Geodesic and Horocyclic Trajectories

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Author: Françoise Dal’Bo

Publisher: Springer Science & Business Media

ISBN: 9780857290731

Category: Mathematics

Page: 176

View: 3152

Geodesic and Horocyclic Trajectories presents an introduction to the topological dynamics of two classical flows associated with surfaces of curvature −1, namely the geodesic and horocycle flows. Written primarily with the idea of highlighting, in a relatively elementary framework, the existence of gateways between some mathematical fields, and the advantages of using them, historical aspects of this field are not addressed and most of the references are reserved until the end of each chapter in the Comments section. Topics within the text cover geometry, and examples, of Fuchsian groups; topological dynamics of the geodesic flow; Schottky groups; the Lorentzian point of view and Trajectories and Diophantine approximations.

Ergodic Theory and Negative Curvature

CIRM Jean-Morlet Chair, Fall 2013

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Author: Boris Hasselblatt

Publisher: Springer

ISBN: 3319430599

Category: Mathematics

Page: 328

View: 8552

Focussing on the mathematics related to the recent proof of ergodicity of the (Weil–Petersson) geodesic flow on a nonpositively curved space whose points are negatively curved metrics on surfaces, this book provides a broad introduction to an important current area of research. It offers original textbook-level material suitable for introductory or advanced courses as well as deep insights into the state of the art of the field, making it useful as a reference and for self-study. The first chapters introduce hyperbolic dynamics, ergodic theory and geodesic and horocycle flows, and include an English translation of Hadamard's original proof of the Stable-Manifold Theorem. An outline of the strategy, motivation and context behind the ergodicity proof is followed by a careful exposition of it (using the Hopf argument) and of the pertinent context of Teichmüller theory. Finally, some complementary lectures describe the deep connections between geodesic flows in negative curvature and Diophantine approximation.

The Spectrum of Hyperbolic Surfaces

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Author: Nicolas Bergeron

Publisher: Springer

ISBN: 3319276662

Category: Mathematics

Page: 370

View: 4099

This text is an introduction to the spectral theory of the Laplacian on compact or finite area hyperbolic surfaces. For some of these surfaces, called “arithmetic hyperbolic surfaces”, the eigenfunctions are of arithmetic nature, and one may use analytic tools as well as powerful methods in number theory to study them. After an introduction to the hyperbolic geometry of surfaces, with a special emphasis on those of arithmetic type, and then an introduction to spectral analytic methods on the Laplace operator on these surfaces, the author develops the analogy between geometry (closed geodesics) and arithmetic (prime numbers) in proving the Selberg trace formula. Along with important number theoretic applications, the author exhibits applications of these tools to the spectral statistics of the Laplacian and the quantum unique ergodicity property. The latter refers to the arithmetic quantum unique ergodicity theorem, recently proved by Elon Lindenstrauss. The fruit of several graduate level courses at Orsay and Jussieu, The Spectrum of Hyperbolic Surfaces allows the reader to review an array of classical results and then to be led towards very active areas in modern mathematics.

A Course on Mathematical Logic

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Author: Shashi Mohan Srivastava

Publisher: Springer Science & Business Media

ISBN: 1461457467

Category: Mathematics

Page: 198

View: 2181

This is a short, modern, and motivated introduction to mathematical logic for upper undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in getting acquainted with logic and would like to learn Gödel’s incompleteness theorems should find this book particularly useful. The treatment is thoroughly mathematical and prepares students to branch out in several areas of mathematics related to foundations and computability, such as logic, axiomatic set theory, model theory, recursion theory, and computability. In this new edition, many small and large changes have been made throughout the text. The main purpose of this new edition is to provide a healthy first introduction to model theory, which is a very important branch of logic. Topics in the new chapter include ultraproduct of models, elimination of quantifiers, types, applications of types to model theory, and applications to algebra, number theory and geometry. Some proofs, such as the proof of the very important completeness theorem, have been completely rewritten in a more clear and concise manner. The new edition also introduces new topics, such as the notion of elementary class of structures, elementary diagrams, partial elementary maps, homogeneous structures, definability, and many more.

A Course in Constructive Algebra

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Author: Ray Mines,Fred Richman,Wim Ruitenburg

Publisher: Springer Science & Business Media

ISBN: 1441986405

Category: Mathematics

Page: 344

View: 8213

The constructive approach to mathematics has enjoyed a renaissance, caused in large part by the appearance of Errett Bishop's book Foundations of constr"uctiue analysis in 1967, and by the subtle influences of the proliferation of powerful computers. Bishop demonstrated that pure mathematics can be developed from a constructive point of view while maintaining a continuity with classical terminology and spirit; much more of classical mathematics was preserved than had been thought possible, and no classically false theorems resulted, as had been the case in other constructive schools such as intuitionism and Russian constructivism. The computers created a widespread awareness of the intuitive notion of an effecti ve procedure, and of computation in principle, in addi tion to stimulating the study of constructive algebra for actual implementation, and from the point of view of recursive function theory. In analysis, constructive problems arise instantly because we must start with the real numbers, and there is no finite procedure for deciding whether two given real numbers are equal or not (the real numbers are not discrete) . The main thrust of constructive mathematics was in the direction of analysis, although several mathematicians, including Kronecker and van der waerden, made important contributions to construc tive algebra. Heyting, working in intuitionistic algebra, concentrated on issues raised by considering algebraic structures over the real numbers, and so developed a handmaiden'of analysis rather than a theory of discrete algebraic structures.

Algebras of Linear Transformations

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Author: Douglas R. Farenick

Publisher: Springer Science & Business Media

ISBN: 1461300975

Category: Mathematics

Page: 240

View: 6283

This book studies algebras and linear transformations acting on finite-dimensional vector spaces over arbitrary fields. It is written for readers who have prior knowledge of algebra and linear algebra. The goal is to present a balance of theory and example in order for readers to gain a firm understanding of the basic theory of finite-dimensional algebras and to provide a foundation for subsequent advanced study in a number of areas of mathematics.

Ergodic Theory and Dynamical Systems

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Author: Yves Coudène

Publisher: Springer

ISBN: 1447172876

Category: Mathematics

Page: 190

View: 6880

This textbook is a self-contained and easy-to-read introduction to ergodic theory and the theory of dynamical systems, with a particular emphasis on chaotic dynamics. This book contains a broad selection of topics and explores the fundamental ideas of the subject. Starting with basic notions such as ergodicity, mixing, and isomorphisms of dynamical systems, the book then focuses on several chaotic transformations with hyperbolic dynamics, before moving on to topics such as entropy, information theory, ergodic decomposition and measurable partitions. Detailed explanations are accompanied by numerous examples, including interval maps, Bernoulli shifts, toral endomorphisms, geodesic flow on negatively curved manifolds, Morse-Smale systems, rational maps on the Riemann sphere and strange attractors. Ergodic Theory and Dynamical Systems will appeal to graduate students as well as researchers looking for an introduction to the subject. While gentle on the beginning student, the book also contains a number of comments for the more advanced reader.

Fuchsian Groups

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Author: Svetlana Katok

Publisher: University of Chicago Press

ISBN: 9780226425825

Category: Mathematics

Page: 175

View: 722

This introductory text provides a thoroughly modern treatment of Fuchsian groups that addresses both the classical material and recent developments in the field. A basic example of lattices in semisimple groups, Fuchsian groups have extensive connections to the theory of a single complex variable, number theory, algebraic and differential geometry, topology, Lie theory, representation theory, and group theory.

Using the Borsuk-Ulam Theorem

Lectures on Topological Methods in Combinatorics and Geometry

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Author: Jiri Matousek

Publisher: Springer Science & Business Media

ISBN: 3540766499

Category: Mathematics

Page: 214

View: 1775

To the uninitiated, algebraic topology might seem fiendishly complex, but its utility is beyond doubt. This brilliant exposition goes back to basics to explain how the subject has been used to further our understanding in some key areas. A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. This book is the first textbook treatment of a significant part of these results. It focuses on so-called equivariant methods, based on the Borsuk-Ulam theorem and its generalizations. The topological tools are intentionally kept on a very elementary level. No prior knowledge of algebraic topology is assumed, only a background in undergraduate mathematics, and the required topological notions and results are gradually explained.

Functional Spaces for the Theory of Elliptic Partial Differential Equations

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Author: Françoise Demengel,Gilbert Demengel

Publisher: Springer Science & Business Media

ISBN: 1447128079

Category: Mathematics

Page: 465

View: 5503

The theory of elliptic boundary problems is fundamental in analysis and the role of spaces of weakly differentiable functions (also called Sobolev spaces) is essential in this theory as a tool for analysing the regularity of the solutions. This book offers on the one hand a complete theory of Sobolev spaces, which are of fundamental importance for elliptic linear and non-linear differential equations, and explains on the other hand how the abstract methods of convex analysis can be combined with this theory to produce existence results for the solutions of non-linear elliptic boundary problems. The book also considers other kinds of functional spaces which are useful for treating variational problems such as the minimal surface problem. The main purpose of the book is to provide a tool for graduate and postgraduate students interested in partial differential equations, as well as a useful reference for researchers active in the field. Prerequisites include a knowledge of classical analysis, differential calculus, Banach and Hilbert spaces, integration and the related standard functional spaces, as well as the Fourier transformation on the Schwartz space. There are complete and detailed proofs of almost all the results announced and, in some cases, more than one proof is provided in order to highlight different features of the result. Each chapter concludes with a range of exercises of varying levels of difficulty, with hints to solutions provided for many of them.

Vitushkin’s Conjecture for Removable Sets

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Author: James Dudziak

Publisher: Springer Science & Business Media

ISBN: 9781441967091

Category: Mathematics

Page: 332

View: 1356

Vitushkin's conjecture, a special case of Painlevé's problem, states that a compact subset of the complex plane with finite linear Hausdorff measure is removable for bounded analytic functions if and only if it intersects every rectifiable curve in a set of zero arclength measure. Chapters 1-5 of the book provide important background material on removability, analytic capacity, Hausdorff measure, arclength measure, and Garabedian duality that will appeal to many analysts with interests independent of Vitushkin's conjecture. The fourth chapter contains a proof of Denjoy's conjecture that employs Melnikov curvature. A brief postscript reports on a deep theorem of Tolsa and its relevance to going beyond Vitushkin's conjecture. This text can be used for a topics course or seminar in complex analysis. To understand it, the reader should have a firm grasp of basic real and complex analysis.

Geometry I

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Author: Marcel Berger,M. Cole

Publisher: Springer Science & Business Media

ISBN: 9783540116585

Category: Mathematics

Page: 432

View: 1649

The first part of a two-volume text providing a readable and lively presentation of large parts of geometry in the classical sense, this book appeals systematically to the reader's intuition and vision, and illustrates the mathematical text with many figures.

Automorphic Forms

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Author: Anton Deitmar

Publisher: Springer Science & Business Media

ISBN: 144714435X

Category: Mathematics

Page: 252

View: 850

Automorphic forms are an important complex analytic tool in number theory and modern arithmetic geometry. They played for example a vital role in Andrew Wiles's proof of Fermat's Last Theorem. This text provides a concise introduction to the world of automorphic forms using two approaches: the classic elementary theory and the modern point of view of adeles and representation theory. The reader will learn the important aims and results of the theory by focussing on its essential aspects and restricting it to the 'base field' of rational numbers. Students interested for example in arithmetic geometry or number theory will find that this book provides an optimal and easily accessible introduction into this topic.

Lattices and Ordered Algebraic Structures

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Author: T.S. Blyth

Publisher: Springer Science & Business Media

ISBN: 184628127X

Category: Mathematics

Page: 304

View: 4957

"The text can serve as an introduction to fundamentals in the respective areas from a residuated-maps perspective and with an eye on coordinatization. The historical notes that are interspersed are also worth mentioning....The exposition is thorough and all proofs that the reviewer checked were highly polished....Overall, the book is a well-done introduction from a distinct point of view and with exposure to the author’s research expertise." --MATHEMATICAL REVIEWS

A Course in Formal Languages, Automata and Groups

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Author: Ian M. Chiswell

Publisher: Springer Science & Business Media

ISBN: 1848009402

Category: Mathematics

Page: 157

View: 8557

This book is based on notes for a master’s course given at Queen Mary, University of London, in the 1998/9 session. Such courses in London are quite short, and the course consisted essentially of the material in the ?rst three chapters, together with a two-hour lecture on connections with group theory. Chapter 5 is a considerably expanded version of this. For the course, the main sources were the books by Hopcroft and Ullman ([20]), by Cohen ([4]), and by Epstein et al. ([7]). Some use was also made of a later book by Hopcroft and Ullman ([21]). The ulterior motive in the ?rst three chapters is to give a rigorous proof that various notions of recursively enumerable language are equivalent. Three such notions are considered. These are: generated by a type 0 grammar, recognised by a Turing machine (deterministic or not) and de?ned by means of a Godel ̈ numbering, having de?ned “recursively enumerable” for sets of natural numbers. It is hoped that this has been achieved without too many ar- ments using complicated notation. This is a problem with the entire subject, and it is important to understand the idea of the proof, which is often quite simple. Two particular places that are heavy going are the proof at the end of Chapter 1 that a language recognised by a Turing machine is type 0, and the proof in Chapter 2 that a Turing machine computable function is partial recursive.

