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Topology and Geometry for Physicists
by Charles Nash Siddhartha SenDifferential geometry and topology are essential tools for many theoretical physicists, particularly in the study of condensed matter physics, gravity, and particle physics. Written by physicists for physics students, this text introduces geometrical and topological methods in theoretical physics and applied mathematics. It assumes no detailed background in topology or geometry, and it emphasizes physical motivations, enabling students to apply the techniques to their physics formulas and research. "Thoroughly recommended" by The Physics Bulletin, this volume's physics applications range from condensed matter physics and statistical mechanics to elementary particle theory. Its main mathematical topics include differential forms, homotopy, homology, cohomology, fiber bundles, connection and covariant derivatives, and Morse theory.
Topology and Geometry for Physics
by Helmut EschrigA concise but self-contained introduction of the central concepts of modern topology and differential geometry on a mathematical level is given specifically with applications in physics in mind. All basic concepts are systematically provided including sketches of the proofs of most statements. Smooth finite-dimensional manifolds, tensor and exterior calculus operating on them, homotopy, (co)homology theory including Morse theory of critical points, as well as the theory of fiber bundles and Riemannian geometry, are treated. Examples from physics comprise topological charges, the topology of periodic boundary conditions for solids, gauge fields, geometric phases in quantum physics and gravitation.
Topology and Its Applications
by William F. BasenerDiscover a unique and modern treatment of topology employing a cross-disciplinary approachImplemented recently to understand diverse topics, such as cell biology, superconductors, and robot motion, topology has been transformed from a theoretical field that highlights mathematical theory to a subject that plays a growing role in nearly all fields of scientific investigation. Moving from the concrete to the abstract, Topology and Its Applications displays both the beauty and utility of topology, first presenting the essentials of topology followed by its emerging role within the new frontiers in research.Filling a gap between the teaching of topology and its modern uses in real-world phenomena, Topology and Its Applications is organized around the mathematical theory of topology, a framework of rigorous theorems, and clear, elegant proofs.This book is the first of its kind to present applications in computer graphics, economics, dynamical systems, condensed matter physics, biology, robotics, chemistry, cosmology, material science, computational topology, and population modeling, as well as other areas of science and engineering. Many of these applications are presented in optional sections, allowing an instructor to customize the presentation.The author presents a diversity of topological areas, including point-set topology, geometric topology, differential topology, and algebraic/combinatorial topology. Topics within these areas include:Open sets Compactness Homotopy Surface classification Index theory on surfaces Manifolds and complexes Topological groups The fundamental group and homologySpecial "core intuition" segments throughout the book briefly explain the basic intuition essential to understanding several topics. A generous number of figures and examples, many of which come from applications such as liquid crystals, space probe data, and computer graphics, are all available from the publisher's Web site.
Topology and K-Theory: Lectures by Daniel Quillen (Lecture Notes in Mathematics #2262)
by Robert PennerThese are notes from a graduate student course on algebraic topology and K-theory given by Daniel Quillen at the Massachusetts Institute of Technology during 1979-1980. He had just received the Fields Medal for his work on these topics among others and was funny and playful with a confident humility from the start. These are not meant to be polished lecture notes, rather, things are presented as did Quillen reflected in the hand-written notes, resisting any temptation to change or add notation, details or elaborations. Indeed, the text is faithful to Quillen's own exposition, even respecting the {\sl board-like presentation} of formulae, diagrams and proofs, omitting numbering theorems in favor of names and so on. This is meant to be Quillen on Quillen as it happened forty years ago, an informal text for a second-semester graduate student on topology, category theory and K-theory, a potential preface to studying Quillen's own landmark papers and an informal glimpse of his great mind. The intellectual pace of the lectures, namely fast and lively, is Quillen himself, and part of the point here is to capture some of this intimacy. To be sure, much has happened since then from this categorical perspective started by Grothendieck, and Misha Kapranov has contributed an Afterword in order to make it more useful to current students.
Topology Essentials
by Emil G. MilewskiREA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Topology includes an overview of elementary set theory, relations and functions, ordinals and cardinals, topological spaces, continuous functions, metric spaces and normed spaces, countable spaces, and separation axioms.
