This updated and revised edition of a widely acclaimed and successful text for undergraduates examines topology of recent compact surfaces through the development of simple ideas in plane geometry. Containing over 171 diagrams, the approach allows for a straightforward treatment of its subject area. It is particularly attractive for its wealth of applications and variety of interactions with branches of mathematics, linked with surface topology, graph theory, group theory, vector field theory, and plane Euclidean and non-Euclidean geometry.
Examines topology of recent compact surfaces through the development of simple ideas in plane geometryContains a wealth of applications and a variety of interactions with branches of mathematics, linked with surface topology, graph theory, group theory, vector field theory, and plane Euclidean and non-Euclidean geometry

The central theme of this book is the study of self-dual connections on four-manifolds. The author's aim is to present a lucid introduction to moduli space techniques (for vector bundles with SO (3) as structure group) and to apply them to four-manifolds. The authors have adopted a topologists' perspective. For example, they have included some explicit calculations using the Atiyah-Singer index theorem as well as methods from equivariant topology in the study of the topology of the moduli space. Results covered include Donaldson's Theorem that the only positive definite form which occurs as an intersection form of a smooth four-manifold is the standard positive definite form, as well as those of Fintushel and Stern which show that the integral homology cobordism group of integral homology three-spheres has elements of infinite order. Little previous knowledge of differential geometry is assumed and so postgraduate students and research workers will find this both an accessible and complete introduction to currently one of the most active areas of mathematical research.

This textbook offers an accessible, modern introduction at undergraduate level to an area known variously as general topology, point-set topology or analytic topology with a particular focus on helping students to build theory for themselves. It is the result of several years of the authors' combined university teaching experience stimulated by sustained interest in advanced mathematical thinking and learning, alongside established research careers in analytic topology. Point-set topology is a discipline that needs relatively little background knowledge, but sufficient determination to grasp ideas precisely and to argue with straight and careful logic. Research and long experience in undergraduate mathematics education suggests that an optimal way to learn such a subject is to teach it to yourself, pro-actively, by guided reading of brief skeleton notes and by doing your own spadework to fill in the details and to flesh out the examples. This text will facilitate such an approach for those learners who opt to do it this way and for those instructors who would like to encourage this so-called 'Moore approach', even for a modest segment of the teaching term or for part of the class. In reality, most students simply do not have the combination of time, background and motivation needed to implement such a plan fully. The accessibility, flexibility and completeness of this text enable it to be used equally effectively for more conventional instructor-led courses. Critically, it furnishes a rich variety of exercises and examples, many of which have specimen solutions, through which to gain in confidence and competence.

The theory of buildings was introduced by J Tits in order to focus on geometric and combinatorial aspects of simple groups of Lie type. Since then the theory has blossomed into an extremely active field of mathematical research having deep connections with topics as diverse as algebraic groups, arithmetic groups, finite simple groups, and finite geometries, as well as with graph theory and other aspects of combinatorics. This volume is an up-to-date survey of the theory of buildings with special emphasis on its interaction with related geometries. As such it will be an invaluable guide to all those whose research touches on these themes. The articles presented here are by experts in their respective fields and are based on talks given at the 1988 Buildings and Related Geometries conference at Pingree Park, Colorado. Topics covered include the classification and construction of buildings, finite groups associated with building-like geometries, graphs and association schemes.

This book contains selected chapters on recent research in topology. It bridges the gap between recent trends of topological theories and their applications in areas like social sciences, natural sciences, soft computing, economics, theoretical chemistry, cryptography, pattern recognitions and granular computing. There are 14 chapters, including two chapters on mathematical economics from the perspective of topology. The book discusses topics on function spaces, relator space, preorder, quasi-uniformities, bitopological dynamical systems, b-metric spaces and related fixed point theory. This book is useful to researchers, experts and scientists in studying the cutting-edge research in topology and related areas and helps them applying topology in solving real-life problems the society and science are facing these days.. 

This book presents material in two parts. Part one provides an introduction to crossed modules of groups, Lie algebras and associative algebras with fully written out proofs and is suitable for graduate students interested in homological algebra. In part two, more advanced and less standard topics such as crossed modules of Hopf algebra, Lie groups, and racks are discussed as well as recent developments and research on crossed modules.

