Introduction to the $h$-Principle

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Publisher : American Mathematical Society
ISBN 13 : 1470476177
Total Pages : 384 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Introduction to the $h$-Principle by : K. Cieliebak

Download or read book Introduction to the $h$-Principle written by K. Cieliebak and published by American Mathematical Society. This book was released on 2024-01-30 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Introduction to the H-principle

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Publisher :
ISBN 13 : 9781470417963
Total Pages : 198 pages
Book Rating : 4.4/5 (179 download)

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Book Synopsis Introduction to the H-principle by : Y. Eliashberg

Download or read book Introduction to the H-principle written by Y. Eliashberg and published by . This book was released on 1900 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 fifties 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 in the book is made on applications to symplectic and contact geometry. Gromov's famous book ``Partial Differential Relations'', which is devoted to the same subject, is an encyclopedia of the $h$-principle, written for experts, while the present book is the first broadly accessible exposition of the theory and its applications. The book would be 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.

Introduction to the H-principle

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Publisher : American Mathematical Soc.
ISBN 13 : 0821872273
Total Pages : 226 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Introduction to the H-principle by : Y. Eliashberg

Download or read book Introduction to the H-principle written by Y. Eliashberg and published by American Mathematical Soc.. This book was released on with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Integration Theory

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Publisher : Springer Science & Business Media
ISBN 13 : 3034800606
Total Pages : 219 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Convex Integration Theory by : David Spring

Download or read book Convex Integration Theory written by David Spring and published by Springer Science & Business Media. This book was released on 2010-12-02 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: §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.

Partial Differential Relations

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Publisher : Springer Science & Business Media
ISBN 13 : 3662022672
Total Pages : 372 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Partial Differential Relations by : Misha Gromov

Download or read book Partial Differential Relations written by Misha Gromov and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: The classical theory of partial differential equations is rooted in physics, where equations (are assumed to) describe the laws of nature. Law abiding functions, which satisfy such an equation, are very rare in the space of all admissible functions (regardless of a particular topology in a function space). Moreover, some additional (like initial or boundary) conditions often insure the uniqueness of solutions. The existence of these is usually established with some apriori estimates which locate a possible solution in a given function space. We deal in this book with a completely different class of partial differential equations (and more general relations) which arise in differential geometry rather than in physics. Our equations are, for the most part, undetermined (or, at least, behave like those) and their solutions are rather dense in spaces of functions. We solve and classify solutions of these equations by means of direct (and not so direct) geometric constructions. Our exposition is elementary and the proofs of the basic results are selfcontained. However, there is a number of examples and exercises (of variable difficulty), where the treatment of a particular equation requires a certain knowledge of pertinent facts in the surrounding field. The techniques we employ, though quite general, do not cover all geometrically interesting equations. The border of the unexplored territory is marked by a number of open questions throughout the book.

An Introduction to the Uncertainty Principle

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Publisher : Springer Science & Business Media
ISBN 13 : 0817681647
Total Pages : 189 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis An Introduction to the Uncertainty Principle by : Sundaram Thangavelu

Download or read book An Introduction to the Uncertainty Principle written by Sundaram Thangavelu and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 189 pages. Available in PDF, EPUB and Kindle. Book excerpt: In 1932 Norbert Wiener gave a series of lectures on Fourier analysis at the Univer sity of Cambridge. One result of Wiener's visit to Cambridge was his well-known text The Fourier Integral and Certain of its Applications; another was a paper by G. H. Hardy in the 1933 Journalofthe London Mathematical Society. As Hardy says in the introduction to this paper, This note originates from a remark of Prof. N. Wiener, to the effect that "a f and g [= j] cannot both be very small". ... The theo pair of transforms rems which follow give the most precise interpretation possible ofWiener's remark. Hardy's own statement of his results, lightly paraphrased, is as follows, in which f is an integrable function on the real line and f is its Fourier transform: x 2 m If f and j are both 0 (Ix1e- /2) for large x and some m, then each is a finite linear combination ofHermite functions. In particular, if f and j are x2 x 2 2 2 both O(e- / ), then f = j = Ae- / , where A is a constant; and if one x 2 2 is0(e- / ), then both are null.

