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An authorised reissue of the long out of print classic textbook, Advanced Calculus by the late Dr Lynn Loomis and Dr Shlomo Sternberg both of Harvard University has been a revered but hard to find textbook for the advanced calculus course for decades.This book is based on an honors course in advanced calculus that the authors gave in the 1960's. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis.The prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra. The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication. As possible introductory texts, we mention Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, and Pure Mathematics by G Hardy. The reader should also have some experience with partial derivatives.In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds.
According to the authors of this highly useful compendium, focusing on examples is an extremely effective method of involving undergraduate mathematics students in actual research. It is only as a result of pursuing the details of each example that students experience a significant increment in topological understanding. With that in mind, Professors Steen and Seebach have assembled 143 examples in this book, providing innumerable concrete illustrations of definitions, theorems, and general methods of proof. Far from presenting all relevant examples, however, the book instead provides a fruitful context in which to ask new questions and seek new answers.
Ranging from the familiar to the obscure, the examples are preceded by a succinct exposition of general topology and basic terminology and theory. Each example is treated as a whole, with a highly geometric exposition that helps readers comprehend the material. Over 25 Venn diagrams and reference charts summarize the properties of the examples and allow students to scan quickly for examples with prescribed properties. In addition, discussions of general methods of constructing and changing examples acquaint readers with the art of constructing counterexamples. The authors have included an extensive collection of problems and exercises, all correlated with various examples, and a bibliography of 140 sources, tracing each uncommon example to its origin.
This revised and expanded second edition will be especially useful as a course supplement and reference work for students of general topology. Moreover, it gives the instructor the flexibility to design his own course while providing students with a wealth of historically and mathematically significant examples. 1978 edition.
"As textbook and reference work, this is a valuable addition to the topological literature." — Mathematical Reviews
Designed as a text for a one-year first course in topology, this authoritative volume offers an excellent general treatment of the main ideas of topology. It includes a large number and variety of topics from classical topology as well as newer areas of research activity.
There are four set-theoretic chapters, followed by four primarily algebraic chapters. Chapter I covers the fundamentals of topological and metrical spaces, mappings, compactness, product spaces, the Tychonoff theorem, function spaces, uniform continuity and uniform spaces. The next two chapters are devoted to topics in point-set topology: various separation axioms, continua in Hausdorff spaces, real-valued functions, and more Chapter IV is on homotopy theory. Chapter V covers basic material on geometric and abstract simplicial complexes and their subdivisions. Chapter VI is devoted to simplicial homology theory, Chapter VII covers various topics in algebraic topology, including relative homology, exact sequences, the Mayer-Vietoris sequence, and more. Finally, Chapter VIII discusses Cech homology.
There are a large number of illuminating examples, counter-examples and problems, both those which test the understanding and those which deepen it. The authors have also made a special effort to make this an "open-ended" book, i.e while many topics are covered, there is much beyond the confines of this book. In many instances they have attempted to show the direction in which further material may be found.
Topology is so fundamental, its influence is apparent in almost every other branch of mathematics, as well as such fields as symbolic logic, mechanics, geography, network theory, and even psychology. This well-written text offers a clear and careful exposition of this increasingly important discipline.
Geometry and topology are strongly motivated by the visualization of ideal objects that have certain special characteristics. A clear formulation of a specific property or a logically consistent proof of a theorem often comes only after the mathematician has correctly "seen" what is going on. These pictures which are meant to serve as signposts leading to mathematical understanding, frequently also contain a beauty of their own. The principal aim of this book is to narrate, in an accessible and fairly visual language, about some classical and modern achievements of geometry and topology in both intrinsic mathematical problems and applications to mathematical physics. The book starts from classical notions of topology and ends with remarkable new results in Hamiltonian geometry. Fomenko lays special emphasis upon visual explanations of the problems and results and downplays the abstract logical aspects of calculations. As an example, readers can very quickly penetrate into the new theory of topological descriptions of integrable Hamiltonian differential equations. The book includes numerous graphical sheets drawn by the author, which are presented in special sections of "Visual material". These pictures illustrate the mathematical ideas and results contained in the book. Using these pictures, the reader can understand many modern mathematical ideas and methods. Although "Visual Geometry and Topology" is about mathematics, Fomenko has written and illustrated this book so that students and researchers from all the natural sciences and also artists and art students will find something of interest within its pages.
The French mathematician Élie Cartan (1869–1951) was one of the founders of the modern theory of Lie groups, a subject of central importance in mathematics and also one with many applications. In this volume, he describes the orthogonal groups, either with real or complex parameters including reflections, and also the related groups with indefinite metrics. He develops the theory of spinors (he discovered the general mathematical form of spinors in 1913) systematically by giving a purely geometrical definition of these mathematical entities; this geometrical origin makes it very easy to introduce spinors into Riemannian geometry, and particularly to apply the idea of parallel transport to these geometrical entities.
The book is divided into two parts. The first is devoted to generalities on the group of rotations in n-dimensional space and on the linear representations of groups, and to the theory of spinors in three-dimensional space. Finally, the linear representations of the group of rotations in that space (of particular importance to quantum mechanics) are also examined. The second part is devoted to the theory of spinors in spaces of any number of dimensions, and particularly in the space of special relativity (Minkowski space). While the basic orientation of the book as a whole is mathematical, physicists will be especially interested in the final chapters treating the applications of spinors in the rotation and Lorentz groups. In this connection, Cartan shows how to derive the "Dirac" equation for any group, and extends the equation to general relativity.
One of the greatest mathematicians of the 20th century, Cartan made notable contributions in mathematical physics, differential geometry, and group theory. Although a profound theorist, he was able to explain difficult concepts with clarity and simplicity. In this detailed, explicit treatise, mathematicians specializing in quantum mechanics will find his lucid approach a great value.
The purpose of this monograph, which is aimed at the graduate level and beyond, is to obtain a deeper understanding of Quillen's model categories. A model category is a category together with three distinguished classes of maps, called weak equivalences, cofibrations, and fibrations. Model categories have become a standard tool in algebraic topology and homological algebra and, increasingly, in other fields where homotopy theoretic ideas are becoming important, such as algebraic $K$-theory and algebraic geometry. The authors' approach is to define the notion of a homotopical category, which is more general than that of a model category, and to consider model categories as special cases of this. A homotopical category is a category with only a single distinguished class of maps, called weak equivalences, subject to an appropriate axiom. This enables one to define ``homotopical'' versions of such basic categorical notions as initial and terminal objects, colimit and limit functors, cocompleteness and completeness, adjunctions, Kan extensions, and universal properties. There are two essentially self-contained parts, and part II logically precedes part I. Part II defines and develops the notion of a homotopical category and can be considered as the beginnings of a kind of ``relative'' category theory. The results of part II are used in part I to obtain a deeper understanding of model categories. The authors show in particular that model categories are homotopically cocomplete and complete in a sense stronger than just the requirement of the existence of small homotopy colimit and limit functors. A reader of part II is assumed to have only some familiarity with the above-mentioned categorical notions. Those who read part I, and especially its introductory chapter, should also know something about model categories.
A highly valued resource for those who wish to move from the introductory and preliminary understandings and the measurement of chaotic behavior to a more sophisticated and precise understanding of chaotic systems. The authors provide a deep understanding of the structure of strange attractors, how they are classified, and how the information required to identify and classify a strange attractor can be extracted from experimental data.
In its first edition, the Topology of Chaos has been a valuable resource for physicist and mathematicians interested in the topological analysis of dynamical systems. Since its publication in 2002, important theoretical and experimental advances have put the topological analysis program on a firmer basis. This second edition includes relevant results and connects the material to other recent developments. Following significant improvements will be included:
* A gentler introduction to the topological analysis of chaotic systems for the non expert which introduces the problems and questions that one commonly encounters when observing a chaotic dynamics and which are well addressed by a topological approach: existence of unstable periodic orbits, bifurcation sequences, multistability etc.
* A new chapter is devoted to bounding tori which are essential for achieving generality as well as for understanding the influence of boundary conditions.
* The new edition also reflects the progress which had been made towards extending topological analysis to higher-dimensional systems by proposing a new formalism where evolving triangulations replace braids.
* There has also been much progress in the understanding of what is a good representation of a chaotic system, and therefore a new chapter is devoted to embeddings.
* The chapter on topological analysis program will be expanded to cover traditional measures of chaos. This will help to connect those readers who are familiar with those measures and tests to the more sophisticated methodologies discussed in detail in this book.
* The addition of the Appendix with both frequently asked and open questions with answers gathers the most essential points readers should keep in mind and guides to corresponding sections in the book. This will be of great help to those who want to selectively dive into the book and its treatments rather than reading it cover to cover.

