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An Invitation To Computational Homotopy
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Book Synopsis An Invitation to Computational Homotopy by : Graham Ellis
Download or read book An Invitation to Computational Homotopy written by Graham Ellis and published by Oxford University Press, USA. This book was released on 2019-08-13 with total page 550 pages. Available in PDF, EPUB and Kindle. Book excerpt: An Invitation to Computational Homotopy is an introduction to elementary algebraic topology for those with an interest in computers and computer programming. It expertly illustrates how the basics of the subject can be implemented on a computer through its focus on fully-worked examples designed to develop problem solving techniques. The transition from basic theory to practical computation raises a range of non-trivial algorithmic issues which will appeal to readers already familiar with basic theory and who are interested in developing computational aspects. The book covers a subset of standard introductory material on fundamental groups, covering spaces, homology, cohomology and classifying spaces as well as some less standard material on crossed modules. These topics are covered in a way that hints at potential applications of topology in areas of computer science and engineering outside the usual territory of pure mathematics, and also in a way that demonstrates how computers can be used to perform explicit calculations within the domain of pure algebraic topology itself. The initial chapters include in-depth examples from data mining, biology and digital image analysis, while the later chapters cover a range of computational examples on the cohomology of classifying spaces that are likely beyond the reach of a purely paper-and-pen approach to the subject. An Invitation to Computational Homotopy serves as a self-contained and informal introduction to these topics and their implementation in the sphere of computer science. Written in a dynamic and engaging style, it skilfully showcases a range of useful machine computations, and will serve as an invaluable aid to graduate students working with algebraic topology.
Book Synopsis A Short Course in Computational Geometry and Topology by : Herbert Edelsbrunner
Download or read book A Short Course in Computational Geometry and Topology written by Herbert Edelsbrunner and published by Springer Science & Business. This book was released on 2014-04-28 with total page 105 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents a short course in computational geometry and topology. In the first part the book covers Voronoi diagrams and Delaunay triangulations, then it presents the theory of alpha complexes which play a crucial role in biology. The central part of the book is the homology theory and their computation, including the theory of persistence which is indispensable for applications, e.g. shape reconstruction. The target audience comprises researchers and practitioners in mathematics, biology, neuroscience and computer science, but the book may also be beneficial to graduate students of these fields.
Book Synopsis Computing the Homology of the Lambda Algebra by : Martin C. Tangora
Download or read book Computing the Homology of the Lambda Algebra written by Martin C. Tangora and published by American Mathematical Soc.. This book was released on 1985 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: For each prime [italic]p there is a 'lambda algebra' which serves, among other things, as an [italic]E1 term for the (stable or unstable) Adams spectral sequence. This paper describes algorithms and programs for computing the homology of these algebras using a mainframe computer. Tables of the results are included for [italic]p = 2 and [italic]p = 3.
Book Synopsis Computational Topology for Data Analysis by : Tamal Krishna Dey
Download or read book Computational Topology for Data Analysis written by Tamal Krishna Dey and published by Cambridge University Press. This book was released on 2022-03-10 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: Topological 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.
Book Synopsis Computational Topology by : Herbert Edelsbrunner
Download or read book Computational Topology written by Herbert Edelsbrunner and published by American Mathematical Soc.. This book was released on 2010 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: Combining 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.
Book Synopsis Parallel Scientific Computation by : Rob H. Bisseling
Download or read book Parallel Scientific Computation written by Rob H. Bisseling and published by Oxford University Press, USA. This book was released on 2020-09-30 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: Parallel Scientific Computation presents a methodology for designing parallel algorithms and writing parallel computer programs for modern computer architectures with multiple processors.
Book Synopsis Complex Cobordism and Stable Homotopy Groups of Spheres by : Douglas C. Ravenel
Download or read book Complex Cobordism and Stable Homotopy Groups of Spheres written by Douglas C. Ravenel and published by American Mathematical Society. This book was released on 2023-02-09 with total page 417 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the publication of its first edition, this book has served as one of the few available on the classical Adams spectral sequence, and is the best account on the Adams-Novikov spectral sequence. This new edition has been updated in many places, especially the final chapter, which has been completely rewritten with an eye toward future research in the field. It remains the definitive reference on the stable homotopy groups of spheres. The first three chapters introduce the homotopy groups of spheres and take the reader from the classical results in the field though the computational aspects of the classical Adams spectral sequence and its modifications, which are the main tools topologists have to investigate the homotopy groups of spheres. Nowadays, the most efficient tools are the Brown-Peterson theory, the Adams-Novikov spectral sequence, and the chromatic spectral sequence, a device for analyzing the global structure of the stable homotopy groups of spheres and relating them to the cohomology of the Morava stabilizer groups. These topics are described in detail in Chapters 4 to 6. The revamped Chapter 7 is the computational payoff of the book, yielding a lot of information about the stable homotopy group of spheres. Appendices follow, giving self-contained accounts of the theory of formal group laws and the homological algebra associated with Hopf algebras and Hopf algebroids. The book is intended for anyone wishing to study computational stable homotopy theory. It is accessible to graduate students with a knowledge of algebraic topology and recommended to anyone wishing to venture into the frontiers of the subject.
