic topology needed to read the book. All of this background is pretty standard, although the appearance of spin structures may not be as familiar to the mathematician-reader, but completely familiar to the physicist reader. Detailed proofs of the main results are not given, but reference to these are quoted. Also, the theory of characteristic classes is outlined only briefly so no insight is given as to why they work so well. In particular, the reason for the vanishing of the second Stiefel-Whitney class as a precondition for the manifold having a spin structure is not given.

In chapter 2, the author goes into the spin geometry of 4-manifolds in more detail. After discussing the role of quaternions in this regard, spin structures are defined. A spin structure on a manifold M, via its cocycle condition, give two complex vector bundles of rank two over M. The complexified tangent bundle can thus be represented in terms of these vector bundles, which are themselves quaternionic line bundles over M. The author also defines spin(c) structures, and shows how, using an almost complex structure, to obtain a canonical spin(c) structure on a complex manifold of complex dimension two. The spin(c) structure also allows a construction of the "virtual vector bundles" W+, W-, and L, for manifolds that do not have a spin structure. These bundles play a central role in the book. Clifford algebra becomes meaningful on the direct sum W of W+ and W-, and spin connections can be defined on W. In particular given a unitary connection on a complex line bundle L over a spin manifold M, one can obtain a connection on the tensor product of W and L. When M is not a spin manifold, this is still possible but one must use the "square" L^2 of L. One can then define the Dirac operator over the sections of this tensor product, which the author does and extends it to one with coefficients in a general vector bundle. The author then discusses, but does not prove, the Atiyah-Singer index theorem and the Hirzebruch signature theorem. These theorems, the author emphasizes, are proved in the context of linear partial differential equations, and give invariants of 4-manifolds.

This sets up the discussion in chapter 3, which deals with the problem of how to find invariants of 4-manifolds if one works in the context of nonlinear partial differential equations. Those familiar with the Donaldson theory, which was done using the (nonlinear!) Yang-Mills equations, will understand the difficulties of this approach. The strategy of the nonlinear approach as outlined by the author is to show that the solution set of a nonlinear PDE is compact and a finite-dimensional compact manifold. The solution set depends on the Riemannian metric, but its cobordism class does not, and this may give a topological invariant. The fact that it is defined in terms of a PDE might give a way of distinguishing smooth structures.

The Seiberg-Witten theory is one method for doing this. The Seiberg-Witten equations are nonlinear, but the nonlinearity is "soft" enough that it can be dealt with. They arise in the context of oriented 4-dimensional Riemannian manifolds with a spin(c) structure and a positive spinor bundle W+ tensored with L. A connection on L^2 and a section of this spinor bundle are chosen to satisfy these equations, which involve the self-dual part of the connection. One also needs to work with the "perturbed" Seiberg Witten equations, where a self-dual two-form is added. The moduli space of the solutions to the perturbed Seiberg-Witten equations is shown to form a compact finite-dimensional manifold. The proof follows essentially from the Weitzenbock formula, the Sobolev embedding theorem, and Rellich's theorem. Sard's theorem shows that the moduli space is smooth and the Fredholm theory shows it is oriented. The Seiberg-Witten invariants are associated to virtual complex line bundles over the 4-manifold, and when the dimension of the self-dual harmonic two-forms is greater than or equal to 2, and the dimension of the moduli space is even. Their definition does involve the Riemannian metric, but changing this metric only alters the moduli space by a cobordism. It is proved that oriented Riemannian manifolds with positive scalar curvature have vanishing Seiberg-Witten invariants. Kahler surfaces are shown to have positive Seiberg-Witten invariants, and the author proves that there is a compact topological manifold with infinitely many distinct smooth structures. Unfortunately though, an explicit example of one of these is not given. Such an example may be very important from the standpoint of physics, for the behavior of dynamical systems or quantum field theories might be very different for different smooth structures. ... Read more

Book Description

Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development. The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial, related articles. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is relatively recent and appears for the first time in book format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth settings. ... Read more

Reviews (3)

A well-written book
This is a well-written book. While containing
cutting-edge topics in analysis on metric spaces,
it is also accessible to students who have had
a good course in measure theory, a reasonable course
in topology, and a nodding aquaintance with
functional analysis!

this book is the greatest
this book may well save the universe

this book is the greatest book ever written
this book may well save the universe ... Read more

Book Description

Digital topology deals with properties of digital images that correspond to topological properties of the objects represented in those images. Concepts and results of digital topology are used to specify and justify important image-processing operations, such as thinning and boundary surface extraction.This book is a coherent introduction to and a reference on digital topology.The book includes a concise but self-contained introduction that covers the parts of graph theory and topology that are needed in the sequel. There is substantial coverage of finite and locally finite topological spaces, which are discussed in very few other texts. Among topics covered are: *The boundary tracking problem with applications and methodology*Reductive andreductive-augmentative shrinking algorithms*Topology-preserving deletion of 1's in 2-4 dimensional digital pictures*Topological soundness verification of parallelthinning algorithms*Continuous and metric analogs*The approximation of compact Hausdorff spaces by finite spacesIts readership includes both computer scientists concerned with applications of the subject and mathematicians interested in thenumerous theoretical questions that arise in connection with those applications. ... Read more

