Geometrical Methods in Variational Problems PDF Download

Author: N.A. Bobylov
Publisher: Springer Science & Business Media
ISBN: 9401146292
Category : Mathematics
Languages : en
Pages : 556

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Book Description
This self-contained monograph presents methods for the investigation of nonlinear variational problems. These methods are based on geometric and topological ideas such as topological index, degree of a mapping, Morse-Conley index, Euler characteristics, deformation invariant, homotopic invariant, and the Lusternik-Shnirelman category. Attention is also given to applications in optimisation, mathematical physics, control, and numerical methods. Audience: This volume will be of interest to specialists in functional analysis and its applications, and can also be recommended as a text for graduate and postgraduate-level courses in these fields.

Author: Vladimir Boltyanski
Publisher: Springer Science & Business Media
ISBN: 1461553199
Category : Mathematics
Languages : en
Pages : 438

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Book Description
VII Preface In many fields of mathematics, geometry has established itself as a fruitful method and common language for describing basic phenomena and problems as well as suggesting ways of solutions. Especially in pure mathematics this is ob vious and well-known (examples are the much discussed interplay between lin ear algebra and analytical geometry and several problems in multidimensional analysis). On the other hand, many specialists from applied mathematics seem to prefer more formal analytical and numerical methods and representations. Nevertheless, very often the internal development of disciplines from applied mathematics led to geometric models, and occasionally breakthroughs were b~ed on geometric insights. An excellent example is the Klee-Minty cube, solving a problem of linear programming by transforming it into a geomet ric problem. Also the development of convex programming in recent decades demonstrated the power of methods that evolved within the field of convex geometry. The present book focuses on three applied disciplines: control theory, location science and computational geometry. It is our aim to demonstrate how methods and topics from convex geometry in a wider sense (separation theory of convex cones, Minkowski geometry, convex partitionings, etc.) can help to solve various problems from these disciplines.

Author: Seiki Nishikawa
Publisher: Springer Science & Business Media
ISBN: 4431684026
Category : Mathematics
Languages : en
Pages : 160

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Book Description
In this volume are collected notes of lectures delivered at the First In ternational Research Institute of the Mathematical Society of Japan. This conference, held at Tohoku University in July 1993, was devoted to geometry and global analysis. Subsequent to the conference, in answer to popular de mand from the participants, it was decided to publish the notes of the survey lectures. Written by the lecturers themselves, all experts in their respective fields, these notes are here presented in a single volume. It is hoped that they will provide a vivid account of the current research, from the introduc tory level up to and including the most recent results, and will indicate the direction to be taken by future researeh. This compilation begins with Jean-Pierre Bourguignon's notes entitled "An Introduction to Geometric Variational Problems," illustrating the gen eral framework of the field with many examples and providing the reader with a broad view of the current research. Following this, Kenji Fukaya's notes on "Geometry of Gauge Fields" are concerned with gauge theory and its applications to low-dimensional topology, without delving too deeply into technical detail. Special emphasis is placed on explaining the ideas of infi nite dimensional geometry that, in the literature, are often hidden behind rigorous formulations or technical arguments.

Author: Roger Bielawski
Publisher: Cambridge University Press
ISBN: 1139504118
Category : Mathematics
Languages : en
Pages : 216

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Book Description
With a mix of expository and original papers this volume is an excellent reference for experienced researchers in geometric variational problems, as well as an ideal introduction for graduate students. It presents all the varied methods and techniques used in attacking geometric variational problems and includes many up-to-date results.

Author: Dmitry V. Zenkov
Publisher: Springer
ISBN: 9462391092
Category : Mathematics
Languages : en
Pages : 296

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Book Description
The aim of the present book is to give a systematic treatment of the inverse problem of the calculus of variations, i.e. how to recognize whether a system of differential equations can be treated as a system for extremals of a variational functional (the Euler-Lagrange equations), using contemporary geometric methods. Selected applications in geometry, physics, optimal control, and general relativity are also considered. The book includes the following chapters: - Helmholtz conditions and the method of controlled Lagrangians (Bloch, Krupka, Zenkov) - The Sonin-Douglas's problem (Krupka) - Inverse variational problem and symmetry in action: The Ostrogradskyj relativistic third order dynamics (Matsyuk.) - Source forms and their variational completion (Voicu) - First-order variational sequences and the inverse problem of the calculus of variations (Urban, Volna) - The inverse problem of the calculus of variations on Grassmann fibrations (Urban).

