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Applications Of Centre Manifold Theory

Author: J. Carr
Publisher: Springer Science & Business Media
ISBN: 1461259290
Size: 65.37 MB
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These notes are based on a series of lectures given in the Lefschetz Center for Dynamical Systems in the Division of Applied Mathematics at Brown University during the academic year 1978-79. The purpose of the lectures was to give an introduction to the applications of centre manifold theory to differential equations. Most of the material is presented in an informal fashion, by means of worked examples in the hope that this clarifies the use of centre manifold theory. The main application of centre manifold theory given in these notes is to dynamic bifurcation theory. Dynamic bifurcation theory is concerned with topological changes in the nature of the solutions of differential equations as para meters are varied. Such an example is the creation of periodic orbits from an equilibrium point as a parameter crosses a critical value. In certain circumstances, the application of centre manifold theory reduces the dimension of the system under investigation. In this respect the centre manifold theory plays the same role for dynamic problems as the Liapunov-Schmitt procedure plays for the analysis of static solutions. Our use of centre manifold theory in bifurcation problems follows that of Ruelle and Takens [57) and of Marsden and McCracken [51).

Manifold Theory

Author: D. Martin
Publisher: Elsevier
ISBN: 0857099639
Size: 29.36 MB
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This account of basic manifold theory and global analysis, based on senior undergraduate and post-graduate courses at Glasgow University for students and researchers in theoretical physics, has been proven over many years. The treatment is rigorous yet less condensed than in books written primarily for pure mathematicians. Prerequisites include knowledge of basic linear algebra and topology. Topology is included in two appendices because many courses on mathematics for physics students do not include this subject. Provides a comprehensive account of basic manifold theory for post-graduate students Introduces the basic theory of differential geometry to students in theoretical physics and mathematics Contains more than 130 exercises, with helpful hints and solutions

Four Manifold Theory

Author: Cameron Gordon
Publisher: American Mathematical Soc.
ISBN: 0821850334
Size: 16.97 MB
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These are the proceedings of the Summer Research Conference on 4-manifolds held at Durham, New Hampshire, July 1982, under the auspices of the American Mathematical Society and National Science Foundation. The conference was highlighted by the breakthroughs of Michael Freedman and S. K. Donaldson and by Frank Quinn's completion at the conference of the proof of the annulus conjecture. (We commend the AMS committee, particularly Julius Shaneson, who had the foresight in Spring 1981 to choose the subject, 4-manifolds, in which such remarkable activity was imminent.) Freedman and several others spoke on his work; some of their talks are represented by papers in this volume. Donaldson and Clifford H. Taubes gave surveys of their work on gauge theory and 4-manifolds and their papers are also included herein. There were a variety of other lectures, including Quinn's surprise, and a couple of problem sessions which led to the problem list. A background of basic differential topology is adequate for potential readers.

Invariant Manifold Theory For Hydrodynamic Transition

Author: S.S. Sritharan
Publisher: Dover Publications
ISBN: 048682828X
Size: 45.60 MB
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Invariant manifold theory serves as a link between dynamical systems theory and turbulence phenomena. This volume consists of research notes by author S. S. Sritharan that develop a theory for the Navier-Stokes equations in bounded and certain unbounded geometries. The main results include spectral theorems and analyticity theorems for semigroups and invariant manifolds. "This monograph contains a lot of useful information, including much that cannot be found in the standard texts on the Navier-Stokes equations," observed MathSciNet, adding "the book is well worth the reader's attention." The treatment is suitable for researchers and graduate students in the areas of chaos and turbulence theory, hydrodynamic stability, dynamical systems, partial differential equations, and control theory. Topics include the governing equations and the functional framework, the linearized operator and its spectral properties, the monodromy operator and its properties, the nonlinear hydrodynamic semigroup, invariant cone theorem, and invariant manifold theorem. Two helpful appendixes conclude the text.

