Convex Integration Theory

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Convex Integration Theory

David Spring
Convex Integration Theory Book PDF

Author : David Spring
Publisher : Birkhäuser
Page : 217 pages
File Size : 47,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034889407

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Convex Integration Theory by David Spring Pdf

§1. Historical Remarks Convex Integration theory, first introduced by M. Gromov [17], is one of three general methods in immersion-theoretic topology for solving a broad range of problems in geometry and topology. The other methods are: (i) Removal of Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii) the covering homotopy method which, following M. Gromov's thesis [16], is also referred to as the method of sheaves. The covering homotopy method is due originally to S. Smale [36] who proved a crucial covering homotopy result in order to solve the classification problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succes sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Conse quently, each method has a range of applications to problems in topology that are best suited to its particular insight. For example, a distinguishing feature of Convex Integration theory is that it applies to solve closed relations in jet spaces, including certain general classes of underdetermined non-linear systems of par tial differential equations. As a case of interest, the Nash-Kuiper Cl-isometrie immersion theorem ean be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaees can be proved by means of the other two methods.

Convex Integration Theory

David Spring
Convex Integration Theory Book PDF

Author : David Spring
Publisher : Birkhäuser
Page : 213 pages
File Size : 42,8 Mb
Release : 2010-12-09
Category : Mathematics
ISBN : 3034800592

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Convex Integration Theory by David Spring Pdf

§1. Historical Remarks Convex Integration theory, ?rst introduced by M. Gromov [17], is one of three general methods in immersion-theoretic topology for solving a broad range of problems in geometry and topology. The other methods are: (i) Removal of Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii) the covering homotopy method which, following M. Gromov’s thesis [16], is also referred to as the method of sheaves. The covering homotopy method is due originally to S. Smale [36] who proved a crucial covering homotopy result in order to solve the classi?cation problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succ- sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Con- quently, each method has a range of applications to problems in topology that are best suited to its particular insight. For example, a distinguishing feature of ConvexIntegrationtheoryisthatitappliestosolveclosed relationsinjetspaces, including certain general classes of underdetermined non-linear systems of par- 1 tial di?erential equations. As a case of interest, the Nash-Kuiper C -isometric immersion theorem can be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaces can be proved by means of the other two methods. On the other hand, many classical results in immersion-theoretic topology, such as the classi?cation of immersions, are provable by all three methods.

Introduction to the $h$-Principle

K. Cieliebak,Y. Eliashberg,N. Mishachev
Introduction to the h Principle Book PDF

Author : K. Cieliebak,Y. Eliashberg,N. Mishachev
Publisher : American Mathematical Society
Page : 384 pages
File Size : 48,6 Mb
Release : 2024-01-30
Category : Mathematics
ISBN : 9781470476175

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Introduction to the $h$-Principle by K. Cieliebak,Y. Eliashberg,N. Mishachev Pdf

In differential geometry and topology one often deals with systems of partial differential equations as well as partial differential inequalities that have infinitely many solutions whatever boundary conditions are imposed. It was discovered in the 1950s that the solvability of differential relations (i.e., equations and inequalities) of this kind can often be reduced to a problem of a purely homotopy-theoretic nature. One says in this case that the corresponding differential relation satisfies the $h$-principle. Two famous examples of the $h$-principle, the Nash–Kuiper $C^1$-isometric embedding theory in Riemannian geometry and the Smale–Hirsch immersion theory in differential topology, were later transformed by Gromov into powerful general methods for establishing the $h$-principle. The authors cover two main methods for proving the $h$-principle: holonomic approximation and convex integration. The reader will find that, with a few notable exceptions, most instances of the $h$-principle can be treated by the methods considered here. A special emphasis is made on applications to symplectic and contact geometry. The present book is the first broadly accessible exposition of the theory and its applications, making it an excellent text for a graduate course on geometric methods for solving partial differential equations and inequalities. Geometers, topologists, and analysts will also find much value in this very readable exposition of an important and remarkable topic. This second edition of the book is significantly revised and expanded to almost twice of the original size. The most significant addition to the original book is the new part devoted to the method of wrinkling and its applications. Several other chapters (e.g., on multivalued holonomic approximation and foliations) are either added or completely rewritten.

H-Principles and Flexibility in Geometry

Hansjörg Geiges
H Principles and Flexibility in Geometry Book PDF

Author : Hansjörg Geiges
Publisher : Unknown
Page : 58 pages
File Size : 44,6 Mb
Release : 2014-09-11
Category : Electronic books
ISBN : 1470403773

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H-Principles and Flexibility in Geometry by Hansjörg Geiges Pdf

Introduction Differential relations and $h$-principles The $h$-principle for open, invariant relations Convex integration theory Bibliography

Geometric Integration Theory

Steven G. Krantz,Harold R. Parks
Geometric Integration Theory Book PDF

Author : Steven G. Krantz,Harold R. Parks
Publisher : Springer Science & Business Media
Page : 340 pages
File Size : 43,9 Mb
Release : 2008-12-15
Category : Mathematics
ISBN : 9780817646790

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Geometric Integration Theory by Steven G. Krantz,Harold R. Parks Pdf

This textbook introduces geometric measure theory through the notion of currents. Currents, continuous linear functionals on spaces of differential forms, are a natural language in which to formulate types of extremal problems arising in geometry, and can be used to study generalized versions of the Plateau problem and related questions in geometric analysis. Motivating key ideas with examples and figures, this book is a comprehensive introduction ideal for both self-study and for use in the classroom. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for both graduate students and researchers.

H-principles and Flexibility in Geometry

Hansjšrg Geiges
H principles and Flexibility in Geometry Book PDF

Author : Hansjšrg Geiges
Publisher : American Mathematical Soc.
Page : 76 pages
File Size : 55,5 Mb
Release : 2003-05-30
Category : Mathematics
ISBN : 0821865013

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H-principles and Flexibility in Geometry by Hansjšrg Geiges Pdf

The notion of homotopy principle or $h$-principle is one of the key concepts in an elegant language developed by Gromov to deal with a host of questions in geometry and topology. Roughly speaking, for a certain differential geometric problem to satisfy the $h$-principle is equivalent to saying that a solution to the problem exists whenever certain obvious topological obstructions vanish. The foundational examples for applications of Gromov's ideas include (i) Hirsch-Smale immersion theory, (ii) Nash-Kuiper $C^1$-isometric immersion theory, (iii) existence of symplectic and contact structures on open manifolds. Gromov has developed several powerful methods that allow one to prove $h$-principles. These notes, based on lectures given in the Graduiertenkolleg of Leipzig University, present two such methods which are strong enough to deal with applications (i) and (iii).

Geometric Integration Theory on Supermanifolds

T. Voronov
Geometric Integration Theory on Supermanifolds Book PDF

Author : T. Voronov
Publisher : CRC Press
Page : 152 pages
File Size : 52,9 Mb
Release : 1991
Category : Mathematics
ISBN : 3718651998

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Geometric Integration Theory on Supermanifolds by T. Voronov Pdf

The author presents the first detailed and original account of his theory of forms on supermanifolds-a correct and non-trivial analogue of Cartan-de Rham theory based on new concepts. The paper develops the apparatus of supermanifold differential topology necessary for the integration theory. A key feature is the identification of a class of proper morphisms intimately connected with Berezin integration, which are of fundamental importance in various problems. The work also contains a condensed introduction to superanalysis and supermanifolds, free from algebraic formalism, which sets out afresh such challenging problems as the Berezin intgegral on a bounded domain.

Issues in General Science and Scientific Theory and Method: 2013 Edition

Anonim
Issues in General Science and Scientific Theory and Method 2013 Edition Book PDF

Author : Anonim
Publisher : ScholarlyEditions
Page : 1203 pages
File Size : 50,5 Mb
Release : 2013-05-01
Category : Science
ISBN : 9781490106915

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Issues in General Science and Scientific Theory and Method: 2013 Edition by Anonim Pdf

Issues in General Science and Scientific Theory and Method: 2013 Edition is a ScholarlyEditions™ book that delivers timely, authoritative, and comprehensive information about Mixed Methods Research. The editors have built Issues in General Science and Scientific Theory and Method: 2013 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about Mixed Methods Research in this book to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in General Science and Scientific Theory and Method: 2013 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.

Embeddings and Immersions

Masahisa Adachi
Embeddings and Immersions Book PDF

Author : Masahisa Adachi
Publisher : American Mathematical Soc.
Page : 198 pages
File Size : 42,8 Mb
Release : 2012-11-07
Category : Mathematics
ISBN : 9780821891643

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Embeddings and Immersions by Masahisa Adachi Pdf

This book covers fundamental techniques in the theory of -imbeddings and -immersions, emphasizing clear intuitive understanding and containing many figures and diagrams. Adachi starts with an introduction to the work of Whitney and of Haefliger on -imbeddings and -manifolds. The Smale-Hirsch theorem is presented as a generalization of the classification of -imbeddings by isotopy and is extended by Gromov's work on the subject, including Gromov's convex integration theory. Finally, as an application of Gromov's work, the author introduces Haefliger's classification theorem of foliations on open manifolds. Also described here is the Adachi's work with Landweber on the integrability of almost complex structures on open manifolds. This book would be an excellent text for upper-division undergraduate or graduate courses.Nothing provided

Lectures on Convex Geometry

Daniel Hug,Wolfgang Weil
Lectures on Convex Geometry Book PDF

Author : Daniel Hug,Wolfgang Weil
Publisher : Springer Nature
Page : 287 pages
File Size : 44,7 Mb
Release : 2020-08-27
Category : Mathematics
ISBN : 9783030501808

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Lectures on Convex Geometry by Daniel Hug,Wolfgang Weil Pdf

This book provides a self-contained introduction to convex geometry in Euclidean space. After covering the basic concepts and results, it develops Brunn–Minkowski theory, with an exposition of mixed volumes, the Brunn–Minkowski inequality, and some of its consequences, including the isoperimetric inequality. Further central topics are then treated, such as surface area measures, projection functions, zonoids, and geometric valuations. Finally, an introduction to integral-geometric formulas in Euclidean space is provided. The numerous exercises and the supplementary material at the end of each section form an essential part of the book. Convexity is an elementary and natural concept. It plays a key role in many mathematical fields, including functional analysis, optimization, probability theory, and stochastic geometry. Paving the way to the more advanced and specialized literature, the material will be accessible to students in the third year and can be covered in one semester.

From Topology to Computation: Proceedings of the Smalefest

Morris W. Hirsch,Jerrold E. Marsden,Michael Shub
From Topology to Computation Proceedings of the Smalefest Book PDF

Author : Morris W. Hirsch,Jerrold E. Marsden,Michael Shub
Publisher : Springer Science & Business Media
Page : 620 pages
File Size : 41,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461227403

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From Topology to Computation: Proceedings of the Smalefest by Morris W. Hirsch,Jerrold E. Marsden,Michael Shub Pdf

An extraordinary mathematical conference was held 5-9 August 1990 at the University of California at Berkeley: From Topology to Computation: Unity and Diversity in the Mathematical Sciences An International Research Conference in Honor of Stephen Smale's 60th Birthday The topics of the conference were some of the fields in which Smale has worked: • Differential Topology • Mathematical Economics • Dynamical Systems • Theory of Computation • Nonlinear Functional Analysis • Physical and Biological Applications This book comprises the proceedings of that conference. The goal of the conference was to gather in a single meeting mathemati cians working in the many fields to which Smale has made lasting con tributions. The theme "Unity and Diversity" is enlarged upon in the section entitled "Research Themes and Conference Schedule." The organizers hoped that illuminating connections between seemingly separate mathematical sub jects would emerge from the conference. Since such connections are not easily made in formal mathematical papers, the conference included discussions after each of the historical reviews of Smale's work in different fields. In addition, there was a final panel discussion at the end of the conference.

Convex Integration Applied to the Multi-Dimensional Compressible Euler Equations

Simon Markfelder
Convex Integration Applied to the Multi Dimensional Compressible Euler Equations Book PDF

Author : Simon Markfelder
Publisher : Springer Nature
Page : 244 pages
File Size : 51,6 Mb
Release : 2021-10-20
Category : Mathematics
ISBN : 9783030837853

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Convex Integration Applied to the Multi-Dimensional Compressible Euler Equations by Simon Markfelder Pdf