Author: Michael Struwe
Publisher: Springer Science & Business Media
ISBN: 3662032120
Category : Science
Languages : en
Pages : 288

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Book Description
Hilbert's talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateau's problem by Douglas and Radò. The book gives a concise introduction to variational methods and presents an overview of areas of current research in this field. This new edition has been substantially enlarged, a new chapter on the Yamabe problem has been added and the references have been updated. All topics are illustrated by carefully chosen examples, representing the current state of the art in their field.

Author: R. Bielawski
Publisher:
ISBN: 9781139160551
Category : Geometry, Differential
Languages : en
Pages : 201

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Book Description
"The field of geometric variational problems is fast-moving and influential. These problems interact with many other areas of mathematics and have strong relevance to the study of integrable systems, mathematical physics and PDEs. The workshop 'Variational Problems in Differential Geometry' held in 2009 at the University of Leeds brought together internationally respected researchers from many different areas of the field. Topics discussed included recent developments in harmonic maps and morphisms, minimal and CMC surfaces, extremal Ka;hler metrics, the Yamabe functional, Hamiltonian variational problems and topics related to gauge theory and to the Ricci flow. These articles reflect the whole spectrum of the subject and cover not only current results, but also the varied methods and techniques used in attacking variational problems. With a mix of original and expository papers, this volume forms a valuable reference for more experienced researchers and an ideal introduction for graduate students and postdoctoral researchers"--Provided by publisher.

Author: Alexandru Kristály
Publisher: Cambridge University Press
ISBN: 0521117828
Category : Mathematics
Languages : en
Pages : 385

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Book Description
A comprehensive introduction to modern applied functional analysis. Assumes only basic notions of calculus, real analysis, geometry, and differential equations.

Author: BERESTYCKI
Publisher: Springer Science & Business Media
ISBN: 1475710801
Category : Mathematics
Languages : en
Pages : 468

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Book Description
In the framework of the "Annee non lineaire" (the special nonlinear year) sponsored by the C.N.R.S. (the French National Center for Scien tific Research), a meeting was held in Paris in June 1988. It took place in the Conference Hall of the Ministere de la Recherche and had as an organizing theme the topic of "Variational Problems." Nonlinear analysis has been one of the leading themes in mathemat ical research for the past decade. The use of direct variational methods has been particularly successful in understanding problems arising from physics and geometry. The growth of nonlinear analysis is largely due to the wealth of ap plications from various domains of sciences and industrial applica tions. Most of the papers gathered in this volume have their origin in applications: from mechanics, the study of Hamiltonian systems, from physics, from the recent mathematical theory of liquid crystals, from geometry, relativity, etc. Clearly, no single volume could pretend to cover the whole scope of nonlinear variational problems. We have chosen to concentrate on three main aspects of these problems, organizing them roughly around the following topics: 1. Variational methods in partial differential equations in mathemat ical physics 2. Variational problems in geometry 3. Hamiltonian systems and related topics.

Author: Maïtine Bergounioux
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110430495
Category : Mathematics
Languages : en
Pages : 621

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Book Description
With a focus on the interplay between mathematics and applications of imaging, the first part covers topics from optimization, inverse problems and shape spaces to computer vision and computational anatomy. The second part is geared towards geometric control and related topics, including Riemannian geometry, celestial mechanics and quantum control. Contents: Part I Second-order decomposition model for image processing: numerical experimentation Optimizing spatial and tonal data for PDE-based inpainting Image registration using phase・amplitude separation Rotation invariance in exemplar-based image inpainting Convective regularization for optical flow A variational method for quantitative photoacoustic tomography with piecewise constant coefficients On optical flow models for variational motion estimation Bilevel approaches for learning of variational imaging models Part II Non-degenerate forms of the generalized Euler・Lagrange condition for state-constrained optimal control problems The Purcell three-link swimmer: some geometric and numerical aspects related to periodic optimal controls Controllability of Keplerian motion with low-thrust control systems Higher variational equation techniques for the integrability of homogeneous potentials Introduction to KAM theory with a view to celestial mechanics Invariants of contact sub-pseudo-Riemannian structures and Einstein・Weyl geometry Time-optimal control for a perturbed Brockett integrator Twist maps and Arnold diffusion for diffeomorphisms A Hamiltonian approach to sufficiency in optimal control with minimal regularity conditions: Part I Index