Manifold Learning Theory And Applications

Author: Yunqian Ma
Publisher: CRC Press
ISBN: 1466558873
Size: 36.54 MB
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Trained to extract actionable information from large volumes of high-dimensional data, engineers and scientists often have trouble isolating meaningful low-dimensional structures hidden in their high-dimensional observations. Manifold learning, a groundbreaking technique designed to tackle these issues of dimensionality reduction, finds widespread application in machine learning, neural networks, pattern recognition, image processing, and computer vision. Filling a void in the literature, Manifold Learning Theory and Applications incorporates state-of-the-art techniques in manifold learning with a solid theoretical and practical treatment of the subject. Comprehensive in its coverage, this pioneering work explores this novel modality from algorithm creation to successful implementation—offering examples of applications in medical, biometrics, multimedia, and computer vision. Emphasizing implementation, it highlights the various permutations of manifold learning in industry including manifold optimization, large scale manifold learning, semidefinite programming for embedding, manifold models for signal acquisition, compression and processing, and multi scale manifold. Beginning with an introduction to manifold learning theories and applications, the book includes discussions on the relevance to nonlinear dimensionality reduction, clustering, graph-based subspace learning, spectral learning and embedding, extensions, and multi-manifold modeling. It synergizes cross-domain knowledge for interdisciplinary instructions, offers a rich set of specialized topics contributed by expert professionals and researchers from a variety of fields. Finally, the book discusses specific algorithms and methodologies using case studies to apply manifold learning for real-world problems.

Topological Quantum Field Theory And Four Manifolds

Author: Jose Labastida
Publisher: Springer Science & Business Media
ISBN: 1402031777
Size: 54.27 MB
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The emergence of topological quantum ?eld theory has been one of the most important breakthroughs which have occurred in the context of ma- ematical physics in the last century, a century characterizedbyindependent developments of the main ideas in both disciplines, physics and mathematics, which has concluded with two decades of strong interaction between them, where physics, as in previous centuries, has acted as a source of new mat- matics. Topological quantum ?eld theories constitute the core of these p- nomena, although the main drivingforce behind it has been the enormous e?ort made in theoretical particle physics to understand string theory as a theory able to unify the four fundamental interactions observed in nature. These theories set up a new realm where both disciplines pro?t from each other. Although the most striking results have appeared on the mathema- calside,theoreticalphysicshasclearlyalsobene?tted,sincethecorresponding developments have helped better to understand aspects of the fundamentals of ?eld and string theory.

Manifolds And K Theory

Author: Gregory Arone
Publisher: American Mathematical Soc.
ISBN: 1470417006
Size: 34.58 MB
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This volume contains the proceedings of the conference on Manifolds, -Theory, and Related Topics, held from June 23–27, 2014, in Dubrovnik, Croatia. The articles contained in this volume are a collection of research papers featuring recent advances in homotopy theory, -theory, and their applications to manifolds. Topics covered include homotopy and manifold calculus, structured spectra, and their applications to group theory and the geometry of manifolds. This volume is a tribute to the influence of Tom Goodwillie in these fields.

Bifurcation Theory Of Functional Differential Equations

Author: Shangjiang Guo
Publisher: Springer Science & Business Media
ISBN: 1461469929
Size: 41.24 MB
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This book provides a crash course on various methods from the bifurcation theory of Functional Differential Equations (FDEs). FDEs arise very naturally in economics, life sciences and engineering and the study of FDEs has been a major source of inspiration for advancement in nonlinear analysis and infinite dimensional dynamical systems. The book summarizes some practical and general approaches and frameworks for the investigation of bifurcation phenomena of FDEs depending on parameters with chap. This well illustrated book aims to be self contained so the readers will find in this book all relevant materials in bifurcation, dynamical systems with symmetry, functional differential equations, normal forms and center manifold reduction. This material was used in graduate courses on functional differential equations at Hunan University (China) and York University (Canada).

Elements Of Applied Bifurcation Theory

Author: Yuri Kuznetsov
Publisher: Springer Science & Business Media
ISBN: 9780387219066
Size: 19.30 MB
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Providing readers with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems, the focus here is on efficient numerical implementations of the developed techniques. The book is designed for advanced undergraduates or graduates in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the first while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.