Topology for Analysis (Dover Books on Mathematics)
by Albert WilanskyAppropriate for both students and professionals, this volume starts with the first principles of topology and advances to general analysis. Three levels of examples and problems, ordered and numbered by degree of difficulty, illustrate important concepts. A 40-page appendix, featuring tables of theorems and counter examples, provides a valuable reference.From explorations of topological space, convergence, and separation axioms, the text proceeds to considerations of sup and weak topologies, products and quotients, compactness and compactification, and complete semimetric space. The concluding chapters explore metrization, topological groups, and function spaces. Each subject area is supplemented with examples, problems, and exercises that progress to increasingly rigorous levels. All examples and problems are classified as essential, optional, and advanced.
Topology, Geometry and Gauge fields
by Gregory L. NaberLike any books on a subject as vast as this, this book has to have a point-of-view to guide the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. The reader is asked to join the author on some vague notion of what an electromagnetic field might be, to be willing to accept a few of the more elementary pronouncements of quantum mechanics, and to have a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. In return, the book offers an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1.
The Topology of Chaos: Alice in Stretch and Squeezeland
by Robert Gilmore Marc LefrancA 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. <P><P>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.
Topology of Infinite-Dimensional Manifolds (Springer Monographs in Mathematics)
by Katsuro SakaiAn infinite-dimensional manifold is a topological manifold modeled on some infinite-dimensional homogeneous space called a model space. In this book, the following spaces are considered model spaces: Hilbert space (or non-separable Hilbert spaces), the Hilbert cube, dense subspaces of Hilbert spaces being universal spaces for absolute Borel spaces, the direct limit of Euclidean spaces, and the direct limit of Hilbert cubes (which is homeomorphic to the dual of a separable infinite-dimensional Banach space with bounded weak-star topology). This book is designed for graduate students to acquire knowledge of fundamental results on infinite-dimensional manifolds and their characterizations. To read and understand this book, some background is required even for senior graduate students in topology, but that background knowledge is minimized and is listed in the first chapter so that references can easily be found. Almost all necessary background information is found in Geometric Aspects of General Topology, the author's first book. Many kinds of hyperspaces and function spaces are investigated in various branches of mathematics, which are mostly infinite-dimensional. Among them, many examples of infinite-dimensional manifolds have been found. For researchers studying such objects, this book will be very helpful. As outstanding applications of Hilbert cube manifolds, the book contains proofs of the topological invariance of Whitehead torsion and Borsuk’s conjecture on the homotopy type of compact ANRs. This is also the first book that presents combinatorial ∞-manifolds, the infinite-dimensional version of combinatorial n-manifolds, and proofs of two remarkable results, that is, any triangulation of each manifold modeled on the direct limit of Euclidean spaces is a combinatorial ∞-manifold and the Hauptvermutung for them is true.
A Topology of Mind: Spiral Thought Patterns, the Hyperlinking of Text, Ideas and More (Mathematics in Mind)
by Robert K. Logan Izabella Pruska-OldenhofThis volume covers many diverse topics related in varying degrees to mathematics in mind including the mathematical and topological structures of thought and communication. It examines mathematics in mind from the perspective of the spiral, cyclic and hyperlinked structures of the human mind in terms of its language, its thoughts and its various modes of communication in science, philosophy, literature and the arts including a chapter devoted to the spiral structure of the thought of Marshall McLuhan. In it, the authors examine the topological structures of hypertext, hyperlinking, and hypermedia made possible by the Internet and the hyperlinked structures that existed before its emergence. It also explores the cognitive origins of mathematical thinking of the human mind and its relation to the emergence of spoken language, and studies the emergence of mathematical notation and its impact on education. Topics addressed include:• The historical context of any topic that involves how mathematical thinking emerged, focusing on archaeological and philological evidence. • Connection between math cognition and symbolism, annotation and other semiotic processes. • Interrelationships between mathematical discovery and cultural processes, including technological systems that guide the thrust of cognitive and social evolution. • Whether mathematics is an innate faculty or forged in cultural-historical context• What, if any, structures are shared between mathematics and language
Topology Optimization Design of Heterogeneous Materials and Structures
by Daicong DaThis book pursues optimal design from the perspective of mechanical properties and resistance to failure caused by cracks and fatigue. The book abandons the scale separation hypothesis and takes up phase-field modeling, which is at the cutting edge of research and is of high industrial and practical relevance. Part 1 starts by testing the limits of the homogenization-based approach when the size of the representative volume element is non-negligible compared to the structure. The book then introduces a non-local homogenization scheme to take into account the strain gradient effects. Using a phase field method, Part 2 offers three significant contributions concerning optimal placement of the inclusion phases. Respectively, these contributions take into account fractures in quasi-brittle materials, interface cracks and periodic composites. The topology optimization proposed has significantly increased the fracture resistance of the composites studied.