Book Synopsis Homotopy Theory: Proceedings of the Durham Symposium 1985 by : E. Rees
Download or read book Homotopy Theory: Proceedings of the Durham Symposium 1985 written by E. Rees and published by Cambridge University Press. This book was released on 1987-10-29 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: This 1987 volume presents a collection of papers given at the 1985 Durham Symposium on homotopy theory. They survey recent developments in the subject including localisation and periodicity, computational complexity, and the algebraic K-theory of spaces.
Book Synopsis Computational Chemistry by : Jerzy Leszczynski
Download or read book Computational Chemistry written by Jerzy Leszczynski and published by World Scientific. This book was released on 2000 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume comprises six chapters which explore the development and applications of the methods of computational chemistry. The first chapter is on new developments in coupled-cluster (CC) theory. The homotopy method is used to obtain complete sets of solutions of nonlinear CC equations. The correspondence between multiple solutions to the CCSD, CCSDT, and full CI equations is established, and the applications of the new approach in modeling molecular systems are discussed. The second chapter reviews the computational theory for the time-dependent calculations of a solution to the Schrodinger equation for two electrons and focuses on the development of propagators to the solution. The next chapter features a discussion on a new self-consistent field for molecular interactions (SCF-MI) scheme for modifying Roothaan equations in order to avoid basis set superposition errors (BSSE). This method is especially suitable for computations of intermolecular interactions. Details of the theory, alongwith examples of applications to nucleic acid base pair complexes, are given. This chapter is well complemented by the following chapter, which reports the current status of computational studies of aromatic stacking and hydrogen bonding interactions among nucleic acid bases. The next chapter reveals the possibility of calculating the kinetics of chemical reactions in biological systems from the first principles. The last chapter reviews the results of rigorous ab initio studies of the series of derivatives of methane, silane, and germane. The presented molecular and vibrational parameters complement experimental data for these systems. In addition, the theoretical approach allows the predictionof the effects of halogeno-substitutions on their structures and properties.
Book Synopsis The Nature of Computation by : Cristopher Moore
Download or read book The Nature of Computation written by Cristopher Moore and published by OUP Oxford. This book was released on 2011-08-11 with total page 1498 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational complexity is one of the most beautiful fields of modern mathematics, and it is increasingly relevant to other sciences ranging from physics to biology. But this beauty is often buried underneath layers of unnecessary formalism, and exciting recent results like interactive proofs, phase transitions, and quantum computing are usually considered too advanced for the typical student. This book bridges these gaps by explaining the deep ideas of theoretical computer science in a clear and enjoyable fashion, making them accessible to non-computer scientists and to computer scientists who finally want to appreciate their field from a new point of view. The authors start with a lucid and playful explanation of the P vs. NP problem, explaining why it is so fundamental, and so hard to resolve. They then lead the reader through the complexity of mazes and games; optimization in theory and practice; randomized algorithms, interactive proofs, and pseudorandomness; Markov chains and phase transitions; and the outer reaches of quantum computing. At every turn, they use a minimum of formalism, providing explanations that are both deep and accessible. The book is intended for graduate and undergraduate students, scientists from other areas who have long wanted to understand this subject, and experts who want to fall in love with this field all over again.
Book Synopsis Invitation to Discrete Mathematics by : Jiří Matoušek
Download or read book Invitation to Discrete Mathematics written by Jiří Matoušek and published by Oxford University Press. This book was released on 2009 with total page 462 pages. Available in PDF, EPUB and Kindle. Book excerpt: A clear and self-contained introduction to discrete mathematics for undergraduates and early graduates.
