This book is dedicated to the theory of continuous selections of multi valued mappings, a classical area of mathematics (as far as the formulation of its fundamental problems and methods of solutions are concerned) as well as 'J-n area which has been intensively developing in recent decades and has found various applications in general topology, theory of absolute retracts and infinite-dimensional manifolds, geometric topology, fixed-point theory, functional and convex analysis, game theory, mathematical economics, and other branches of modern mathematics. The fundamental results in this the ory were laid down in the mid 1950's by E. Michael. The book consists of (relatively independent) three parts - Part A: Theory, Part B: Results, and Part C: Applications. (We shall refer to these parts simply by their names). The target audience for the first part are students of mathematics (in their senior year or in their first year of graduate school) who wish to get familiar with the foundations of this theory. The goal of the second part is to give a comprehensive survey of the existing results on continuous selections of multivalued mappings. It is intended for specialists in this area as well as for those who have mastered the material of the first part of the book. In the third part we present important examples of applications of continuous selections. We have chosen examples which are sufficiently interesting and have played in some sense key role in the corresponding areas of mathematics."

Fixed point theory in probabilistic metric spaces can be considered as a part of Probabilistic Analysis, which is a very dynamic area of mathematical research. A primary aim of this monograph is to stimulate interest among scientists and students in this fascinating field. The text is self-contained for a reader with a modest knowledge of the metric fixed point theory. Several themes run through this book. The first is the theory of triangular norms (t-norms), which is closely related to fixed point theory in probabilistic metric spaces. Its recent development has had a strong influence upon the fixed point theory in probabilistic metric spaces. In Chapter 1 some basic properties of t-norms are presented and several special classes of t-norms are investigated. Chapter 2 is an overview of some basic definitions and examples from the theory of probabilistic metric spaces. Chapters 3, 4, and 5 deal with some single-valued and multi-valued probabilistic versions of the Banach contraction principle. In Chapter 6, some basic results in locally convex topological vector spaces are used and applied to fixed point theory in vector spaces. Audience: The book will be of value to graduate students, researchers, and applied mathematicians working in nonlinear analysis and probabilistic metric spaces.

One service mathematics has rendered the human race. It has put common sense back where it belongs. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled discarded nonsense. Eric TBell Every picture tells a story. Advenisement for for Sloan's backache and kidney oils, 1907 The book you have in your hands as you are reading this, is a text on3-dimensional topology. It can serve as a pretty comprehensive text book on the subject. On the other hand, it frequently gets to the frontiers of current research in the topic. If pressed, I would initially classify it as a monograph, but, thanks to the over three hundred illustrations of the geometrical ideas involved, as a rather accessible one, and hence suitable for advanced classes. The style is somewhat informal; more or less like orally presented lectures, and the illustrations more than make up for all the visual aids and handwaving one has at one's command during an actual presentation.

General equilibrium In this book we try to cope with the challenging task of reviewing the so called general equilibrium model and of discussing one specific aspect of the approach underlying it, namely, market completeness. With the denomination "general equilibrium" (from now on in short GE) we shall mainly refer to two different things. On one hand, in particular when using the expression "GE approach", we shall refer to a long established methodolog ical tradition in building and developing economic models, which includes, as of today, an enormous amount of contributions, ranging in number by several 1 thousands * On the other hand, in particular when using the expression "stan dard differentiable GE model", we refer to a very specific version of economic model of exchange and production, to be presented in Chapters 8 and 9, and to be modified in Chapters 10 to 15. Such a version is certainly formulated within the GE approach, but it is generated by making several quite restrictive 2 assumptions * Even to list and review very shortly all the collective work which can be ascribed to the GE approach would be a formidable task for several coauthors in a lifetime perspective. The book instead intends to address just a single issue. Before providing an illustration of its main topic, we feel the obligation to say a word on the controversial character of GE. First of all, we should say that we identify the GE approach as being based 3 on three principles .

This book consists essentially of a collection of papers that were contributed to a national meeting held in Houston, Texas, in 1983. The papers contained herein cover a wide range of electro- chemical engineering topics and should serve as useful starting points in the design of electrochemical cells. The editor would like to thank the authors for their contribu- tions and patience and the typists, Mrs. Susan 'Firth and Mrs. Jeri Saulsbury, for their efforts. Also, the editor would like to thank Mr. T. Nguyen and Ms. M. A. Nictrolson for their help in preparing the index of this book. R. E. White v CONTENTS Design and Development of Electrochemical Chlor-Alkali Cells * * * * * * * * * 1 S. N. Chatterjee A Simple Model of a Diaphragm - Type Chlorine Cell 25 R. E. White, J. S. Beckerdite, and J. Van Zee Design Principles for Chlorine Membrane Cells 61 K. H. Simmrock Hydroxyl Ion Migration, Chemical Reactions, Water Transport and Other Effects as Optimizing Parameters in Cross-, Co- and Countercurrently Operated Membrane Cells for the Chlor/Alkali Electrolysis . *******. **** 89 K. H. Simmrock Hydraulic Modelling as an Aid to Electrochemical Cell Design. 115 I. Wardle Calculating Mechanical Component Voltage Drops in 123 Electrochemical Cells * * * * I. Wardle Electrolysis Cell Design for Ion Exchange Membrane Chlor-Alkali Process * * * * * * * * * * * * 135 M. Seko, A. Yomiyama, and S.

