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Isolated Invariant Sets And The Morse Index

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Isolated Invariant Sets And The Morse Index

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Author : Charles C. Conley
language : en
Publisher: American Mathematical Soc.
Release Date : 1978-12-31

Isolated Invariant Sets And The Morse Index written by Charles C. Conley and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1978-12-31 with Mathematics categories.

This volume contains lectures from the Conference Board of Mathematical Sciences meeting held at the University of Colorado on May 31-June 4, 1976. The lectures consist of an expository discussion of basic results for topological flows and a somewhat more detailed discussion of isolated invariant sets and continuation. The construction of the index for isolated invariant sets is new and allows more general application than previous ones. Also, the index itself is endowed with more structure and the continuation theorem is modified to take this new structure into account. Some elementary applications are given, but the main emphasis is on the abstract theory.

Isolated Invariant Sets And The Morse Index

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Author : Charles C. Conley
language : en
Publisher:
Release Date : 1978

Isolated Invariant Sets And The Morse Index written by Charles C. Conley and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1978 with Differentiable dynamical systems categories.

Isolated Invariant Sets And The Morse Index

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Author : Charles Conley
language : en
Publisher:
Release Date : 1978

Isolated Invariant Sets And The Morse Index written by Charles Conley and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1978 with Differentiable dynamical systems categories.

Isolated Invariant Sets And The Morse Index

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Author : Charles Conley
language : en
Publisher:
Release Date : 1976

Isolated Invariant Sets And The Morse Index written by Charles Conley and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1976 with Differentiable dynamical systems categories.

Topological Methods Variational Methods And Their Applications

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Author : Haim Brzis
language : en
Publisher: World Scientific
Release Date : 2003

Topological Methods Variational Methods And Their Applications written by Haim Brzis and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Mathematics categories.

ICM 2002 Satellite Conference on Nonlinear Analysis was held in the period: August 1418, 2002 at Taiyuan, Shanxi Province, China. This conference was organized by Mathematical School of Peking University, Academy of Mathematics and System Sciences of Chinese Academy of Sciences, Mathematical school of Nankai University, and Department of Mathematics of Shanxi University, and was sponsored by Shanxi Province Education Committee, Tian Yuan Mathematics Foundation, and Shanxi University. 166 mathematicians from 21 countries and areas in the world attended the conference. 53 invited speakers and 30 contributors presented their lectures. This conference aims at an overview of the recent development in nonlinear analysis. It covers the following topics: variational methods, topological methods, fixed point theory, bifurcations, nonlinear spectral theory, nonlinear Schrvdinger equations, semilinear elliptic equations, Hamiltonian systems, central configuration in N-body problems and variational problems arising in geometry and physics.

Morse Theoretic Methods In Nonlinear Analysis And In Symplectic Topology

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Author : Paul Biran
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-02-12

Morse Theoretic Methods In Nonlinear Analysis And In Symplectic Topology written by Paul Biran and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-02-12 with Mathematics categories.

The papers collected in this volume are contributions to the 43rd session of the Seminaire ́ de mathematiques ́ superieures ́ (SMS) on “Morse Theoretic Methods in Nonlinear Analysis and Symplectic Topology.” This session took place at the Universite ́ de Montreal ́ in July 2004 and was a NATO Advanced Study Institute (ASI). The aim of the ASI was to bring together young researchers from various parts of the world and to present to them some of the most signi cant recent advances in these areas. More than 77 mathematicians from 17 countries followed the 12 series of lectures and participated in the lively exchange of ideas. The lectures covered an ample spectrum of subjects which are re ected in the present volume: Morse theory and related techniques in in nite dim- sional spaces, Floer theory and its recent extensions and generalizations, Morse and Floer theory in relation to string topology, generating functions, structure of the group of Hamiltonian di?eomorphisms and related dynamical problems, applications to robotics and many others. We thank all our main speakers for their stimulating lectures and all p- ticipants for creating a friendly atmosphere during the meeting. We also thank Ms. Diane Belanger ́ , our administrative assistant, for her help with the organi- tion and Mr. Andre ́ Montpetit, our technical editor, for his help in the preparation of the volume.

The Dynamic Morse Theory Of Control Systems

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Author : Josiney Souza
language : en
Publisher: Cambridge Scholars Publishing
Release Date : 2020-01-20

The Dynamic Morse Theory Of Control Systems written by Josiney Souza and has been published by Cambridge Scholars Publishing this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-01-20 with Mathematics categories.