Book Synopsis Introduction to Homotopy Theory by : Aneta Hajek
Download or read book Introduction to Homotopy Theory written by Aneta Hajek and published by . This book was released on 2015-08 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Homotopy theory, which is the main part of algebraic topology, studies topological objects up to homotopy equivalence. Homotopy equivalence is weaker relations than topological equivalence, i.e., homotopy classes of spaces are larger than homeomorphism classes. Even though the ultimate goal of topology is to classify various classes of topological spaces up to a homeomorphism, in algebraic topology, homotopy equivalence plays a more important role than homeomorphism, essentially because the basic tools of algebraic topology (homology and homotopy groups) are invariant with respect to homotopy equivalence, and do not distinguish topologically nonequivalent, but homotopic objects. The idea of homotopy can be turned into a formal category of category theory. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent. Then a functor on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category. Based on the concept of the homotopy, computation methods for algebraic and differential equations have been developed. The methods for algebraic equations include the homotopy continuation method and the continuation method. The methods for differential equations include the homotopy analysis method. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. This book deals with homotopy theory, one of the main branches of algebraic topology.
Book Synopsis An Invitation to Applied Category Theory by : Brendan Fong
Download or read book An Invitation to Applied Category Theory written by Brendan Fong and published by Cambridge University Press. This book was released on 2019-07-18 with total page 351 pages. Available in PDF, EPUB and Kindle. Book excerpt: Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.
Book Synopsis Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem by : Michael A Hill
Download or read book Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem written by Michael A Hill and published by . This book was released on 2021-02 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: "This unique book on modern topology looks well beyond traditional treatises, and explores spaces that may, but need not, be Hausdorff. This is essential for domain theory, the cornerstone of semantics of computer languages, where the Scott topology is almost never Hausdorff. For the first time in a single volume, this book covers basic material on metric and topological spaces, advanced material on complete partial orders, Stone duality, stable compactness, quasi-metric spaces, and much more. An early chapter on metric spaces serves as an invitation to the topic (continuity, limits, compactness, completeness) and forms a complete introductory course by itself"--
Book Synopsis An Invitation to Quantum Cohomology by : Joachim Kock
Download or read book An Invitation to Quantum Cohomology written by Joachim Kock and published by Springer Science & Business Media. This book was released on 2007-12-27 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: Elementary introduction to stable maps and quantum cohomology presents the problem of counting rational plane curves Viewpoint is mostly that of enumerative geometry Emphasis is on examples, heuristic discussions, and simple applications to best convey the intuition behind the subject Ideal for self-study, for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory
Book Synopsis Computational Complexity and Statistical Physics by : Allon Percus
Download or read book Computational Complexity and Statistical Physics written by Allon Percus and published by Oxford University Press, USA. This book was released on 2006 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computer science and physics have been closely linked since the birth of modern computing. In recent years, an interdisciplinary area has blossomed at the junction of these fields, connecting insights from statistical physics with basic computational challenges. Researchers have successfully applied techniques from the study of phase transitions to analyze NP-complete problems such as satisfiability and graph coloring. This is leading to a new understanding of the structure of these problems, and of how algorithms perform on them. Computational Complexity and Statistical Physics will serve as a standard reference and pedagogical aid to statistical physics methods in computer science, with a particular focus on phase transitions in combinatorial problems. Addressed to a broad range of readers, the book includes substantial background material along with current research by leading computer scientists, mathematicians, and physicists. It will prepare students and researchers from all of these fields to contribute to this exciting area.
Book Synopsis The Mathematical World of Charles L. Dodgson (Lewis Carroll) by : Robin Wilson
Download or read book The Mathematical World of Charles L. Dodgson (Lewis Carroll) written by Robin Wilson and published by Oxford University Press. This book was released on 2019-02-14 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Charles Lutwidge Dodgson is best known for his 'Alice' books, Alice's Adventures in Wonderland and Through the Looking-Glass, written under his pen name of Lewis Carroll. Yet, whilst lauded for his work in children's fiction and his pioneering work in the world of Victorian photography, his everyday job was a lecturer in Mathematics at Christ Church, Oxford University. The Mathematical World of Charles L. Dodgson (Lewis Carroll) explores the academic background behind this complex individual, outlining his mathematical life, describing his writings in geometry, algebra, logic, the theory of voting, and recreational mathematics, before going on to discuss his mathematical legacy. This is the first academic work that collects the research on Dodgson's wide-ranging mathematical achievements into a single practical volume. Much material appears here for the first time, such as Dodgson's personal letters and drawings, as well as the results of recent investigations into the life and work of Dodgson. Complementing this are many illustrations, both historical and explanatory, as well as a full mathematical bibliography of Dodgson's mathematical publications.
