Variational Problems In Topology

Author : A.T. Fomenko
Publisher : Routledge
Page : 226 pages
File Size : 45,6 Mb
Release : 2019-06-21
Category : Mathematics
ISBN : 9781351405683

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Variational Problems in Topology by A.T. Fomenko Pdf

Many of the modern variational problems of topology arise in different but overlapping fields of scientific study: mechanics, physics and mathematics. In this work, Professor Fomenko offers a concise and clear explanation of some of these problems (both solved and unsolved), using current methods of analytical topology. His book falls into three interrelated sections. The first gives an elementary introduction to some of the most important concepts of topology used in modern physics and mechanics: homology and cohomology, and fibration. The second investigates the significant role of Morse theory in modern aspects of the topology of smooth manifolds, particularly those of three and four dimensions. The third discusses minimal surfaces and harmonic mappings, and presents a number of classic physical experiments that lie at the foundations of modern understanding of multidimensional variational calculus. The author's skilful exposition of these topics and his own graphic illustrations give an unusual motivation to the theory expounded, and his work is recommended reading for specialists and non-specialists alike, involved in the fields of physics and mathematics at both undergraduate and graduate levels.

Author : Thomas Bartsch
Publisher : Springer
Page : 162 pages
File Size : 45,5 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540480990

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Topological Methods for Variational Problems with Symmetries by Thomas Bartsch Pdf

Symmetry has a strong impact on the number and shape of solutions to variational problems. This has been observed, for instance, in the search for periodic solutions of Hamiltonian systems or of the nonlinear wave equation; when one is interested in elliptic equations on symmetric domains or in the corresponding semiflows; and when one is looking for "special" solutions of these problems. This book is concerned with Lusternik-Schnirelmann theory and Morse-Conley theory for group invariant functionals. These topological methods are developed in detail with new calculations of the equivariant Lusternik-Schnirelmann category and versions of the Borsuk-Ulam theorem for very general classes of symmetry groups. The Morse-Conley theory is applied to bifurcation problems, in particular to the bifurcation of steady states and hetero-clinic orbits of O(3)-symmetric flows; and to the existence of periodic solutions nearequilibria of symmetric Hamiltonian systems. Some familiarity with the usualminimax theory and basic algebraic topology is assumed.

Author : Seiki Nishikawa
Publisher : American Mathematical Soc.
Page : 236 pages
File Size : 55,9 Mb
Release : 2002
Category : Mathematics
ISBN : 0821813560

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Kikagakuteki Henbun Mondai by Seiki Nishikawa Pdf

A minimal length curve joining two points in a surface is called a geodesic. One may trace the origin of the problem of finding geodesics back to the birth of calculus. Many contemporary mathematical problems, as in the case of geodesics, may be formulated as variational problems in surfaces or in a more generalized form on manifolds. One may characterize geometric variational problems as a field of mathematics that studies global aspects of variational problems relevant in the geometry and topology of manifolds. For example, the problem of finding a surface of minimal area spanning a given frame of wire originally appeared as a mathematical model for soap films. It has also been actively investigated as a geometric variational problem. With recent developments in computer graphics, totally new aspects of the study on the subject have begun to emerge. This book is intended to be an introduction to some of the fundamental questions and results in geometric variational problems, studying variational problems on the length of curves and the energy of maps. The first two chapters treat variational problems of the length and energy of curves in Riemannian manifolds, with an in-depth discussion of the existence and properties of geodesics viewed as solutions to variational problems. In addition, a special emphasis is placed on the facts that concepts of connection and covariant differentiation are naturally induced from the formula for the first variation in this problem, and that the notion of curvature is obtained from the formula for the second variation. The last two chapters treat the variational problem on the energy of maps between two Riemannian manifolds and its solution, harmonic maps. The concept of a harmonic map includes geodesics and minimal submanifolds as examples. Its existence and properties have successfully been applied to various problems in geometry and topology. The author discusses in detail the existence theorem of Eells-Sampson, which is considered to be the most fundamental among existence theorems for harmonic maps. The proof uses the inverse function theorem for Banach spaces. It is presented to be as self-contained as possible for easy reading. Each chapter may be read independently, with minimal preparation for covariant differentiation and curvature on manifolds. The first two chapters provide readers with basic knowledge of Riemannian manifolds. Prerequisites for reading this book include elementary facts in the theory of manifolds and functional analysis, which are included in the form of appendices. Exercises are given at the end of each chapter. This is the English translation of a book originally published in Japanese. It is an outgrowth of lectures delivered at Tohoku University and at the Summer Graduate Program held at the Institute for Mathematics and its Applications at the University of Minnesota. It would make a suitable textbook for advanced undergraduates and graduate students. This item will also be of interest to those working in analysis.