This book provides insights into the dynamics of control systems with the integration of conceptions such as stability, controllability, attraction, and chain transitivity. It highlights the importance of Morse theory with its feature of describing the global dynamics of systems, presented here for the first time in control theory. The mathematical formulations are comprehensive, designed especially for students, researches, and professionals interested in qualitative studies of control systems. The reader will find the book an accessible source of basic definitions, properties, methods, examples, theorems, references, lists of problems, and open questions. Parts of the book may be used for courses or seminars in mathematics or control-theoretic engineering, and its reference guide will serve as a great resource for research projects and academic dissertations on control theory or dynamical systems.

Algebraic Cycles And Hodge Theory

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Author : Mark L. Green
language : en
Publisher: Springer
Release Date : 2004-09-02

Algebraic Cycles And Hodge Theory written by Mark L. Green and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-09-02 with Mathematics categories.

The main goal of the CIME Summer School on "Algebraic Cycles and Hodge Theory" has been to gather the most active mathematicians in this area to make the point on the present state of the art. Thus the papers included in the proceedings are surveys and notes on the most important topics of this area of research. They include infinitesimal methods in Hodge theory; algebraic cycles and algebraic aspects of cohomology and k-theory, transcendental methods in the study of algebraic cycles.

The Homotopy Index And Partial Differential Equations

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Author : Krzysztof P. Rybakowski
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

The Homotopy Index And Partial Differential Equations written by Krzysztof P. Rybakowski and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

The homotopy index theory was developed by Charles Conley for two sided flows on compact spaces. The homotopy or Conley index, which provides an algebraic-topologi cal measure of an isolated invariant set, is defined to be the ho motopy type of the quotient space N /N , where is a certain 1 2 1 2 compact pair, called an index pair. Roughly speaking, N1 isolates the invariant set and N2 is the "exit ramp" of N . 1 It is shown that the index is independent of the choice of the in dex pair and is invariant under homotopic perturbations of the flow. Moreover, the homotopy index generalizes the Morse index of a nQnde generate critical point p with respect to a gradient flow on a com pact manifold. In fact if the Morse index of p is k, then the homo topy index of the invariant set {p} is Ik - the homotopy type of the pointed k-dimensional unit sphere.

Connection Matrices In Combinatorial Topological Dynamics

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Author : Marian Mrozek
language : en
Publisher: Springer Nature
Release Date : 2025-07-08

Connection Matrices In Combinatorial Topological Dynamics written by Marian Mrozek and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-07-08 with Mathematics categories.

This book provides an introduction to the theory of connection matrices in the context of combinatorial multivector fields. The theory of connection matrices was proposed by Conley and Franzosa for classical continuous-time dynamical systems as a tool for studying connecting orbits between isolated invariant sets. It generalizes the Morse complex in Morse theory, and has found numerous applications in dynamics. Connection matrices have been and still are a challenging topic to study, as there are no complete introductory texts, and both their intricate definition and properties are scattered over numerous research papers. In recent years, dynamical concepts have found their way into a combinatorial context. Starting with combinatorial vector fields, introduced by Forman to generalize classical Morse theory, it has been realized that this transfer of ideas can lead to important applications. Similarly, Conley's theory of isolated invariant sets has been transferred to the combinatorial setting. This, when combined with the concept of multivector fields, opens the door to a complete combinatorial dynamical theory. In this book, we take Conley's theory one step further, by presenting a complete discussion of connection matrices for combinatorial multivector fields. While some of the results in this book are based on known approaches, we show in a detailed way how they can be carried over to the case of multivector fields on general Lefschetz complexes. Along the way, we introduce notions which are new even in the classical setting, such as a formal approach to addressing the nonuniqueness of connection matrices, as well as mechanisms for comparing connection matrices even under poset changes. Finally, we show that specifically for the case of Forman's gradient combinatorial vector fields connection matrices are necessarily unique, and can be determined explicitly in a straightforward way. Focusing on the combinatorial theory of connection matrices has a number of advantages. On the one hand, many of the technical difficulties of the classical continuous-time dynamics situation are not present in the discrete combinatorial context. This allows us to provide a complete and informal introduction to the theory in the second section of the book. This in turn will enable the readers to construct and analyze their own examples easily. On the other hand, the complete theory, including the existence of connecting orbits in the combinatorial setting can be presented in detail, based on an explicit distinction between the algebraic and topological parts of the theory. In this way, it is our hope that this book will be an impetus for further knowledge transfer between dynamics and combinatorics, and even topological data analysis. This text is aimed at researchers in the fields of dynamics and topological data analysis, and it is suitable for advanced graduate students interested in applying connection matrix methods to their own studies.

Isolated Invariant Sets And The Morse Index

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