This book examines an abstract mathematical theory, placing special emphasis on results applicable to formal logic. If a theory is especially abstract, it may find a natural home within several of the more familiar branches of mathematics. This is the case with the theory of closure spaces. It might be considered part of topology, lattice theory, universal algebra or, no doubt, one of several other branches of mathematics as well. In our development we have treated it, conceptually and methodologically, as part of topology, partly because we first thought ofthe basic structure involved (closure space), as a generalization of Frechet's concept V-space. V-spaces have been used in some developments of general topology as a generalization of topological space. Indeed, when in the early '50s, one of us started thinking about closure spaces, we thought ofit as the generalization of Frechet V space which comes from not requiring the null set to be CLOSURE SPACES ANDLOGIC XlI closed(as it is in V-spaces). This generalization has an extreme advantage in connection with application to logic, since the most important closure notion in logic, deductive closure, in most cases does not generate a V-space, since the closure of the null set typically consists of the "logical truths" of the logic being examined."

This book is written in a pedagogical style intelligible for graduate students. It reviews recent progress in black-hole and wormhole theory and in mathematical cosmology within the framework of Einstein's field equations and beyond, including quantum effects. This collection of essays, written by leading scientists of long standing reputation, should become an indispensable source for future research.

Absolute measurable space and absolute null space are very old topological notions, developed from well-known facts of descriptive set theory, topology, Borel measure theory and analysis. This monograph systematically develops and returns to the topological and geometrical origins of these notions. Motivating the development of the exposition are the action of the group of homeomorphisms of a space on Borel measures, the Oxtoby-Ulam theorem on Lebesgue-like measures on the unit cube, and the extensions of this theorem to many other topological spaces. Existence of uncountable absolute null space, extension of the Purves theorem and recent advances on homeomorphic Borel probability measures on the Cantor space, are among the many topics discussed. A brief discussion of set-theoretic results on absolute null space is given, and a four-part appendix aids the reader with topological dimension theory, Hausdorff measure and Hausdorff dimension, and geometric measure theory.

1. 1 Preface Many phenomena from physics, biology, chemistry and economics are modeled by di?erential equations with parameters. When a nonlinear equation is est- lished, its behavior/dynamics should be understood. In general, it is impossible to ?nd a complete dynamics of a nonlinear di?erential equation. Hence at least, either periodic or irregular/chaotic solutions are tried to be shown. So a pr- erty of a desired solution of a nonlinear equation is given as a parameterized boundary value problem. Consequently, the task is transformed to a solvability of an abstract nonlinear equation with parameters on a certain functional space. When a family of solutions of the abstract equation is known for some para- ters, the persistence or bifurcations of solutions from that family is studied as parameters are changing. There are several approaches to handle such nonl- ear bifurcation problems. One of them is a topological degree method, which is rather powerful in cases when nonlinearities are not enough smooth. The aim of this book is to present several original bifurcation results achieved by the author using the topological degree theory. The scope of the results is rather broad from showing periodic and chaotic behavior of non-smooth mechanical systems through the existence of traveling waves for ordinary di?erential eq- tions on in?nite lattices up to study periodic oscillations of undamped abstract waveequationsonHilbertspaceswithapplicationstononlinearbeamandstring partial di?erential equations. 1.

General topology is the domain ofmathematics devoted to the investigation of the concepts of continuity and passage to a limit at their natural level of generality. The most basic concepts of general topology, that of a topological space and a continuous map, were introduced by Hausdorffin 1914. Oneofthecentralproblemsoftopologyisthedeterminationandinvestigation of topological invariants; that is, properties ofspaces which are preserved under homeomorphisms. Topological invariants need not be numbers. Connectedness, compactness, andmetrizability, forexample, arenon-numericaltopologicalinvariants.Dimen sional invariants, on the otherhand, areexamplesofnumericalinvariants which take integervalues on specific topological spaces. Part II ofthis book is devoted to them. Topological invariants which take values in the cardinal numbers play an especially important role, providing the raw material for many useful coin" putations. Weight, density, character, and Suslin number are invariants ofthis type. Certain classes of topological spaces are defined in terms of topological in variants. Particularly important examples include the metrizable spaces, spaces with a countable base, compact spaces, Tikhonov spaces, Polish spaces, Cech complete spaces and the symmetrizable spaces."

Tensor Analysis and Nonlinear Tensor Functions embraces the basic fields of tensor calculus: tensor algebra, tensor analysis, tensor description of curves and surfaces, tensor integral calculus, the basis of tensor calculus in Riemannian spaces and affinely connected spaces, - which are used in mechanics and electrodynamics of continua, crystallophysics, quantum chemistry etc.

The book suggests a new approach to definition of a tensor in space R3, which allows us to show a geometric representation of a tensor and operations on tensors. Based on this approach, the author gives a mathematically rigorous definition of a tensor as an individual object in arbitrary linear, Riemannian and other spaces for the first time.

It is the first book to present a systematized theory of tensor invariants, a theory of nonlinear anisotropic tensor functions and a theory of indifferent tensors describing the physical properties of continua.

The book will be useful for students and postgraduates of mathematical, mechanical engineering and physical departments of universities and also for investigators and academic scientists working in continuum mechanics, solid physics, general relativity, crystallophysics, quantum chemistry of solids and material science.

Contents: Introduction. - Fundamental Concepts. - Topological Vector Spaces.- The Quotient Topology. - Completion of Metric Spaces. - Homotopy. - The Two Countability Axioms. - CW-Complexes. - Construction of Continuous Functions on Topological Spaces. - Covering Spaces. - The Theorem of Tychonoff. - Set Theory (by T. Br|cker). - References. - Table of Symbols. -Index.

Property (T) is a rigidity property for topological groups, first formulated by D. Kazhdan in the mid 1960's with the aim of demonstrating that a large class of lattices are finitely generated. Later developments have shown that Property (T) plays an important role in an amazingly large variety of subjects, including discrete subgroups of Lie groups, ergodic theory, random walks, operator algebras, combinatorics, and theoretical computer science. This monograph offers a comprehensive introduction to the theory. It describes the two most important points of view on Property (T): the first uses a unitary group representation approach, and the second a fixed point property for affine isometric actions. Via these the authors discuss a range of important examples and applications to several domains of mathematics. A detailed appendix provides a systematic exposition of parts of the theory of group representations that are used to formulate and develop Property (T).

Many nonlinear problems in physics, engineering, biology and social sciences can be reduced to finding critical points of functionals. While minimax and Morse theories provide answers to many situations and problems on the existence of multiple critical points of a functional, they often cannot provide much-needed additional properties of these critical points. Sign-changing critical point theory has emerged as a new area of rich research on critical points of a differentiable functional with important applications to nonlinear elliptic PDEs.

This book is intended for advanced graduate students and researchers involved in sign-changing critical point theory, PDEs, global analysis, and nonlinear functional analysis.

Quite simply, this book offers the most comprehensive survey to date of the theory of semiparallel submanifolds. It begins with the necessary background material, detailing symmetric and semisymmetric Riemannian manifolds, smooth manifolds in space forms, and parallel submanifolds. The book then introduces semiparallel submanifolds and gives some characterizations for their class as well as several subclasses. The coverage moves on to discuss the concept of main symmetric orbit and presents all known results concerning umbilic-like main symmetric orbits. With more than 40 published papers under his belt on the subject, Lumiste provides readers with the most authoritative treatment.

This book is about an investigation of recent developments in the field of sympletic and contact structures on four- and three-dimensional manifolds from a topologist 's point of view. In it, two main issues are addressed: what kind of sympletic and contact structures we can construct via surgery theory and what kind of sympletic and contact structures are not allowed via gauge theory and the newly invented Heegaard-Floer theory.

This account of the History of General Topology has grown out of the special session on this topic at the American Mathematical Soeiety meeting in San Antonio, Texas, 1993. It was there that the idea grew to publish a book on the historical development of General Topology. Moreover it was feit that it was important to undertake this project while topologists who knew some of the early researchers were still active. Since the first paper by Frechet, "Generalisation d'un theoreme de Weier- strass", C.R.Acad. Sei. 139, 1904, 848-849, and Hausdorff's c1assic book, "Grundzuge der Mengenlehre", Leipzig, 1914, there have been numerous de- velopments in a multitude of directions and there have been many interactions with a great number of other mathematical fields. We have tried to cover as many of these as possible. Most contributions concern either individual topologists, speeific schools, speeific periods, speeific topics or a combination of these.

The book gives a systematic exposition of the diverse ideas and methods in the area, from algebraic topology of manifolds to invariants arising from quantum field theories. The main topics covered include: constructions and classification of homology 3-spheres, Rokhlin invariant, Casson invariant and its extensions, and Floer homology and gauge-theoretical invariants of homology cobordism. Many of the topics covered in the book appear in monograph form for the first time. The book gives a rather broad overview of ideas and methods and provides a comprehensive bibliography. The text will be a valuable source for both the graduate student and researcher in mathematics and theoretical physics.