Author : Alexander J. Zaslavski
Publisher : Springer Science & Business Media
Page : 382 pages
File Size : 54,9 Mb
Release : 2013-06-12
Category : Mathematics
ISBN : 9781461473787

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Nonconvex Optimal Control and Variational Problems by Alexander J. Zaslavski Pdf

Nonconvex Optimal Control and Variational Problems is an important contribution to the existing literature in the field and is devoted to the presentation of progress made in the last 15 years of research in the area of optimal control and the calculus of variations. This volume contains a number of results concerning well-posedness of optimal control and variational problems, nonoccurrence of the Lavrentiev phenomenon for optimal control and variational problems, and turnpike properties of approximate solutions of variational problems. Chapter 1 contains an introduction as well as examples of select topics. Chapters 2-5 consider the well-posedness condition using fine tools of general topology and porosity. Chapters 6-8 are devoted to the nonoccurrence of the Lavrentiev phenomenon and contain original results. Chapter 9 focuses on infinite-dimensional linear control problems, and Chapter 10 deals with “good” functions and explores new understandings on the questions of optimality and variational problems. Finally, Chapters 11-12 are centered around the turnpike property, a particular area of expertise for the author. This volume is intended for mathematicians, engineers, and scientists interested in the calculus of variations, optimal control, optimization, and applied functional analysis, as well as both undergraduate and graduate students specializing in those areas. The text devoted to Turnpike properties may be of particular interest to the economics community.

Author : Dumitru Motreanu,Viorica Venera Motreanu,Nikolaos Papageorgiou
Publisher : Springer Science & Business Media
Page : 465 pages
File Size : 53,7 Mb
Release : 2013-11-19
Category : Mathematics
ISBN : 9781461493235

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Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems by Dumitru Motreanu,Viorica Venera Motreanu,Nikolaos Papageorgiou Pdf

This book focuses on nonlinear boundary value problems and the aspects of nonlinear analysis which are necessary to their study. The authors first give a comprehensive introduction to the many different classical methods from nonlinear analysis, variational principles, and Morse theory. They then provide a rigorous and detailed treatment of the relevant areas of nonlinear analysis with new applications to nonlinear boundary value problems for both ordinary and partial differential equations. Recent results on the existence and multiplicity of critical points for both smooth and nonsmooth functional, developments on the degree theory of monotone type operators, nonlinear maximum and comparison principles for p-Laplacian type operators, and new developments on nonlinear Neumann problems involving non-homogeneous differential operators appear for the first time in book form. The presentation is systematic, and an extensive bibliography and a remarks section at the end of each chapter highlight the text. This work will serve as an invaluable reference for researchers working in nonlinear analysis and partial differential equations as well as a useful tool for all those interested in the topics presented.

Author : Seiki Nishikawa,Richard Schoen
Publisher : Springer Science & Business Media
Page : 160 pages
File Size : 52,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9784431684022

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Lectures on Geometric Variational Problems by Seiki Nishikawa,Richard Schoen Pdf

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 : Paolo Maria Mariano
Publisher : Springer Nature
Page : 315 pages
File Size : 54,9 Mb
Release : 2022-02-08
Category : Mathematics
ISBN : 9783030900519

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Variational Views in Mechanics by Paolo Maria Mariano Pdf

This volume provides a timely survey of interactions between the calculus of variations and theoretical and applied mechanics. Chapters have been significantly expanded since preliminary versions appeared in a special issue of the Journal of Optimization Theory and Applications (184(1), 2020) on “Calculus of Variations in Mechanics and Related Fields”. The variety of topics covered offers researchers an overview of problems in mechanics that can be analyzed with variational techniques, making this a valuable reference for researchers in the field. It also presents ideas for possible future areas of research, showing how the mastery of these foundational mathematical techniques can be used for many exciting applications. Specific topics covered include: Topology optimization Identification of material properties Optimal control Plastic flows Gradient polyconvexity Obstacle problems Quasi-monotonicity Variational Views in Mechanics will appeal to researchers in mathematics, solid-states physics, and mechanical, civil, and materials engineering.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 45,7 Mb
Release : 2024-06-30
Category : Electronic
ISBN : 9789814474504

