The complexity of nature's shapes differs in kind, not merely degree, from that of the shapes of ordinary geometry, the geometry of fractal shapes.Now that the field has expanded greatly with many active researchers, Mandelbrot presents the definitive overview of the origins of his ideas and their new applications. The Fractal Geometry of Nature is based on his highly acclaimed earlier work, but has much broader and deeper coverage and more extensive illustrations.

    A remarkable pictorial discussion of the curved space-time we call home, it achieves even greater impact through the use of 141 excellent illustrations. This is the first sustained visual account of many important topics in relativity theory that up till now have only been treated separately.Finding a perfect analogy in the situation of the geometrical characters in Flatland, Professor Rucker continues the adventures of the two-dimensional world visited by a three-dimensional being to explain our three-dimensional world in terms of the fourth dimension. Following this adventure into the fourth dimension, the author discusses non-Euclidean geometry, curved space, time as a higher dimension, special relativity, time travel, and the shape of space-time. The mathematics is sound throughout, but the casual reader may skip those few sections that seem too purely mathematical and still follow the line of argument. Readable and interesting in itself, the annotated bibliography is a valuable guide to further study.Professor Rucker teaches mathematics at the State University of New York in Geneseo. Students and laymen will find his discussion to be unusually stimulating. Experienced mathematicians and physicists will find a great deal of original material here and many unexpected novelties. Annotated bibliography. 44 problems.

    Containing information necessary for teachers to teach math through problem solving, this resource is filled with engaging activities from every strand of mathematics.

    The volume includes an Appendix that helps bridge the gap between metric and topological spaces, a Selected Bibliography, and an Index.

    Recently revised andexpanded, the second edition will be a valuable resource for upper level undergraduate and graduate students.

    Methods range from plotting picture-drawing techniques to rather elaborate numerical summaries. Several of the methods are the original creations of the author, and all can be carried out either with pencil or aided by hand-held calculator.

    In this second English edition all three volumes have been put together with a new, combined index and bibliography. Some corrections and revisions have been made in the text, primarily in Volume II. Volumes II and III contain numerous references to the earlier volumes, so that the reader is reminded of the exact statements (and proofs) of the more elementary results made use of. The three-volume-in-one format makes it easy to flip back the pages, refresh one's memory, and proceed. The proofs chosen are those that give the student the best `feel' for the subject. The watchword is clarity and straightforwardness. The author was a leading Soviet function-theorist: It is seldom that an expert of his stature puts himself so wholly at the service of the student. This book includes over 150 illustrations and 700 exercises.

    In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning.

    

    It provides the necessary background for a more abstract course in differential geometry. The inclusion of diagrams is done without sacrificing the rigor of the material. For all readers interested in differential geometry.

    I have taught this course several times and always find it problematic. The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books

    A Way of Thinking This activity-centered program contains lessons and blackline masters that visually present concepts to students struggling with math. Teaching ideas and hands-on activities cover problem solving, computation, geometry, measurement, probability and graphing.

    Enlarged and Corrected Printing J. DIEUDONNE This book is the first volume of a treatise which will eventually consist offour volumes. It is also an enlarged and corrected printing, essentially without changes, of my Foundations of Modern Analysis. Many readers, colleagues, and friends have urged me to write a sequelto that book, and in the end I became convinced that there was a place fora survey of modern analysis, somewhere between the minimum tool kitof an elementary nature which I had intended to write, and specialist monographs leading to the frontiers of research. My experience of teachinghas also persuaded me that the mathematical apprentice, after taking the firststep of Foundations, needs further guidance and a kind of general birdseye view of his subject before he is launched onto the ocean of mathematical literature or set on the narrow path of his own topic of research.Thus I have finally been led to attempt to write an equivalent, for the mathematicians of 1970, of what the Cours dAnalyse of Jordan, Picard, and Goursat were for mathematical students between 1880 and 1920.It is manifestly out of the question to attempt encyclopedic coverage, and certainly superfluous to rewrite the works of N. Bourbaki. I have therefore been obliged to cut ruthlessly in order to keep within limits comparable tothose of the classical treatises. I have opted for breadth rather than depth, inthe opinion that it is better to show the reader rudiments of many branchesof modern analysis rather than to provide him with a complete and detailedexposition of a small number of topics.Experience seems to show that the student usually finds a new theorydifficult to grasp at a first reading. He needs to return to it several times beforehe becomes really familiar with it and can distinguish for himself whichare the essential ideas and which results are of minor importance, and onlythen will he be able to apply it intelligently. The chapters of this treatise are therefore samples rather than complete theories: indeed, I have systematically tried not to be exhaustive. The works quoted in the bibliography willalways enable the reader to go deeper into any particular theory.However, I have refused to distort the main ideas of analysis by presentingthem in too specialized a form, and thereby obscuring their power andgenerality. It gives a false impression, for example, if differential geometryis restricted to two or three dimensions, or if integration is restricted to Lebesgue measure, on the pretext of making these subjects more accessible orintuitive.On the other hand I do not believe that the essential content of the ideasinvolved is lost, in a first study, by restricting attention to separable metrizabletopological spaces. The mathematicians of my own generation were certainlyright to banish, hypotheses of countability wherever they were not needed: thiswas the only way to get a clear understanding

    Includes 150 problems, many with answers. Indispensable to mathematicians and natural scientists alike.