H-Principles and Flexibility in Geometry

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Publisher :
ISBN 13 : 9781470403775
Total Pages : 58 pages
Book Rating : 4.4/5 (37 download)

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Book Synopsis H-Principles and Flexibility in Geometry by : Hansjörg Geiges

Download or read book H-Principles and Flexibility in Geometry written by Hansjörg Geiges and published by . This book was released on 2014-09-11 with total page 58 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction Differential relations and $h$-principles The $h$-principle for open, invariant relations Convex integration theory Bibliography

Principles of Computer System Design

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Publisher : Morgan Kaufmann
ISBN 13 : 0080959423
Total Pages : 561 pages
Book Rating : 4.0/5 (89 download)

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Book Synopsis Principles of Computer System Design by : Jerome H. Saltzer

Download or read book Principles of Computer System Design written by Jerome H. Saltzer and published by Morgan Kaufmann. This book was released on 2009-05-21 with total page 561 pages. Available in PDF, EPUB and Kindle. Book excerpt: Principles of Computer System Design is the first textbook to take a principles-based approach to the computer system design. It identifies, examines, and illustrates fundamental concepts in computer system design that are common across operating systems, networks, database systems, distributed systems, programming languages, software engineering, security, fault tolerance, and architecture. Through carefully analyzed case studies from each of these disciplines, it demonstrates how to apply these concepts to tackle practical system design problems. To support the focus on design, the text identifies and explains abstractions that have proven successful in practice such as remote procedure call, client/service organization, file systems, data integrity, consistency, and authenticated messages. Most computer systems are built using a handful of such abstractions. The text describes how these abstractions are implemented, demonstrates how they are used in different systems, and prepares the reader to apply them in future designs. The book is recommended for junior and senior undergraduate students in Operating Systems, Distributed Systems, Distributed Operating Systems and/or Computer Systems Design courses; and professional computer systems designers. Concepts of computer system design guided by fundamental principles Cross-cutting approach that identifies abstractions common to networking, operating systems, transaction systems, distributed systems, architecture, and software engineering Case studies that make the abstractions real: naming (DNS and the URL); file systems (the UNIX file system); clients and services (NFS); virtualization (virtual machines); scheduling (disk arms); security (TLS) Numerous pseudocode fragments that provide concrete examples of abstract concepts Extensive support. The authors and MIT OpenCourseWare provide on-line, free of charge, open educational resources, including additional chapters, course syllabi, board layouts and slides, lecture videos, and an archive of lecture schedules, class assignments, and design projects

H-principles and Flexibility in Geometry

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Publisher : American Mathematical Soc.
ISBN 13 : 9780821865019
Total Pages : 76 pages
Book Rating : 4.8/5 (65 download)

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Book Synopsis H-principles and Flexibility in Geometry by : Hansjšrg Geiges

Download or read book H-principles and Flexibility in Geometry written by Hansjšrg Geiges and published by American Mathematical Soc.. This book was released on 2003-05-30 with total page 76 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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).

Basic Category Theory

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Publisher : Cambridge University Press
ISBN 13 : 1107044243
Total Pages : 193 pages
Book Rating : 4.1/5 (7 download)

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Book Synopsis Basic Category Theory by : Tom Leinster

Download or read book Basic Category Theory written by Tom Leinster and published by Cambridge University Press. This book was released on 2014-07-24 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: A short introduction ideal for students learning category theory for the first time.

Problems in Operator Theory

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Publisher : American Mathematical Soc.
ISBN 13 : 0821821474
Total Pages : 402 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Problems in Operator Theory by : Yuri A. Abramovich

Download or read book Problems in Operator Theory written by Yuri A. Abramovich and published by American Mathematical Soc.. This book was released on 2002 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains complete solutions to the more than six hundred exercises in the authors' book: Invitation to operator theory--foreword.