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Variational Methods for Strongly Indefinite Problems by Anonim Pdf

Author : Daniel Goeleven
Publisher : CRC Press
Page : 186 pages
File Size : 47,9 Mb
Release : 1996-10-10
Category : Mathematics
ISBN : 0582304024

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Noncoercive Variational Problems and Related Results by Daniel Goeleven Pdf

In establishing a general theory of the existence of solutions for noncoercive variational problems and constrained problems formulated as variational inequalities or hemivariational inequalities, this Research Note illustrates recent mathematical approaches and results with various examples from mathematics and mechanics. The book unifies ideas for the treatment of various noncoercive problems and provides previously unpublished results for variational inequalities and hemivariational inequalities. The author points out important applications in mechanics and their mathfematical tratment using recession tools. This book will be of particular interest to researchers in pure and aplied mathematics and mechanics.

Author : N.A. Bobylov,S.V. Emel'yanov,S. Korovin
Publisher : Springer Science & Business Media
Page : 556 pages
File Size : 55,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401146296

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Geometrical Methods in Variational Problems by N.A. Bobylov,S.V. Emel'yanov,S. Korovin Pdf

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 : Frederick J. Almgren
Publisher : American Mathematical Soc.
Page : 212 pages
File Size : 50,7 Mb
Release : 1976
Category : Calculus of variations
ISBN : 9780821818657

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Existence and Regularity Almost Everywhere of Solutions to Elliptic Variational Problems with Constraints by Frederick J. Almgren Pdf

Author : L. A. Ljusternik
Publisher : American Mathematical Soc.
Page : 106 pages
File Size : 52,7 Mb
Release : 1967-12-31
Category : Mathematics
ISBN : 9780821815663

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The Topology of Function Spaces and the Calculus of Variations in the Large by L. A. Ljusternik Pdf

Author : Roger Bielawski,Kevin Houston,Martin Speight
Publisher : Cambridge University Press
Page : 216 pages
File Size : 55,9 Mb
Release : 2011-10-20
Category : Mathematics
ISBN : 9781139504119

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Variational Problems in Differential Geometry by Roger Bielawski,Kevin Houston,Martin Speight Pdf

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 : Noncompact Va Brezis-Browder Conference,Brezis-Browder Conference, Noncompact Variational Problems and General Relativity,Abbas Bahri,Sergiu Klainerman,Michael Vogelius
Publisher : American Mathematical Soc.
Page : 252 pages
File Size : 42,8 Mb
Release : 2004
Category : Science
ISBN : 9780821836354

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Noncompact Problems at the Intersection of Geometry, Analysis, and Topology by Noncompact Va Brezis-Browder Conference,Brezis-Browder Conference, Noncompact Variational Problems and General Relativity,Abbas Bahri,Sergiu Klainerman,Michael Vogelius Pdf

This proceedings volume contains articles from the conference held at Rutgers University in honor of Haim Brezis and Felix Browder, two mathematicians who have had a profound impact on partial differential equations, functional analysis, and geometry. The material is suitable for graduate students and researchers interested in problems in analysis and differential equations on noncompact manifolds.

Author : Zhitao Zhang
Publisher : Springer Science & Business Media
Page : 333 pages
File Size : 41,7 Mb
Release : 2012-09-17
Category : Mathematics
ISBN : 9783642307096

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Variational, Topological, and Partial Order Methods with Their Applications by Zhitao Zhang Pdf

Nonlinear functional analysis is an important branch of contemporary mathematics. It's related to topology, ordinary differential equations, partial differential equations, groups, dynamical systems, differential geometry, measure theory, and more. In this book, the author presents some new and interesting results on fundamental methods in nonlinear functional analysis, namely variational, topological and partial order methods, which have been used extensively to solve existence of solutions for elliptic equations, wave equations, Schrödinger equations, Hamiltonian systems etc., and are also used to study the existence of multiple solutions and properties of solutions. This book is useful for researchers and graduate students in the field of nonlinear functional analysis.