Topology Optimization of Compliant Mechanisms
by Xianmin Zhang Benliang ZhuThis book covers various topics regarding the design of compliant mechanisms using topology optimization that have attracted a great deal of attention in recent decades. After comprehensively describing state-of-the-art methods for designing compliant mechanisms, it provides a new topology optimization method for finding new flexure hinges. It then presents several attempts to obtain distributed compliant mechanisms using the topology optimization method. Further, it discusses a Jacobian-based topology optimization method for compliant parallel mechanisms, and introduces readers to the topology optimization of compliant mechanisms, taking into account geometrical nonlinearity and reliability. Providing a systematic method for topology optimization of flexure hinges, which are essential for designing compliant mechanisms, the book offers a valuable resource for all readers who are interested in designing compliant mechanism-based positioning stages. In addition, the methods for solving the de facto hinges in topology optimized compliant mechanisms will benefit all engineers seeking to design micro-electro-mechanical system (MEMS) structures.
Topos Theory (Dover Books on Mathematics)
by P. T. JohnstoneOne of the best books on a relatively new branch of mathematics, this volume focuses on how topos theory integrates geometric and logical ideas into the foundations of mathematics and theoretical computer science. Topics include internal category theory, topologies and sheaves, geometric morphisms, natural number objects, cohomology, set theory, and more. 1977 edition.
Toric Topology and Polyhedral Products (Fields Institute Communications #89)
by Anthony Bahri Lisa Jeffrey Taras Panov Donald Stanley Stephen TheriaultThis book explores toric topology, polyhedral products and related mathematics from a wide range of perspectives, collectively giving an overview of the potential of the areas while contributing original research to drive the subject forward in interesting new directions. Contributions to this volume were written in connection to the thematic program Toric Topology and Polyhedral Products held at the Fields Institute from January-June 2020. 16 original conributions were inspired or influenced by the program.Toric Topology arose as a subject in its own right about twenty-five years ago. It sits at the intersection of commutative algebra, topology, combinatorics, algebraic geometry, and symplectic and convex geometry. Polyhedral products are a functorial generalization of a construction that is at the centre of Toric Topology. They are of independent interest and unify several constructions that arise in a diverse range of areas, such as geometric group theory, homotopy theory, algebraic combinatorics and subspace arrangements.
Tortuosity and Microstructure Effects in Porous Media: Classical Theories, Empirical Data and Modern Methods (Springer Series in Materials Science #333)
by Lorenz Holzer Philip Marmet Mathias Fingerle Andreas Wiegmann Matthias Neumann Volker SchmidtThis open access book presents a thorough look at tortuosity and microstructure effects in porous materials. The book delivers a comprehensive review of the subject, summarizing all key results in the field with respect to the underlying theories, empirical data available in the literature, modern methodologies and calculation approaches, and quantitative relationships between microscopic and macroscopic properties. It thoroughly discusses up to 20 different types of tortuosity and introduces a new classification scheme and nomenclature based on direct geometric tortuosities, indirect physics-based tortuosities, and mixed tortuosities (geometric and physics-based). The book also covers recent progress in 3D imaging and image modeling for studying novel aspects of tortuosity and associated transport properties in materials, while providing a comprehensive list of available software packages for practitioners in the community.This book is a must-read for researchers and students in materials science and engineering interested in a deeper understanding of microstructure–property relationships in porous materials. For energy materials in particular, such as lithium-ion batteries, tortuosity is a key microstructural parameter that can greatly impact long-term material performance. Thus, the information laid out in this book will also greatly benefit researchers interested in computational modeling and design of next-generation materials, especially those for sustainability and energy applications.
Torture and the Twilight of Empire: From Algiers to Baghdad (Human Rights and Crimes against Humanity #26)
by Marnia LazregTorture and the Twilight of Empire looks at the intimate relationship between torture and colonial domination through a close examination of the French army's coercive tactics during the Algerian war from 1954 to 1962. By tracing the psychological, cultural, and political meanings of torture at the end of the French empire, Marnia Lazreg also sheds new light on the United States and its recourse to torture in Iraq and Afghanistan. This book is nothing less than an anatomy of torture--its methods, justifications, functions, and consequences. Drawing extensively from archives, confessions by former torturers, interviews with former soldiers, and war diaries, as well as writings by Jean-Paul Sartre, Albert Camus, and others, Lazreg argues that occupying nations justify their systematic use of torture as a regrettable but necessary means of saving Western civilization from those who challenge their rule. She shows how torture was central to guerre révolutionnaire, a French theory of modern warfare that called for total war against the subject population and which informed a pacification strategy founded on brutal psychological techniques borrowed from totalitarian movements. Lazreg seeks to understand torture's impact on the Algerian population--especially women--and also on the French troops who became their torturers. She explores the roles Christianity and Islam played in rationalizing these acts, and the ways in which torture became not only routine but even acceptable. Written by a preeminent historical sociologist, Torture and the Twilight of Empire holds particularly disturbing lessons for us today as we carry out the War on Terror.
Total Addiction
by Kate RussoSeeing a total solar eclipse is often described as a once-in-a-lifetime experience. However, for many who have experienced totality, once-in-a-lifetime is simply not enough. They want more, and are willing to go to great lengths often at great expense to repeat the experience. What is it like to experience totality? What is it about the experience that motivates these eclipse chasers? Is there an eclipse chaser personality? Can eclipse chasing actually be described as an addiction? This book describes the people who dedicate their lives to chasing their dream.
Total Domination in Graphs
by Michael A. Henning Anders YeoTotal Domination in Graphs gives a clear understanding of this topic to any interested reader who has a modest background in graph theory. This book provides and explores the fundamentals of total domination in graphs. Some of the topics featured include the interplay between total domination in graphs and transversals in hypergraphs, and the association with total domination in graphs and diameter-2-critical graphs. Several proofs are included in this text which enables readers to acquaint themselves with a toolbox of proof techniques and ideas with which to attack open problems in the field. This work is an excellent resource for students interested in beginning their research in this field. Additionally, established researchers will find the book valuable to have as it contains the latest developments and open problems.
Total Manufacturing Assurance: Controlling Product Quality, Reliability, and Safety
by Douglas Brauer John CesaroneThis new edition presents an enhanced perspective for the innovative concept of Total Manufacturing Assurance (TMA) and the holistic means by which such assurance can be attained. In fulfilling this objective, this textbook discusses the management and engineering techniques and tools, as well as the practical implementation required to achieve TMA. Using a holistic approach to manufacturing operations, Total Manufacturing Assurance: Controlling Product Quality, Reliability, and Safety, Second Edition focuses on analytics and performance assessment, along with Industry 4.0 and the role it plays in advanced manufacturing. The textbook covers strategic planning, innovation, and engineering economics, as well as the manufacturing process, materials, and operations. Product manufacturing system reliability, maintainability, availability, quality, and safety, along with financial issues in decision-making and engineering analysis, are all captured in this new edition. Students at undergraduate and graduate levels studying engineering management, mechanical, industrial, and manufacturing engineering, as well as business students will find this new edition an invaluable instructional resource. At the same time, working professionals, including management, engineers, and others who are intimately involved in the manufacturing system sector will also find this textbook very useful in their day-to-day work. PowerPoint slides and a solutions manual are available to instructors for qualified course adoptions.
Total Survey Error in Practice: Improving Quality In The Era Of Big Data (Wiley Series In Survey Methodology Ser.)
by Frauke Kreuter Paul P. Biemer Edith De Leeuw Brad Edwards Brady T. West Lars E. Lyberg N. Clyde Tucker Stephanie EckmanFeaturing a timely presentation of total survey error (TSE), this edited volume introduces valuable tools for understanding and improving survey data quality in the context of evolving large-scale data sets This book provides an overview of the TSE framework and current TSE research as related to survey design, data collection, estimation, and analysis. It recognizes that survey data affects many public policy and business decisions and thus focuses on the framework for understanding and improving survey data quality. The book also addresses issues with data quality in official statistics and in social, opinion, and market research as these fields continue to evolve, leading to larger and messier data sets. This perspective challenges survey organizations to find ways to collect and process data more efficiently without sacrificing quality. The volume consists of the most up-to-date research and reporting from over 70 contributors representing the best academics and researchers from a range of fields. The chapters are broken out into five main sections: The Concept of TSE and the TSE Paradigm, Implications for Survey Design, Data Collection and Data Processing Applications, Evaluation and Improvement, and Estimation and Analysis. Each chapter introduces and examines multiple error sources, such as sampling error, measurement error, and nonresponse error, which often offer the greatest risks to data quality, while also encouraging readers not to lose sight of the less commonly studied error sources, such as coverage error, processing error, and specification error. The book also notes the relationships between errors and the ways in which efforts to reduce one type can increase another, resulting in an estimate with larger total error. This book: • Features various error sources, and the complex relationships between them, in 25 high-quality chapters on the most up-to-date research in the field of TSE • Provides comprehensive reviews of the literature on error sources as well as data collection approaches and estimation methods to reduce their effects • Presents examples of recent international events that demonstrate the effects of data error, the importance of survey data quality, and the real-world issues that arise from these errors • Spans the four pillars of the total survey error paradigm (design, data collection, evaluation and analysis) to address key data quality issues in official statistics and survey research Total Survey Error in Practice is a reference for survey researchers and data scientists in research areas that include social science, public opinion, public policy, and business. It can also be used as a textbook or supplementary material for a graduate-level course in survey research methods. Paul P. Biemer, PhD, is distinguished fellow at RTI International and associate director of Survey Research and Development at the Odum Institute, University of North Carolina, USA. Edith de Leeuw, PhD, is professor of survey methodology in the Department of Methodology and Statistics at Utrecht University, the Netherlands. Stephanie Eckman, PhD, is fellow at RTI International, USA. Brad Edwards is vice president, director of Field Services, and deputy area director at Westat, USA. Frauke Kreuter, PhD, is professor and director of the Joint Program in Survey Methodology, University of Maryland, USA; professor of statistics and methodology at the University of Mannheim, Germany; and head of the Statistical Methods Research Department at the Institute for Employment Research, Germany. Lars E. Lyberg, PhD, is senior advisor at Inizio, Sweden. N. Clyde Tucker, PhD, is principal survey methodologist at the American Institutes for Research, USA. Brady T. West, PhD, is research associate professor in the Survey Resea
Totality and Infinity: An Essay on Exteriority
by Emmanuel Levinas Alphonso LingisFirst published in English by Duquesne in 1969, this has become one of the classics of modern philosophy.
Totally Nonnegative Matrices (Princeton Series in Applied Mathematics #35)
by Shaun M. Fallat Charles R. JohnsonTotally nonnegative matrices arise in a remarkable variety of mathematical applications. This book is a comprehensive and self-contained study of the essential theory of totally nonnegative matrices, defined by the nonnegativity of all subdeterminants. It explores methodological background, historical highlights of key ideas, and specialized topics.The book uses classical and ad hoc tools, but a unifying theme is the elementary bidiagonal factorization, which has emerged as the single most important tool for this particular class of matrices. Recent work has shown that bidiagonal factorizations may be viewed in a succinct combinatorial way, leading to many deep insights. Despite slow development, bidiagonal factorizations, along with determinants, now provide the dominant methodology for understanding total nonnegativity. The remainder of the book treats important topics, such as recognition of totally nonnegative or totally positive matrices, variation diminution, spectral properties, determinantal inequalities, Hadamard products, and completion problems associated with totally nonnegative or totally positive matrices. The book also contains sample applications, an up-to-date bibliography, a glossary of all symbols used, an index, and related references.
A Tour of the Calculus
by David BerlinskiIn its largest aspect, the calculus functions as a celestial measuring tape, able to order the infinite expanse of the universe. Time and space are given names, points, and limits; seemingly intractable problems of motion, growth, and form are reduced to answerable questions. Calculus was humanity's first attempt to represent the world and perhaps its greatest meditation on the theme of continuity. Charts and graphs throughout.From the Hardcover edition.
A Tour of the Calculus
by David BerlinskiIn its largest aspect, the calculus functions as a celestial measuring tape, able to order the infinite expanse of the universe. Time and space are given names, points, and limits; seemingly intractable problems of motion, growth, and form are reduced to answerable questions. Calculus was humanity's first attempt to represent the world and perhaps its greatest meditation on the theme of continuity. Charts and graphs throughout.From the Hardcover edition.
A Tour of the Human Body: Amazing Numbers--Fantastic Facts (Number Tours for Curious Kids)
by Jennifer BerneJennifer Berne takes children on a tour of the human body to reveal the wonders of how it works -- with some astonishing numbers and fascinating facts along the way.From our eyes to our toes, kids will find out what makes the human body tick. They&’ll discover that our hearts beat 100,000 times a day, which equals 36 MILLION times a year. And that our tongue&’s 8,000 taste buds can detect only 5 flavors. And that we have 60,000 miles of blood vessels, enough to circle the world more than twice!With such remarkable facts and numbers, and vivid informative illustrations by Dawn DeVries Sokol, this book takes your child on an entertainingly educational journey through the wonders of the human body.