This book comes out of need and urgency (expressed especially in areas of Information Retrieval with respect to Image, Audio, Internet and Biology) to have a working tool to compare data.
The book will provide powerful resource for all researchers using Mathematics as well as for mathematicians themselves. In the time when over-specialization and terminology fences isolate researchers, this Dictionary try to be "centripedal" and "oikoumeni," providing some access and altitude of vision but without taking the route of scientific vulgarisation. This attempted balance is the main philosophy of this Dictionary which defined its structure and style.
Key features:
- Unicity: it is the first book treating the basic notion of Distance in whole generality.
- Interdisciplinarity: this Dictionary is larger in scope than majority of thematic dictionaries.
- Encyclopedicity: while an Encyclopedia of Distances seems now too difficult to produce, this book (by its scope, short introductions and organization) provides the main material for it and for future tutorials on some parts of this material.
- Applicability: the distances, as well as distance-related notions and paradigms, are provided in ready-to-use fashion.
- Worthiness: the need and urgency for such dictionary was great in several huge areas, esp. Information Retrieval, Image Analysis, Speech Recognition and Biology.
- Accessibility: the definitions are easy to locate by subject or, in Index, by alphabetic order; the introductions and definitions are reader-friendly and maximally independent one from another; still the text is structured, in the 3D HTML style, by hyperlink-like boldfaced references to similar definitions.
* Covers a large range of subjects in pure and applied mathematics
* Designed to be easily applied--the distances and distance-related notions and paradigms are ready to use
* Helps users quickly locate definitions by subject or in alphabetical order; stand-alone entries include references to other entries and sources for further investigation

This monograph is the first and an initial introduction to the theory of bitopological spaces and its applications. In particular, different families of subsets of bitopological spaces are introduced and various relations between two topologies are analyzed on one and the same set; the theory of dimension of bitopological spaces and the theory of Baire bitopological spaces are constructed, and various classes of mappings of bitopological spaces are studied. The previously known results as well the results obtained in this monograph are applied in analysis, potential theory, general topology, and theory of ordered topological spaces. Moreover, a high level of modern knowledge of bitopological spaces theory has made it possible to introduce and study algebra of new type, the corresponding representation of which brings one to the special class of bitopological spaces.
It is beyond any doubt that in the nearest future the areas of essential applications will be the theories of linear topological spaces and topological groups, algebraic and differential topologies, the homotopy theory, not to mention other fundamental areas of modern mathematics such as geometry, mathematical logic, the probability theory and many other areas, including those of applied nature.
Key Features:
- First monograph is "Generalized Lattices"
* The first introduction to the theory of bitopological spaces and its applications.

The book is devoted to universality problems.
A new approach to these problems is given using some specific spaces. Since the construction of these specific spaces is set-theoretical, the given theory can be applied to different topics of Topology such as:
universal mappings, dimension theory, action of groups, inverse spectra, isometrical embeddings, and so on.
.Universal spaces
.Universal mappings
.Dimension theory
.Actions of groups
.Isometric Universal Spaces

Topology, for many years, has been one of the most exciting and influential fields of research in modern mathematics. Although its origins may be traced back several hundred years, it was Poincare who "gave topology wings" in a classic series of articles published around the turn of the century. While the earlier history, sometimes called the prehistory, is also considered, this volume is mainly concerned with the more recent history of topology, from Poincare onwards.
As will be seen from the list of contents the articles cover a wide range of topics. Some are more technical than others, but the reader without a great deal of technical knowledge should still find most of the articles accessible. Some are written by professional historians of mathematics, others by historically-minded mathematicians, who tend to have a different viewpoint.

I The fixed point theorems of Brouwer and Schauder.- 1 The fixed point theorem of Brouwer and applications.- 2 The fixed point theorem of Schauder and applications.- II Measures of noncompactness.- 1 The general notion of a measure of noncompactness.- 2 The Kuratowski and Hausdorff measures of noncompactness.- 3 The separation measure of noncompactness.- 4 Measures of noncompactness in Banach sequences spaces.- 5 Theorem of Darbo and Sadovskii and applications.- III Minimal sets for a measure of noncompactness.- 1 o-minimal sets.- 2 Minimalizable measures of noncompactness.- IV Convexity and smoothness.- 1 Strict convexity and smoothness.- 2 k-uniform convexity.- 3 k-uniform smoothness.- V Nearly uniform convexity and nearly uniform smoothness.- 1 Nearly uniformly convex Banach spaces.- 2 Nearly uniformly smooth Banach spaces.- 3 Uniform Opial condition.- VI Fixed points for nonexpansive mappings and normal structure.- 1 Existence of fixed points for nonexpansive mappings: Kirk's theorem.- 2 The coefficient N(X) and its connection with uniform convexity.- 3 The weakly convergent sequence coefficient.- 4 Uniform smoothness, near uniform convexity and normal structure.- 5 Normal structure in direct sum spaces.- 6 Computation of the normal structure coefficients in Lp-spaces.- VII Fixed point theorems in the absence of normal structure.- 1 Goebel-Karlovitz's lemma and Lin's lemma.- 2 The coefficient M(X) and the fixed point property.- VIII Uniformly Lipschitzian mappings.- 1 Lifshitz characteristic and fixed points.- 2 Connections between the Lifshitz characteristic and certain geometric coefficients.- 3 The normal structure coefficient and fixed points.- IX Asymptotically regular mappings.- 1 A fixed point theorem for asymptotically regular mappings.- 2 Connections between the ?-characteristic and some other geometric coefficients.- 3 The weakly convergent sequence coefficient and fixed points.- X Packing rates and o-contractiveness constants.- 1 Comparable measures of noncompactness.- 2 Packing rates of a metric space.- 3 Connections between the packing rates and the normal structure coefficients.- 4 Packing rates in lp-spaces.- 5 Packing rates in Lpspaces.- 6 Packing rates in direct sum spaces.- References.- List of Symbols and Notations.