Handbook of Dynamical Systems

Volume 1B

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Author: A. Katok,B. Hasselblatt

Publisher: Elsevier

ISBN: 9780080478227

Category: Mathematics

Page: 1234

View: 4637

This second half of Volume 1 of this Handbook follows Volume 1A, which was published in 2002. The contents of these two tightly integrated parts taken together come close to a realization of the program formulated in the introductory survey “Principal Structures of Volume 1A. The present volume contains surveys on subjects in four areas of dynamical systems: Hyperbolic dynamics, parabolic dynamics, ergodic theory and infinite-dimensional dynamical systems (partial differential equations). . Written by experts in the field. . The coverage of ergodic theory in these two parts of Volume 1 is considerably more broad and thorough than that provided in other existing sources. . The final cluster of chapters discusses partial differential equations from the point of view of dynamical systems.

Advanced Techniques in Diagnostic Microbiology

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Author: Yi-Wei Tang,Charles W. Stratton

Publisher: Springer Science & Business Media

ISBN: 1461439701

Category: Medical

Page: 957

View: 8093

In the United States, hospitals annually report over 5 million cases of infectious-disease-related illnesses: clinical microbiology laboratories in these hospitals are engaged in detecting and identifying the pathogenic microorganisms in clinical specimens collected from these patients with suspected infections. Clearly, the timely and accurate detection/identification of these microbial pathogens is critical for patient treatment decisions and outcomes for millions of patients each year. Despite an appreciation that the outcome of an infectious-disease-related illness is directly related to the time required to detect and identify a microbial pathogen, clinical microbiology laboratories in the United States as well as worldwide have long been hampered by traditional culture-based assays, which may require prolonged incubation time for slowly growing microorganisms such as Mycobacterium tuberculosis. Moreover, traditional culture-based assays often require multiple steps with additional time needed for discernment of species and/or detection of antimicrobial resistance. Finally, these traditional, slow multistep culture-based assays are labor-intensive and required skilled clinical microbiologists at the bench. Over the past several decades, advanced molecular techniques in diagnostic microbiology quietly have been revolutionizing the practice of clinical microbiology in the hospital setting. Indeed, molecular diagnostic testing in general and nucleic-acid-based amplification methods in particular have been heralded as diagnostic tools for the new millennium. There is no question that the development of rapid molecular techniques for nucleic acid amplification/characterization combined with automation and user-friendly software has greatly broadened the diagnostic capabilities of the clinical microbiology laboratory. These technical advances in molecular microbiology over the first decade of the 21st Century have profoundly influenced the physical structure of clinical microbiology laboratories as well as their staffing patterns, workflow, and turnaround time. These molecular microbiology advances have also resulted in the need for a revised and updated second edition of Advanced Techniques in Diagnostic Microbiology. This second edition again provides an updated and comprehensive description of the ongoing evolution of molecular methods for the diagnosis of infectious diseases. In addition, many new chapters have been added, including a chapter on the clinical interpretation and relevance of advanced technique results. The second edition, like the first edition, includes both a “techniques” section describing the latest molecular techniques and an “applications” section describing how these advanced molecular techniques are being used in the clinical setting. Finally, the second edition, like the first edition, utilizes a diverse team of authors who have compiled chapters that provide the reader with comprehensive and useable information on advanced molecular microbiology techniques.

Analytic and Probabilistic Approaches to Dynamics in Negative Curvature

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Author: Françoise Dal'Bo,Marc Peigné,Andrea Sambusetti

Publisher: Springer

ISBN: 3319048074

Category: Mathematics

Page: 138

View: 2744

The work consists of two introductory courses, developing different points of view on the study of the asymptotic behaviour of the geodesic flow, namely: the probabilistic approach via martingales and mixing (by Stéphane Le Borgne); the semi-classical approach, by operator theory and resonances (by Frédéric Faure and Masato Tsujii). The contributions aim to give a self-contained introduction to the ideas behind the three different approaches to the investigation of hyperbolic dynamics. The first contribution focus on the convergence towards a Gaussian law of suitably normalized ergodic sums (Central Limit Theorem). The second one deals with Transfer Operators and the structure of their spectrum (Ruelle-Pollicott resonances), explaining the relation with the asymptotics of time correlation function and the periodic orbits of the dynamics.

Topological Dynamics

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Author: Walter Helbig Gottschalk,Gustav Arnold Hedlund

Publisher: American Mathematical Soc.

ISBN: 9780821874691

Category: Mathematics

Page: 167

View: 9721

Geometry and Spectra of Compact Riemann Surfaces

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Author: Peter Buser

Publisher: Springer Science & Business Media

ISBN: 9780817649920

Category: Mathematics

Page: 456

View: 8526

This monograph is a self-contained introduction to the geometry of Riemann Surfaces of constant curvature –1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces. Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.