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Author: Stephen Barr
Publisher: Courier Corporation
Published: 2012-12-04
Total Pages: 244
ISBN-13: 048615274X
DOWNLOAD EBOOKClassic, lively explanation of one of the byways of mathematics. Klein bottles, Moebius strips, projective planes, map coloring, problem of the Koenigsberg bridges, much more, described with clarity and wit.
Author: Stephen Barr
Publisher: Courier Corporation
Published: 1989-01-01
Total Pages: 242
ISBN-13: 9780486259338
DOWNLOAD EBOOK"A mathematician named Klein Thought the Moebius band was divine. Said he: 'If you glue The edges of two, You'll get a weird bottle like mine.' " — Stephen Barr In this lively book, the classic in its field, a master of recreational topology invites readers to venture into such tantalizing topological realms as continuity and connectedness via the Klein bottle and the Moebius strip. Beginning with a definition of topology and a discussion of Euler's theorem, Mr. Barr brings wit and clarity to these topics: New Surfaces (Orientability, Dimension, The Klein Bottle, etc.) The Shortest Moebius Strip The Conical Moebius Strip The Klein Bottle The Projective Plane (Symmetry) Map Coloring Networks (Koenigsberg Bridges, Betti Numbers, Knots) The Trial of the Punctured Torus Continuity and Discreteness ("Next Number," Continuity, Neighborhoods, Limit Points) Sets (Valid or Merely True? Venn Diagrams, Open and Closed Sets, Transformations, Mapping, Homotopy) With this book and a square sheet of paper, the reader can make paper Klein bottles, step by step; then, by intersecting or cutting the bottle, make Moebius strips. Conical Moebius strips, projective planes, the principle of map coloring, the classic problem of the Koenigsberg bridges, and many more aspects of topology are carefully and concisely illuminated by the author's informal and entertaining approach. Now in this inexpensive paperback edition, Experiments in Topology belongs in the library of any math enthusiast with a taste for brainteasing adventures
Author: Roderich Moessner
Publisher: Cambridge University Press
Published: 2021-04-29
Total Pages: 393
ISBN-13: 1107105536
DOWNLOAD EBOOKThis important graduate level text unites the physical mechanisms behind the phenomena of topological matter within a theoretical framework.
Author: V. I. Arnold
Publisher: American Mathematical Soc.
Published: 2015-07-14
Total Pages: 170
ISBN-13: 0821894161
DOWNLOAD EBOOKOne of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the best-known examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years. This book, based on the author's lectures, presents several new directions of mathematical research. All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments). Written in Arnold's unique style, the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession. Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).
Author: Tamal Krishna Dey
Publisher: Cambridge University Press
Published: 2022-03-10
Total Pages: 456
ISBN-13: 1009103199
DOWNLOAD EBOOKTopological data analysis (TDA) has emerged recently as a viable tool for analyzing complex data, and the area has grown substantially both in its methodologies and applicability. Providing a computational and algorithmic foundation for techniques in TDA, this comprehensive, self-contained text introduces students and researchers in mathematics and computer science to the current state of the field. The book features a description of mathematical objects and constructs behind recent advances, the algorithms involved, computational considerations, as well as examples of topological structures or ideas that can be used in applications. It provides a thorough treatment of persistent homology together with various extensions – like zigzag persistence and multiparameter persistence – and their applications to different types of data, like point clouds, triangulations, or graph data. Other important topics covered include discrete Morse theory, the Mapper structure, optimal generating cycles, as well as recent advances in embedding TDA within machine learning frameworks.
Author: Erik Torgersen
Publisher: Cambridge University Press
Published: 1991-03-14
Total Pages: 706
ISBN-13: 9780521250306
DOWNLOAD EBOOKThere are a number of important questions associated with statistical experiments: when does one given experiment yield more information than another; how can we measure the difference in information; how fast does information accumulate by repeating the experiment? The means of answering such questions has emerged from the work of Wald, Blackwell, LeCam and others and is based on the ideas of risk and deficiency. The present work which is devoted to the various methods of comparing statistical experiments, is essentially self-contained, requiring only some background in measure theory and functional analysis. Chapters introducing statistical experiments and the necessary convex analysis begin the book and are followed by others on game theory, decision theory and vector lattices. The notion of deficiency, which measures the difference in information between two experiments, is then introduced. The relation between it and other concepts, such as sufficiency, randomisation, distance, ordering, equivalence, completeness and convergence are explored. This is a comprehensive treatment of the subject and will be an essential reference for mathematical statisticians.
Author: Afra J. Zomorodian
Publisher: Cambridge University Press
Published: 2005-01-10
Total Pages: 264
ISBN-13: 9781139442633
DOWNLOAD EBOOKThe emerging field of computational topology utilizes theory from topology and the power of computing to solve problems in diverse fields. Recent applications include computer graphics, computer-aided design (CAD), and structural biology, all of which involve understanding the intrinsic shape of some real or abstract space. A primary goal of this book is to present basic concepts from topology and Morse theory to enable a non-specialist to grasp and participate in current research in computational topology. The author gives a self-contained presentation of the mathematical concepts from a computer scientist's point of view, combining point set topology, algebraic topology, group theory, differential manifolds, and Morse theory. He also presents some recent advances in the area, including topological persistence and hierarchical Morse complexes. Throughout, the focus is on computational challenges and on presenting algorithms and data structures when appropriate.
Author: Herbert Edelsbrunner
Publisher: American Mathematical Society
Published: 2022-01-31
Total Pages: 241
ISBN-13: 1470467690
DOWNLOAD EBOOKCombining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.
Author: Martin Gardner
Publisher: Courier Corporation
Published: 2013-06-10
Total Pages: 130
ISBN-13: 0486319113
DOWNLOAD EBOOKA prominent popular science writer presents simple instructions for 100 illustrated experiments. Memorable, easily understood experiments illuminate principles related to astronomy, chemistry, physiology, psychology, mathematics, topology, probability, acoustics, other areas.
Author: Deane Montgomery
Publisher: Courier Dover Publications
Published: 2018-06-13
Total Pages: 305
ISBN-13: 0486831582
DOWNLOAD EBOOKAn advanced monograph on the subject of topological transformation groups, this volume summarizes important research conducted during a period of lively activity in this area of mathematics. The book is of particular note because it represents the culmination of research by authors Deane Montgomery and Leo Zippin, undertaken in collaboration with Andrew Gleason of Harvard University, that led to their solution of a well-known mathematical conjecture, Hilbert's Fifth Problem. The treatment begins with an examination of topological spaces and groups and proceeds to locally compact groups and groups with no small subgroups. Subsequent chapters address approximation by Lie groups and transformation groups, concluding with an exploration of compact transformation groups.