This book provides an introduction to topological groups and the structure theory of locally compact abelian groups, with a special emphasis on Pontryagin-van Kampen duality, including a completely self-contained elementary proof of the duality theorem. Further related topics and applications are treated in separate chapters and in the appendix.

The fascinating world of canonical moments--a unique look at this practical, powerful statistical and probability tool
Unusual in its emphasis, this landmark monograph on canonical moments describes the theory and application of canonical moments of probability measures on intervals of the real line and measures on the circle. Stemming from the discovery that canonical moments appear to be more intrinsically related to the measure than ordinary moments, the book's main focus is the broad application of canonical moments in many areas of statistics, probability, and analysis, including problems in the design of experiments, simple random walks or birth and death chains, and in approximation theory.
The book begins with an explanation of the development of the theory of canonical moments for measures on intervals [a, b] and then describes the various practical applications of canonical moments. The book's topical range includes:
* Definition of canonical moments both geometrically and as ratios of Hankel determinants
* Orthogonal polynomials viewed geometrically as hyperplanes to moment spaces
* Continued fractions and their link between ordinary moments and canonical moments
* The determination of optimal designs for polynomial regression
* The relationships between canonical moments, random walks, and orthogonal polynomials
* Canonical moments for the circle or trigonometric functions
Finally, this volume clearly illustrates the powerful mathematical role of canonical moments in a chapter arrangement that is as logical and interdependent as is the relationship of canonical moments to statistics, probability, and analysis.

The subject of nonlinear partial differential equations is experiencing a period of intense activity in the study of systems underlying basic theories in geometry, topology and physics. These mathematical models share the property of being derived from variational principles. Understanding the structure of critical configurations and the dynamics of the corresponding evolution problems is of fundamental importance for the development of the physical theories and their applications. This volume contains survey lectures in four different areas, delivered by leading resarchers at the 1995 Barrett Lectures held at The University of Tennessee: nonlinear hyperbolic systems arising in field theory and relativity (S. Klainerman); harmonic maps from Minkowski spacetime (M. Struwe); dynamics of vortices in the Ginzburg-Landau model of superconductivity (F.-H. Lin); the Seiberg-Witten equations and their application to problems in four-dimensional topology (R. Fintushel). Most of this material has not previously been available in survey form. These lectures provide an up-to-date overview and an introduction to the research literature in each of these areas, which should prove useful to researchers and graduate students in mathematical physics, partial differential equations, differential geometry and topology.

Handbook of Convex Geometry, Volume B offers a survey of convex geometry and its many ramifications and connections with other fields of mathematics, including convexity, lattices, crystallography, and convex functions. The selection first offers information on the geometry of numbers, lattice points, and packing and covering with convex sets. Discussions focus on packing in non-Euclidean spaces, problems in the Euclidean plane, general convex bodies, computational complexity of lattice point problem, centrally symmetric convex bodies, reduction theory, and lattices and the space of lattices. The text then examines finite packing and covering and tilings, including plane tilings, monohedral tilings, bin packing, and sausage problems. The manuscript takes a look at valuations and dissections, geometric crystallography, convexity and differential geometry, and convex functions. Topics include differentiability, inequalities, uniqueness theorems for convex hypersurfaces, mixed discriminants and mixed volumes, differential geometric characterization of convexity, reduction of quadratic forms, and finite groups of symmetry operations. The selection is a dependable source of data for mathematicians and researchers interested in convex geometry.