    In this new edition, much of the material is new or rewritten, but the purpose and style of the first edition are retained. Particular emphasis is given to information obtained by experiment and observation in addition to analysis of the equations of motion. The book's primary concern is to convey a fundamental understanding of the behavior of fluids in motion. Special features include an introductory non-mathematical treatment of three particular flow configurations; extensive consideration of geophysical topics; and detailed coverage of topics that are known primarily through experimental data. Numerous photographs illustrate the phenomena discussed, and a concluding chapter demonstrates the wide applicability of fluid mechanics. New topics in the second edition include double diffusive convection and modern ideas about dynamical chaos. The discussion of instabilities has been restructured and the treatments of separation and of convection in horizontal layers considerably expanded.

    As a consequence, there is a wide demand for a survey of the results of mathematics, for an unconventional approach that would also make it possible to fill gaps in one's knowledge. We do not think that a mere juxtaposition of theorems or a collection of formulae would be suitable for this purpose, because this would over emphasize the symbolic language of signs and letters rather than the mathematical idea, the only thing that really matters. Our task was to describe mathematical interrelations as briefly and precisely as possible. In view of the overwhelming amount of material it goes without saying that we did not just compile details from the numerous text-books for individual branches: what we were aiming at is to smooth out the access to the specialist literature for as many readers as possible. Since well over 700000 copies of the German edition of this book have been sold, we hope to have achieved our difficult goal. Colours are used extensively to help the reader. Important definitions and groups of formulae are on a yellow background, examples on blue, and theorems on red."

    It thus avoids local methods, for example, and presents proofs in a way that highlights the important parts of the arguments. Readers are assumed to be able to fill in the details, which in many places are left as exercises.

    A mathematical prodigy since the age of three, the author travels the world giving exhibitions of her powers.

    Volume 1, Fluid Mechanics, summarizes the key experiments that show how polymeric fluids differ from structurally simple fluids, then presents, in rough historical order, various methods for solving polymer fluid dynamics problems. Volume 2, Kinetic Theory, uses molecular models and the methods of statistical mechanics to obtain relations between bulk flow behavior and polymer structure. Includes end-of-chapter problems and extensive appendixes.

    The authors have taken trouble to make the treatment self-contained. It (is) suitable required reading for a PhD student. Although the material has been developed from lectures at Stanford, it has developed into an almost systematic coverage that is much longer than could be covered in a year's lectures." Newsletter, New Zealand Mathematical Society, 1985 "Primarily addressed to graduate students this elegant book is accessible and useful to a broad spectrum of applied mathematicians." Revue Roumaine de Mathematiques Pures et Appliquees,1985"

    It develops the key techniques of nonstandard analysis at the outset from a single, powerful construction; then, beginning with a nonstandard construction of the real number system, it leads students through a nonstandard treatment of the basic topics of elementary real analysis, topological spaces, and Hilbert space.Important topics include nonstandard treatments of equicontinuity, nonmeasurable sets, and the existence of Haar measure. The focus on compact operators on a Hilbert space includes the Bernstein-Robinson theorem on invariant subspaces, which was first proved with nonstandard methods. Ever mindful of the needs of readers with little background in these subjects, the text offers a straightforward treatment that provides a strong foundation for advanced studies of analysis

    Because of the amount of material accumulated on unitary representations of groups, the latter pages of the book are devoted to a brief account of some aspects of this subject, particularly since the theory of groups provides some of the most interesting examples of C*-algebras. The theory of C*-algebras is still expanding rapidly, but this work remains a clear and accessible introduction to the fundamentals of the subject.

    It also puts together in one volume many scattered, original works, on the use of group theory in general relativity theory." "Each chapter is concluded with a set of problems. The entire book is self-contained in both group theory and general relativity theory, and no prior knowledge of either is assumed." "The subject of this book constitutes a relevant link between field theoreticians and general relativity theoreticians, who usually work rather independently of each other. The treatise is highly topical and of real interest to theoretical physicists, general relativists and applied mathematicians. It is valuable to graduate students and research workers in quantum field theory, general relativity and elementary particle theory." "Readership: Graduate students and researchers in theoretical and mathematical physics, quantum field theory, general relativity, elementary particle theory and applied mathematics."--BOOK JACKET.

    This extensive treatment of the subject offers the advantage of a thorough integration of linear algebra and materials, which aids readers in the development of geometric intuition. An introductory chapter presents background information on vectors in the plane, plane curves, and functions of two variables. Subsequent chapters address differentiation, transformations, and integration. Each chapter concludes with problem sets, and answers to selected exercises appear at the end of the book.

    "The exposition is fresh and sophisticated, and will engage the interest of accomplished mathematicians." — Sci-Tech Book News. 1966 edition.

    

    The integral is initially presented in the context of n-dimensional Euclidean space, following a thorough study of the concepts of outer measure and measure. A more general treatment of the integral, based on an axiomatic approach, is later given.Closely related topics in real variables, such as functions of bounded variation, the Riemann-Stieltjes integral, Fubini's theorem, L(p)) classes, and various results about differentiation are examined in detail. Several applications of the theory to a specific branch of analysis--harmonic analysis--are also provided. Among these applications are basic facts about convolution operators and Fourier series, including results for the conjugate function and the Hardy-Littlewood maximal function.Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis for student interested in mathematics, statistics, or probability. Requiring only a basic familiarity with advanced calculus, this volume is an excellent textbook for advanced undergraduate or first-year graduate student in these areas.