During the past 25 years, set theory has developed in several interesting directions. The most outstanding results cover the application of sophisticated techniques to problems in analysis, topology, infinitary combinatorics and other areas of mathematics. This book contains a selection of contributions, some of which are expository in nature, embracing various aspects of the latest developments. Amongst topics treated are forcing axioms and their applications, combinatorial principles used to construct models, and a variety of other set theoretical tools including inner models, partitions and trees. Audience: This book will be of interest to graduate students and researchers in foundational problems of mathematics.

From the reviews of the 1st edition:

"This book provides a comprehensive and detailed account of different topics in algorithmic 3-dimensional topology, culminating with the recognition procedure for Haken manifolds and including the up-to-date results in computer enumeration of 3-manifolds. Originating from lecture notes of various courses given by the author over a decade, the book is intended to combine the pedagogical approach of a graduate textbook (without exercises) with the completeness and reliability of a research monograph...

All the material, with few exceptions, is presented from the peculiar point of view of special polyhedra and special spines of 3-manifolds. This choice contributes to keep the level of the exposition really elementary.

In conclusion, the reviewer subscribes to the quotation from the back cover: "the book fills a gap in the existing literature and will become a standard reference for algorithmic 3-dimensional topology both for graduate students and researchers."

Zentralblatt fur Mathematik 2004

For this 2nd edition, new results, new proofs, and commentaries for a better orientation of the reader have been added. In particular, in Chapter 7 several new sections concerning applications of the computer program "3-Manifold Recognizer" have been included. "

Classicalexamples of moreand more oscillatingreal-valued functions on a domain N ?of R are the functions u (x)=sin(nx)with x=(x ,...,x ) or the so-called n 1 1 n n+1 Rademacherfunctionson]0,1[,u (x)=r (x) = sgn(sin(2 ?x))(seelater3.1.4). n n They may appear as the gradients?v of minimizing sequences (v ) in some n n n?N variationalproblems. Intheseexamples,thefunctionu convergesinsomesenseto n ameasure on ? xR, called Young measure. In Functional Analysis formulation, this is the narrow convergence to of the image of the Lebesgue measure on ? by ? ? (?,u (?)). In the disintegrated form ( ) ,the parametrized measure n ? ??? ? captures the possible scattering of the u around ?. n Curiously if (X ) is a sequence of random variables deriving from indep- n n?N dent ones, the n-th one may appear more and more far from the k ?rst ones as 2 if it was oscillating (think of orthonormal vectors in L which converge weakly to 0). More precisely when the laws L(X ) narrowly converge to some probability n measure , it often happens that for any k and any A in the algebra generated by X ,...,X , the conditional law L(X|A) still converges to (see Chapter 9) 1 k n which means 1 ??? C (R) ?(X (?))dP(?)?? ?d b n P(A) A R or equivalently, ? denoting the image of P by ? ? (?,X (?)), n X n (1l ??)d? ?? (1l ??)d[P? ].

This IMA Volume in Mathematics and its Applications FRACTALS IN MULTIMEDIA is a result of a very successful three-day minisymposium on the same title. The event was an integral part of the IMA annual program on Mathemat ics in Multimedia, 2000-2001. We would like to thank Michael F. Barnsley (Department of Mathematics and Statistics, University of Melbourne), Di etmar Saupe (Institut fUr Informatik, UniversiUit Leipzig), and Edward R. Vrscay (Department of Applied Mathematics, University of Waterloo) for their excellent work as organizers of the meeting and for editing the proceedings. We take this opportunity to thank the National Science Foundation for their support of the IMA. Series Editors Douglas N. Arnold, Director of the IMA Fadil Santosa, Deputy Director of the IMA v PREFACE This volume grew out of a meeting on Fractals in Multimedia held at the IMA in January 2001. The meeting was an exciting and intense one, focused on fractal image compression, analysis, and synthesis, iterated function systems and fractals in education. The central concerns of the meeting were to establish within these areas where we are now and to develop a vision for the future."

This text on contact topology is a comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg's classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology. Here the focus is on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. This book serves both as a self-contained introduction to the subject for advanced graduate students and as a reference for researchers.

Based on a course given to talented high-school students at Ohio University in 1988, this book is essentially an advanced undergraduate textbook about the mathematics of fractal geometry. It nicely bridges the gap between traditional books on topology/analysis and more specialized treatises on fractal geometry. The book treats such topics as metric spaces, measure theory, dimension theory, and even some algebraic topology. It takes into account developments in the subject matter since 1990. Sections are clear and focused. The book contains plenty of examples, exercises, and good illustrations of fractals, including 16 color plates.

The book covers topics in the theory of algebraic transformation groups and algebraic varieties which are very much at the frontier of mathematical research.