This book applies the convex integration method to multi-dimensional compressible Euler equations in the barotropic case as well as the full system with temperature. The convex integration technique, originally developed in the context of differential inclusions, was applied in the groundbreaking work of De Lellis and Székelyhidi to the incompressible Euler equations, leading to infinitely many solutions. This theory was later refined to prove non-uniqueness of solutions of the compressible Euler system, too. These non-uniqueness results all use an ansatz which reduces the equations to a kind of incompressible system to which a slight modification of the incompressible theory can be applied. This book presents, for the first time, a generalization of the De Lellis–Székelyhidi approach to the setting of compressible Euler equations. The structure of this book is as follows: after providing an accessible introduction to the subject, including the essentials of hyperbolic conservation laws, the idea of convex integration in the compressible framework is developed. The main result proves that under a certain assumption there exist infinitely many solutions to an abstract initial boundary value problem for the Euler system. Next some applications of this theorem are discussed, in particular concerning the Riemann problem. Finally there is a survey of some related results. This self-contained book is suitable for both beginners in the field of hyperbolic conservation laws as well as for advanced readers who already know about convex integration in the incompressible framework.

$h$-Principles and Flexibility in Geometry

Hansjörg Geiges
h Principles and Flexibility in Geometry Book PDF

Author : Hansjörg Geiges
Publisher : American Mathematical Soc.
Page : 74 pages
File Size : 54,6 Mb
Release : 2003
Category : Global differential geometry
ISBN : 9780821833155

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$h$-Principles and Flexibility in Geometry by Hansjörg Geiges Pdf

The notion of homotopy principle or $h$-principle is one of the key concepts in an elegant language developed by Gromov to deal with a host of questions in geometry and topology. Roughly speaking, for a certain differential geometric problem to satisfy the $h$-principle is equivalent to saying that a solution to the problem exists whenever certain obvious topological obstructions vanish. The foundational examples for applications of Gromov's ideas include (i) Hirsch-Smale immersion theory, (ii) Nash-Kuiper $C^1$-isometric immersion theory, (iii) existence of symplectic and contact structures on open manifolds. Gromov has developed several powerful methods that allow one to prove $h$-principles. These notes, based on lectures given in the Graduiertenkolleg of Leipzig University, present two such methods which are strong enough to deal with applications (i) and (iii).

Introduction to the H-principle

Y. Eliashberg,Nikolai M. Mishachev
Introduction to the H principle Book PDF

Author : Y. Eliashberg,Nikolai M. Mishachev
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 50,6 Mb
Release : 2024-07-02
Category : Mathematics
ISBN : 9780821872277

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Introduction to the H-principle by Y. Eliashberg,Nikolai M. Mishachev Pdf

One of the most powerful modern methods of solving partial differential equations is Gromov's $h$-principle. It has also been, traditionally, one of the most difficult to explain. This book is the first broadly accessible exposition of the principle and its applications. The essence of the $h$-principle is the reduction of problems involving partial differential relations to problems of a purely homotopy-theoretic nature. Two famous examples of the $h$-principle are the Nash-Kuiper$C1$-isometric embedding theory in Riemannian geometry and the Smale-Hirsch immersion theory in differential topology. Gromov transformed these examples into a powerful general method for proving the $h$-principle. Both of these examples and their explanations in terms of the $h$-principle arecovered in detail in the book. The authors cover two main embodiments of the principle: holonomic approximation and convex integration. The first is a version of the method of continuous sheaves. The reader will find that, with a few notable exceptions, most instances of the $h$-principle can be treated by the methods considered here. There are, naturally, many connections to symplectic and contact geometry. The book would be an excellent text for a graduate course on modern methods for solvingpartial differential equations. Geometers and analysts will also find much value in this very readable exposition of an important and remarkable technique.

Convex and Stochastic Optimization

J. Frédéric Bonnans
Convex and Stochastic Optimization Book PDF

Author : J. Frédéric Bonnans
Publisher : Springer
Page : 311 pages
File Size : 47,6 Mb
Release : 2019-04-24
Category : Mathematics
ISBN : 9783030149772

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Convex and Stochastic Optimization by J. Frédéric Bonnans Pdf

This textbook provides an introduction to convex duality for optimization problems in Banach spaces, integration theory, and their application to stochastic programming problems in a static or dynamic setting. It introduces and analyses the main algorithms for stochastic programs, while the theoretical aspects are carefully dealt with. The reader is shown how these tools can be applied to various fields, including approximation theory, semidefinite and second-order cone programming and linear decision rules. This textbook is recommended for students, engineers and researchers who are willing to take a rigorous approach to the mathematics involved in the application of duality theory to optimization with uncertainty.