Book Description

GEOMETRY: Plane and Fancy offers students a fascinating tour through parts of geometry they are unlikely to see in the rest of their studies while, at the same time, anchoring their excursions to the well known parallel postulate of Euclid. The author shows how alternatives to Euclid's fifth postulate lead to interesting and different patterns and symmetries. In the process of examining geometric objects, the author incorporates the algebra of complex (and hypercomplex) numbers, some graph theory, and some topology. Nevertheless, the book has only mild prerequisites. Readers are assumed to have had a course in Euclidean geometry (including some analytic geometry and some algebra) at the high school level. While many concepts introduced are advanced, the mathematical techniques are not. Singer's lively exposition and off-beat approach will greatly appeal both to students and mathematicians. Interesting problems are nicely scattered throughout the text. The contents of the book can be covered in a one-semester course, perhaps as a sequel to a Euclidean geometry course. ... Read more

Reviews (1)

Beware!
While some of these complaints may be due to the course that used this book, I do believe that this is one of the worst math books I've ever read.

The book has several problems. The most glaring problem is its vagueness. None of the (117) figures are labeled. None of the questions are numbered, or given any other identifying mark. Some questions refer to figures on different pages, but simply say "look at [the figure]," only to leave the reader with the thought "what figure." Or to suggest one figure, though to use a different figure all together. Coupled with the fact that many theorems are not stated, or are not identified and are rarely proven. Stylistically, this book is just a mess.

If that weren't bad enough, many of the figures look like clipart images from 1991. The book is copyright 1998, suggesting to me that if he spent 7 (or more) years writing this book, maybe he should have redone some of the figures so they're not such an eyesore. Maybe this is a minor point, but if you're going to advertise the fact that you have 117 images, maybe you should make those images look nice.

As for the books good points, the information presented is in a logical fashion, and is correct. The text is easy to read (in that the font is not too small, or strange), and the pages are sturdy enough to write on them if you need to play with the figures.

For any teachers/professors reading this, I would not recommend using this book as a primary source for your course. Maybe it would be ok as a supplementary source.

Finally, if you were considering using this book for the topic on graph theory, then I would suggest instead using "Introduction to Graph Theory" by Robin J. Wilson (ISBN: 0-582-24993-7) ... Read more

Book Description

This is the first systematic study of best approximation theory in inner product spaces and, in particular, in Hilbert space. Geometric considerations play a prominent role in developing and understanding the theory. The only prerequisite for reading the book is some knowledge of advanced calculus and linear algebra. Throughout the book, examples and applications have been interspersed with the theory. Each chapter concludes with numerous exercises and a section in which the author puts the results of that chapter into a historical perspective. The book is based on lecture notes for a graduate course on best approximation which the author has taught for over 25 years. ... Read more

Book Description

Translated and Revised by C. Cecilia Krieger. Critically acclaimed text presents detailed theory of Fréchet (V) spaces and a comprehensive examination of their relevance to topological spaces, plus in-depth discussions of metric and complete spaces. Numerous exercises reinforce teachings of each chapter. "...an elegant piece of work suitable for the beginning student and the mature mathematician."—Scripta Mathematica. 2nd ed.
... Read more

Book Description

Hardbound. These papers survey the developments in General Topology and the applications of it which have taken place since the mid 1980s. The book may be regarded as an update of some of the papers in the Handbook of Set-Theoretic Topology (eds. Kunen/Vaughan, North-Holland, 1984), which gives an almost complete picture of the state of the art of Set Theoretic Topology before 1984. In the present volume several important developments are surveyed that surfaced in the period 1984-1991.

This volume may also be regarded as a partial update of Open Problems in Topology (eds. van Mill/Reed, North-Holland, 1990). Solutions to some of the original 1100 open problems are discussed and new problems are posed. ... Read more

Book Description

This significantly expanded second edition of Riemann, Topology and Physics combines a fascinating account of the life and work of Bernhard Riemann with a lucid discussion of current interaction between topology and physics. The author, a distinguished mathematical physicist, takes into account his own research at the Riemann archives of Gttingen University and developments over the last decade that connect Riemann with numerous significant ideas and methods reflected throughout contemporary mathematics and physics.

Special attention is paid in part one to new results on the Riemann-Hilbert problem and, in part two, to recent discoveries in field theory and condensed matter such as the quantum Hall effect, quasicrystals, membranes with nontrivial topology, 'fake' differential structures on 4-dimensional Euclidean space, new invariants of knots and more. In his relatively short lifetime, this great mathematician made outstanding contributions to nearly all branches of mathematics; today his name appears prominently throughout the literature. The references to the literature have been substantially updated in this new edition. ... Read more

Reviews (1)

Excellent
This is a translation of two quite small Russian books. The first is a biography of Riemann, the second a review of "Geometrical" physics, recently updated. Even the biography includes some mathematics which I did not fully understand, but it is a great introduction, and it's good to get your feet a little wet! The book is small and very readable. I recommend it to students interested in understanding the relationships between physics, mathematics, and geometry. A gentle introduction to the "technical" journey. ... Read more