What makes this book special is its attempt to classify real physical systems (e.g. lasers) using topological techniques applied to real date (e.g. time series). Hence it has become the experimenter?s guidebook to reliable and sophisticated studies of experimental data for comparison with candidate relevant theoretical models, inevitable to physicists, mathematicians, and engineers studying low-dimensional chaotic systems.
This text is intended as a one semester introduction to algebraic topology at the undergraduate and beginning graduate levels. Basically, it covers simplicial homology theory, the fundamental group, covering spaces, the higher homotopy groups and introductory singular homology theory. The text follows a broad historical outline and uses the proofs of the discoverers of the important theorems when this is consistent with the elementary level of the course. This method of presentation is intended to reduce the abstract nature of algebraic topology to a level that is palatable for the beginning student and to provide motivation and cohesion that are often lacking in abstact treatments. The text emphasizes the geometric approach to algebraic topology and attempts to show the importance of topological concepts by applying them to problems of geometry and analysis. The prerequisites for this course are calculus at the sophomore level, a one semester introduction to the theory of groups, a one semester introduc tion to point-set topology and some familiarity with vector spaces. Outlines of the prerequisite material can be found in the appendices at the end of the text. It is suggested that the reader not spend time initially working on the appendices, but rather that he read from the beginning of the text, referring to the appendices as his memory needs refreshing. The text is designed for use by college juniors of normal intelligence and does not require "mathematical maturity" beyond the junior level.
Knot theory is a concept in algebraic topology that has found applications to a variety of mathematical problems as well as to problems in computer science, biological and medical research, and mathematical physics. This book is directed to a broad audience of researchers, beginning graduate students, and senior undergraduate students in these fields.

The book contains most of the fundamental classical facts about the theory, such as knot diagrams, braid representations, Seifert surfaces, tangles, and Alexander polynomials; also included are key newer developments and special topics such as chord diagrams and covering spaces. The work introduces the fascinating study of knots and provides insight into applications to such studies as DNA research and graph theory. In addition, each chapter includes a supplement that consists of interesting historical as well as mathematical comments.

The author clearly outlines what is known and what is not known about knots. He has been careful to avoid advanced mathematical terminology or intricate techniques in algebraic topology or group theory. There are numerous diagrams and exercises relating the material. The study of Jones polynomials and the Vassiliev invariants are closely examined.

"The book ...develops knot theory from an intuitive geometric-combinatorial point of view, avoiding completely more advanced concepts and techniques from algebraic topology...Thus the emphasis is on a lucid and intuitive exposition accessible to a broader audience... The book, written in a stimulating and original style, will serve as a first approach to this interesting field for readers with various backgrounds in mathematics, physics, etc. It is the first text developing recent topics as the Jones polynomial and Vassiliev invariants on a level accessible also for non-specialists in the field." -Zentralblatt Math

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