Handbook of Convex Geometry, Volume A offers a survey of convex geometry and its many ramifications and relations with other areas of mathematics, including convexity, geometric inequalities, and convex sets. The selection first offers information on the history of convexity, characterizations of convex sets, and mixed volumes. Topics include elementary convexity, equality in the Aleksandrov-Fenchel inequality, mixed surface area measures, characteristic properties of convex sets in analysis and differential geometry, and extensions of the notion of a convex set. The text then reviews the standard isoperimetric theorem and stability of geometric inequalities. The manuscript takes a look at selected affine isoperimetric inequalities, extremum problems for convex discs and polyhedra, and rigidity. Discussions focus on include infinitesimal and static rigidity related to surfaces, isoperimetric problem for convex polyhedral, bounds for the volume of a convex polyhedron, curvature image inequality, Busemann intersection inequality and its relatives, and Petty projection inequality. The book then tackles geometric algorithms, convexity and discrete optimization, mathematical programming and convex geometry, and the combinatorial aspects of convex polytopes. The selection is a valuable source of data for mathematicians and researchers interested in convex geometry.

According to Grothendieck, the notion of topos is "the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures". It is what he had "conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an "essence" which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things". The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision by building on the notion of classifying topos educed by categorical logicians. Mathematical theories (formalized within first-order logic) give rise to geometric objects called sites; the passage from sites to their associated toposes embodies the passage from the logical presentation of theories to their mathematical content, i.e. from syntax to semantics. The essential ambiguity given by the fact that any topos is associated in general with an infinite number of theories or different sites allows to study the relations between different theories, and hence the theories themselves, by using toposes as 'bridges' between these different presentations. The expression or calculation of invariants of toposes in terms of the theories associated with them or their sites of definition generates a great number of results and notions varying according to the different types of presentation, giving rise to a veritable mathematical morphogenesis.

The book presents surveys describing recent developments in most of the primary subfields of
General Topology and its applications to Algebra and Analysis during the last decade. It follows freely
the previous edition (North Holland, 1992), Open Problems in Topology (North Holland, 1990) and
Handbook of Set-Theoretic Topology (North Holland, 1984). The book was prepared in
connection with the Prague Topological Symposium, held in 2001. During the last 10 years the focus
in General Topology changed and therefore the selection of topics differs slightly from those
chosen in 1992. The following areas experienced significant developments: Topological Groups, Function Spaces, Dimension Theory, Hyperspaces, Selections, Geometric Topology (including
Infinite-Dimensional Topology and the Geometry of Banach Spaces). Of course, not every important topic
could be included in this book.

Except surveys, the book contains several historical essays written by such eminent topologists as:
R.D. Anderson, W.W. Comfort, M. Henriksen, S. Mardeŝić, J. Nagata, M.E. Rudin, J.M. Smirnov (several reminiscences of L. Vietoris are added). In addition to extensive author and subject indexes, a list of all problems and questions posed in this book are added.

List of all authors of surveys:
A. Arhangel'skii, J. Baker and K. Kunen, H. Bennett and D. Lutzer, J. Dijkstra and J. van Mill, A. Dow, E. Glasner, G. Godefroy, G. Gruenhage, N. Hindman and D. Strauss, L. Hola and J. Pelant, K. Kawamura, H.-P. Kuenzi, W. Marciszewski, K. Martin and M. Mislove and M. Reed, R. Pol and H. Torunczyk, D. Repovs and P. Semenov, D. Shakhmatov, S. Solecki, M. Tkachenko.

For a senior undergraduate or first year graduate-level course in Introduction to Topology. Appropriate for a one-semester course on both general and algebraic topology or separate courses treating each topic separately. This text is designed to provide instructors with a convenient single text resource for bridging between general and algebraic topology courses. Two separate, distinct sections (one on general, point set topology, the other on algebraic topology) are each suitable for a one-semester course and are based around the same set of basic, core topics. Optional, independent topics and applications can be studied and developed in depth depending on course needs and preferences.

Topological surgery is a mathematical technique used for creating new manifolds out of known ones. In this book the authors observe that it also occurs in natural phenomena of all scales: 1-dimensional surgery happens during DNA recombination and when cosmic magnetic lines reconnect; 2-dimensional surgery happens during tornado formation and cell mitosis; and they conjecture that 3-dimensional surgery happens during the formation of black holes from cosmic strings, offering an explanation for the existence of a black hole's singularity. Inspired by such phenomena, the authors present a new topological model that extends the formal definition to a continuous process caused by local forces. Lastly, they describe an intrinsic connection between topological surgery and a chaotic dynamical system exhibiting a "hole drilling" behavior. The authors' model indicates where to look for the forces causing surgery and what deformations should be observed in the local submanifolds involved. These predictions are significant for the study of phenomena exhibiting surgery and they also open new research directions. This novel study enables readers to gain a better understanding of the topology and dynamics of various natural phenomena, as well as topological surgery itself and serves as a basis for many more insightful observations and new physical implications.