Proofs from THE BOOK

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Publisher : Springer Science & Business Media
ISBN 13 : 3662223430
Total Pages : 194 pages
Book Rating : 4.6/5 (622 download)

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Book Synopsis Proofs from THE BOOK by : Martin Aigner

Download or read book Proofs from THE BOOK written by Martin Aigner and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.

The Mathematical Theory of Communication

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Publisher : University of Illinois Press
ISBN 13 : 025209803X
Total Pages : 144 pages
Book Rating : 4.2/5 (52 download)

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Book Synopsis The Mathematical Theory of Communication by : Claude E Shannon

Download or read book The Mathematical Theory of Communication written by Claude E Shannon and published by University of Illinois Press. This book was released on 1998-09-01 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: Scientific knowledge grows at a phenomenal pace--but few books have had as lasting an impact or played as important a role in our modern world as The Mathematical Theory of Communication, published originally as a paper on communication theory more than fifty years ago. Republished in book form shortly thereafter, it has since gone through four hardcover and sixteen paperback printings. It is a revolutionary work, astounding in its foresight and contemporaneity. The University of Illinois Press is pleased and honored to issue this commemorative reprinting of a classic.

Category Theory in Context

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Publisher : Courier Dover Publications
ISBN 13 : 0486820807
Total Pages : 272 pages
Book Rating : 4.4/5 (868 download)

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Book Synopsis Category Theory in Context by : Emily Riehl

Download or read book Category Theory in Context written by Emily Riehl and published by Courier Dover Publications. This book was released on 2017-03-09 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.

Resolution of Singularities

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Publisher : American Mathematical Soc.
ISBN 13 : 0821835556
Total Pages : 198 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Resolution of Singularities by : Steven Dale Cutkosky

Download or read book Resolution of Singularities written by Steven Dale Cutkosky and published by American Mathematical Soc.. This book was released on 2004 with total page 198 pages. Available in PDF, EPUB and Kindle. Book excerpt: The notion of singularity is basic to mathematics. In algebraic geometry, the resolution of singularities by simple algebraic mappings is truly a fundamental problem. It has a complete solution in characteristic zero and partial solutions in arbitrary characteristic. The resolution of singularities in characteristic zero is a key result used in many subjects besides algebraic geometry, such as differential equations, dynamical systems, number theory, the theory of $\mathcal{D $-modules, topology, and mathematical physics. This book is a rigorous, but instructional, look at resolutions. A simplified proof, based on canonical resolutions, is given for characteristic zero. There are several proofs given for resolution of curves and surfaces in characteristic zero and arbitrary characteristic. Besides explaining the tools needed for understanding resolutions, Cutkosky explains the history and ideas, providing valuable insight and intuition for the novice (or expert). There are many examples and exercises throughout the text. The book is suitable for a second course on an exciting topic in algebraic geometry. A core course on resolutions is contained in Chapters 2 through 6. Additional topics are covered in the final chapters. The prerequisite is a course covering the basic notions of schemes and sheaves.

A Book of Set Theory

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Publisher : Courier Corporation
ISBN 13 : 0486497089
Total Pages : 259 pages
Book Rating : 4.4/5 (864 download)

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Book Synopsis A Book of Set Theory by : Charles C Pinter

Download or read book A Book of Set Theory written by Charles C Pinter and published by Courier Corporation. This book was released on 2014-07-23 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This accessible approach to set theory for upper-level undergraduates poses rigorous but simple arguments. Each definition is accompanied by commentary that motivates and explains new concepts. A historical introduction is followed by discussions of classes and sets, functions, natural and cardinal numbers, the arithmetic of ordinal numbers, and related topics. 1971 edition with new material by the author"--

Homotopy Type Theory: Univalent Foundations of Mathematics

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Publisher : Univalent Foundations
ISBN 13 :
Total Pages : 484 pages
Book Rating : 4./5 ( download)

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Book Synopsis Homotopy Type Theory: Univalent Foundations of Mathematics by :

Download or read book Homotopy Type Theory: Univalent Foundations of Mathematics written by and published by Univalent Foundations. This book was released on with total page 484 pages. Available in PDF, EPUB and Kindle. Book excerpt: