Shock Formation In Small Data Solutions To 3d Quasilinear Wave Equations

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Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations

Jared Speck
Shock Formation in Small Data Solutions to 3D Quasilinear Wave Equations Book PDF

Author : Jared Speck
Publisher : American Mathematical Soc.
Page : 544 pages
File Size : 51,5 Mb
Release : 2016-12-07
Category : Differential equations, Nonlinear
ISBN : 9781470428570

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Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations by Jared Speck Pdf

In 1848 James Challis showed that smooth solutions to the compressible Euler equations can become multivalued, thus signifying the onset of a shock singularity. Today it is known that, for many hyperbolic systems, such singularities often develop. However, most shock-formation results have been proved only in one spatial dimension. Serge Alinhac's groundbreaking work on wave equations in the late 1990s was the first to treat more than one spatial dimension. In 2007, for the compressible Euler equations in vorticity-free regions, Demetrios Christodoulou remarkably sharpened Alinhac's results and gave a complete description of shock formation. In this monograph, Christodoulou's framework is extended to two classes of wave equations in three spatial dimensions. It is shown that if the nonlinear terms fail to satisfy the null condition, then for small data, shocks are the only possible singularities that can develop. Moreover, the author exhibits an open set of small data whose solutions form a shock, and he provides a sharp description of the blow-up. These results yield a sharp converse of the fundamental result of Christodoulou and Klainerman, who showed that small-data solutions are global when the null condition is satisfied. Readers who master the material will have acquired tools on the cutting edge of PDEs, fluid mechanics, hyperbolic conservation laws, wave equations, and geometric analysis.

Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations

N. V. Krylov
Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations Book PDF

Author : N. V. Krylov
Publisher : American Mathematical Soc.
Page : 441 pages
File Size : 46,6 Mb
Release : 2018-09-07
Category : Differential equations, Parabolic
ISBN : 9781470447403

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Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations by N. V. Krylov Pdf

This book concentrates on first boundary-value problems for fully nonlinear second-order uniformly elliptic and parabolic equations with discontinuous coefficients. We look for solutions in Sobolev classes, local or global, or for viscosity solutions. Most of the auxiliary results, such as Aleksandrov's elliptic and parabolic estimates, the Krylov–Safonov and the Evans–Krylov theorems, are taken from old sources, and the main results were obtained in the last few years. Presentation of these results is based on a generalization of the Fefferman–Stein theorem, on Fang-Hua Lin's like estimates, and on the so-called “ersatz” existence theorems, saying that one can slightly modify “any” equation and get a “cut-off” equation that has solutions with bounded derivatives. These theorems allow us to prove the solvability in Sobolev classes for equations that are quite far from the ones which are convex or concave with respect to the Hessians of the unknown functions. In studying viscosity solutions, these theorems also allow us to deal with classical approximating solutions, thus avoiding sometimes heavy constructions from the usual theory of viscosity solutions.

Nonlinear Dirac Equation: Spectral Stability of Solitary Waves

Nabile Boussaïd,Andrew Comech
Nonlinear Dirac Equation Spectral Stability of Solitary Waves Book PDF

Author : Nabile Boussaïd,Andrew Comech
Publisher : American Mathematical Soc.
Page : 297 pages
File Size : 52,8 Mb
Release : 2019-11-21
Category : Education
ISBN : 9781470443955

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Nonlinear Dirac Equation: Spectral Stability of Solitary Waves by Nabile Boussaïd,Andrew Comech Pdf

This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves. The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrödinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.

Partial Dynamical Systems, Fell Bundles and Applications

Ruy Exel
Partial Dynamical Systems Fell Bundles and Applications Book PDF

Author : Ruy Exel
Publisher : American Mathematical Soc.
Page : 321 pages
File Size : 46,6 Mb
Release : 2017-09-20
Category : Banach spaces
ISBN : 9781470437855

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Partial Dynamical Systems, Fell Bundles and Applications by Ruy Exel Pdf

Partial dynamical systems, originally developed as a tool to study algebras of operators in Hilbert spaces, has recently become an important branch of algebra. Its most powerful results allow for understanding structural properties of algebras, both in the purely algebraic and in the C*-contexts, in terms of the dynamical properties of certain systems which are often hiding behind algebraic structures. The first indication that the study of an algebra using partial dynamical systems may be helpful is the presence of a grading. While the usual theory of graded algebras often requires gradings to be saturated, the theory of partial dynamical systems is especially well suited to treat nonsaturated graded algebras which are in fact the source of the notion of “partiality”. One of the main results of the book states that every graded algebra satisfying suitable conditions may be reconstructed from a partial dynamical system via a process called the partial crossed product. Running in parallel with partial dynamical systems, partial representations of groups are also presented and studied in depth. In addition to presenting main theoretical results, several specific examples are analyzed, including Wiener–Hopf algebras and graph C*-algebras.

Sixteenth Marcel Grossmann Meeting, The: On Recent Developments In Theoretical And Experimental General Relativity, Astrophysics, And Relativistic Field Theories - Proceedings Of The Mg16 Meeting On General Relativity (In 4 Volumes)

Remo Ruffini,Gregory Vereshchagin
Sixteenth Marcel Grossmann Meeting The On Recent Developments In Theoretical And Experimental General Relativity Astrophysics And Relativistic Field Theories Proceedings Of The Mg16 Meeting On General Relativity In 4 Volumes Book PDF

Author : Remo Ruffini,Gregory Vereshchagin
Publisher : World Scientific
Page : 4880 pages
File Size : 44,5 Mb
Release : 2022-12-15
Category : Science
ISBN : 9789811269783

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Sixteenth Marcel Grossmann Meeting, The: On Recent Developments In Theoretical And Experimental General Relativity, Astrophysics, And Relativistic Field Theories - Proceedings Of The Mg16 Meeting On General Relativity (In 4 Volumes) by Remo Ruffini,Gregory Vereshchagin Pdf

The proceedings of MG16 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 46 plenary presentations, 3 public lectures, 5 round tables and 81 parallel sessions arranged during the intense six-day online meeting. All talks were recorded and are available on the ICRANet YouTube channel at the following link: www.icranet.org/video_mg16.These proceedings are a representative sample of the very many contributions made at the meeting. They contain 383 papers, among which 14 come from the plenary sessions.The material represented in these proceedings cover the following topics: accretion, active galactic nuclei, alternative theories of gravity, black holes (theory, observations and experiments), binaries, boson stars, cosmic microwave background, cosmic strings, dark energy and large scale structure, dark matter, education, exact solutions, early universe, fundamental interactions and stellar evolution, fast transients, gravitational waves, high energy physics, history of relativity, neutron stars, precision tests, quantum gravity, strong fields, and white dwarf; all of them represented by a large number of contributions.The online e-proceedings are published in an open access format.

The Classification of the Finite Simple Groups, Number 8: Part III, Chapters 12–17: The Generic Case, Completed

Daniel Gorenstein,Richard Lyons,Ronald Solomon
The Classification of the Finite Simple Groups Number 8 Part III Chapters 12 17 The Generic Case Completed Book PDF

Author : Daniel Gorenstein,Richard Lyons,Ronald Solomon
Publisher : American Mathematical Soc.
Page : 488 pages
File Size : 52,9 Mb
Release : 2018-12-12
Category : Electronic
ISBN : 9781470441890

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The Classification of the Finite Simple Groups, Number 8: Part III, Chapters 12–17: The Generic Case, Completed by Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

This book completes a trilogy (Numbers 5, 7, and 8) of the series The Classification of the Finite Simple Groups treating the generic case of the classification of the finite simple groups. In conjunction with Numbers 4 and 6, it allows us to reach a major milestone in our series—the completion of the proof of the following theorem: Theorem O: Let G be a finite simple group of odd type, all of whose proper simple sections are known simple groups. Then either G is an alternating group or G is a finite group of Lie type defined over a field of odd order or G is one of six sporadic simple groups. Put another way, Theorem O asserts that any minimal counterexample to the classification of the finite simple groups must be of even type. The work of Aschbacher and Smith shows that a minimal counterexample is not of quasithin even type, while this volume shows that a minimal counterexample cannot be of generic even type, modulo the treatment of certain intermediate configurations of even type which will be ruled out in the next volume of our series.

Jordan Triple Systems in Complex and Functional Analysis

José M. Isidro
Jordan Triple Systems in Complex and Functional Analysis Book PDF

Author : José M. Isidro
Publisher : American Mathematical Soc.
Page : 560 pages
File Size : 42,6 Mb
Release : 2019-12-09
Category : Education
ISBN : 9781470450830

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Jordan Triple Systems in Complex and Functional Analysis by José M. Isidro Pdf

This book is a systematic account of the impressive developments in the theory of symmetric manifolds achieved over the past 50 years. It contains detailed and friendly, but rigorous, proofs of the key results in the theory. Milestones are the study of the group of holomomorphic automorphisms of bounded domains in a complex Banach space (Vigué and Upmeier in the late 1970s), Kaup's theorem on the equivalence of the categories of symmetric Banach manifolds and that of hermitian Jordan triple systems, and the culminating point in the process: the Riemann mapping theorem for complex Banach spaces (Kaup, 1982). This led to the introduction of wide classes of Banach spaces known as JB∗-triples and JBW∗-triples whose geometry has been thoroughly studied by several outstanding mathematicians in the late 1980s. The book presents a good example of fruitful interaction between different branches of mathematics, making it attractive for mathematicians interested in various fields such as algebra, differential geometry and, of course, complex and functional analysis.

Sugawara Operators for Classical Lie Algebras

Alexander Molev:
Sugawara Operators for Classical Lie Algebras Book PDF

Author : Alexander Molev:
Publisher : American Mathematical Soc.
Page : 304 pages
File Size : 41,7 Mb
Release : 2018-02-28
Category : Affine algebraic groups
ISBN : 9781470436599

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Sugawara Operators for Classical Lie Algebras by Alexander Molev: Pdf

The celebrated Schur-Weyl duality gives rise to effective ways of constructing invariant polynomials on the classical Lie algebras. The emergence of the theory of quantum groups in the 1980s brought up special matrix techniques which allowed one to extend these constructions beyond polynomial invariants and produce new families of Casimir elements for finite-dimensional Lie algebras. Sugawara operators are analogs of Casimir elements for the affine Kac-Moody algebras. The goal of this book is to describe algebraic structures associated with the affine Lie algebras, including affine vertex algebras, Yangians, and classical -algebras, which have numerous ties with many areas of mathematics and mathematical physics, including modular forms, conformal field theory, and soliton equations. An affine version of the matrix technique is developed and used to explain the elegant constructions of Sugawara operators, which appeared in the last decade. An affine analogue of the Harish-Chandra isomorphism connects the Sugawara operators with the classical -algebras, which play the role of the Weyl group invariants in the finite-dimensional theory.

Nilpotent Structures in Ergodic Theory

Bernard Host,Bryna Kra
Nilpotent Structures in Ergodic Theory Book PDF

Author : Bernard Host,Bryna Kra
Publisher : American Mathematical Soc.
Page : 427 pages
File Size : 53,7 Mb
Release : 2018-12-12
Category : Ergodic theory
ISBN : 9781470447809

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Nilpotent Structures in Ergodic Theory by Bernard Host,Bryna Kra Pdf

Nilsystems play a key role in the structure theory of measure preserving systems, arising as the natural objects that describe the behavior of multiple ergodic averages. This book is a comprehensive treatment of their role in ergodic theory, covering development of the abstract theory leading to the structural statements, applications of these results, and connections to other fields. Starting with a summary of the relevant dynamical background, the book methodically develops the theory of cubic structures that give rise to nilpotent groups and reviews results on nilsystems and their properties that are scattered throughout the literature. These basic ingredients lay the groundwork for the ergodic structure theorems, and the book includes numerous formulations of these deep results, along with detailed proofs. The structure theorems have many applications, both in ergodic theory and in related fields; the book develops the connections to topological dynamics, combinatorics, and number theory, including an overview of the role of nilsystems in each of these areas. The final section is devoted to applications of the structure theory, covering numerous convergence and recurrence results. The book is aimed at graduate students and researchers in ergodic theory, along with those who work in the related areas of arithmetic combinatorics, harmonic analysis, and number theory.

Weak Convergence of Measures

Vladimir I. Bogachev
Weak Convergence of Measures Book PDF

Author : Vladimir I. Bogachev
Publisher : American Mathematical Soc.
Page : 286 pages
File Size : 42,5 Mb
Release : 2018-09-27
Category : Convergence
ISBN : 9781470447380

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Weak Convergence of Measures by Vladimir I. Bogachev Pdf

This book provides a thorough exposition of the main concepts and results related to various types of convergence of measures arising in measure theory, probability theory, functional analysis, partial differential equations, mathematical physics, and other theoretical and applied fields. Particular attention is given to weak convergence of measures. The principal material is oriented toward a broad circle of readers dealing with convergence in distribution of random variables and weak convergence of measures. The book contains the necessary background from measure theory and functional analysis. Large complementary sections aimed at researchers present the most important recent achievements. More than 100 exercises (ranging from easy introductory exercises to rather difficult problems for experienced readers) are given with hints, solutions, or references. Historic and bibliographic comments are included. The target readership includes mathematicians and physicists whose research is related to probability theory, mathematical statistics, functional analysis, and mathematical physics.

The Classification of the Finite Simple Groups, Number 7: Part III, Chapters 7–11: The Generic Case, Stages 3b and 4a

Daniel Gorenstein,Richard Lyon,Ronald Solomon
The Classification of the Finite Simple Groups Number 7 Part III Chapters 7 11 The Generic Case Stages 3b and 4a Book PDF

Author : Daniel Gorenstein,Richard Lyon,Ronald Solomon
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 47,6 Mb
Release : 2018-02-15
Category : Finite simple groups
ISBN : 9780821840696

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The Classification of the Finite Simple Groups, Number 7: Part III, Chapters 7–11: The Generic Case, Stages 3b and 4a by Daniel Gorenstein,Richard Lyon,Ronald Solomon Pdf

The classification of finite simple groups is a landmark result of modern mathematics. The multipart series of monographs which is being published by the AMS (Volume 40.1–40.7 and future volumes) represents the culmination of a century-long project involving the efforts of scores of mathematicians published in hundreds of journal articles, books, and doctoral theses, totaling an estimated 15,000 pages. This part 7 of the series is the middle of a trilogy (Volume 40.5, Volume 40.7, and forthcoming Volume 40.8) treating the Generic Case, i.e., the identification of the alternating groups of degree at least 13 and most of the finite simple groups of Lie type and Lie rank at least 4. Moreover, Volumes 40.4–40.8 of this series will provide a complete treatment of the simple groups of odd type, i.e., the alternating groups (with two exceptions) and the groups of Lie type defined over a finite field of odd order, as well as some of the sporadic simple groups. In particular, this volume completes the construction, begun in Volume 40.5, of a collection of neighboring centralizers of a particularly nice form. All of this is then applied to complete the identification of the alternating groups of degree at least 13. The book is suitable for graduate students and researchers interested in the theory of finite groups.

The Dirichlet Space and Related Function Spaces

Nicola Arcozzi,Richard Rochberg,Eric T. Sawyer,Brett D. Wick
The Dirichlet Space and Related Function Spaces Book PDF

Author : Nicola Arcozzi,Richard Rochberg,Eric T. Sawyer,Brett D. Wick
Publisher : American Mathematical Soc.
Page : 536 pages
File Size : 55,9 Mb
Release : 2019-09-03
Category : Dirichlet principle
ISBN : 9781470450823

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The Dirichlet Space and Related Function Spaces by Nicola Arcozzi,Richard Rochberg,Eric T. Sawyer,Brett D. Wick Pdf

The study of the classical Dirichlet space is one of the central topics on the intersection of the theory of holomorphic functions and functional analysis. It was introduced about100 years ago and continues to be an area of active current research. The theory is related to such important themes as multipliers, reproducing kernels, and Besov spaces, among others. The authors present the theory of the Dirichlet space and related spaces starting with classical results and including some quite recent achievements like Dirichlet-type spaces of functions in several complex variables and the corona problem. The first part of this book is an introduction to the function theory and operator theory of the classical Dirichlet space, a space of holomorphic functions on the unit disk defined by a smoothness criterion. The Dirichlet space is also a Hilbert space with a reproducing kernel, and is the model for the dyadic Dirichlet space, a sequence space defined on the dyadic tree. These various viewpoints are used to study a range of topics including the Pick property, multipliers, Carleson measures, boundary values, zero sets, interpolating sequences, the local Dirichlet integral, shift invariant subspaces, and Hankel forms. Recurring themes include analogies, sometimes weak and sometimes strong, with the classical Hardy space; and the analogy with the dyadic Dirichlet space. The final chapters of the book focus on Besov spaces of holomorphic functions on the complex unit ball, a class of Banach spaces generalizing the Dirichlet space. Additional techniques are developed to work with the nonisotropic complex geometry, including a useful invariant definition of local oscillation and a sophisticated variation on the dyadic Dirichlet space. Descriptions are obtained of multipliers, Carleson measures, interpolating sequences, and multiplier interpolating sequences; estimates are obtained to prove corona theorems.

Perfectoid Spaces

Bhargav Bhatt,Ana Caraiani,Kiran S. Kedlaya,Peter Scholze,Jared Weinstein
Perfectoid Spaces Book PDF

Author : Bhargav Bhatt,Ana Caraiani,Kiran S. Kedlaya,Peter Scholze,Jared Weinstein
Publisher : American Mathematical Society
Page : 297 pages
File Size : 42,5 Mb
Release : 2022-02-04
Category : Mathematics
ISBN : 9781470465100

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Perfectoid Spaces by Bhargav Bhatt,Ana Caraiani,Kiran S. Kedlaya,Peter Scholze,Jared Weinstein Pdf

Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge between geometry in characteristic 0 and characteristic $p$, and have been used to solve many important problems, including cases of the weight-monodromy conjecture and the association of Galois representations to torsion classes in cohomology. In recognition of the transformative impact perfectoid spaces have had on the field of arithmetic geometry, Scholze was awarded a Fields Medal in 2018. This book, originating from a series of lectures given at the 2017 Arizona Winter School on perfectoid spaces, provides a broad introduction to the subject. After an introduction with insight into the history and future of the subject by Peter Scholze, Jared Weinstein gives a user-friendly and utilitarian account of the theory of adic spaces. Kiran Kedlaya further develops the foundational material, studies vector bundles on Fargues–Fontaine curves, and introduces diamonds and shtukas over them with a view toward the local Langlands correspondence. Bhargav Bhatt explains the application of perfectoid spaces to comparison isomorphisms in $p$-adic Hodge theory. Finally, Ana Caraiani explains the application of perfectoid spaces to the construction of Galois representations associated to torsion classes in the cohomology of locally symmetric spaces for the general linear group. This book will be an invaluable asset for any graduate student or researcher interested in the theory of perfectoid spaces and their applications.

Algebraic Geometry Codes: Advanced Chapters

Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin
Algebraic Geometry Codes Advanced Chapters Book PDF

Author : Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin
Publisher : American Mathematical Soc.
Page : 453 pages
File Size : 47,8 Mb
Release : 2019-07-02
Category : Coding theory
ISBN : 9781470448653

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Algebraic Geometry Codes: Advanced Chapters by Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin Pdf

Algebraic Geometry Codes: Advanced Chapters is devoted to the theory of algebraic geometry codes, a subject related to local_libraryBook Catalogseveral domains of mathematics. On one hand, it involves such classical areas as algebraic geometry and number theory; on the other, it is connected to information transmission theory, combinatorics, finite geometries, dense packings, and so on. The book gives a unique perspective on the subject. Whereas most books on coding theory start with elementary concepts and then develop them in the framework of coding theory itself within, this book systematically presents meaningful and important connections of coding theory with algebraic geometry and number theory. Among many topics treated in the book, the following should be mentioned: curves with many points over finite fields, class field theory, asymptotic theory of global fields, decoding, sphere packing, codes from multi-dimensional varieties, and applications of algebraic geometry codes. The book is the natural continuation of Algebraic Geometric Codes: Basic Notions by the same authors. The concise exposition of the first volume is included as an appendix.

Virtual Fundamental Cycles in Symplectic Topology

John W. Morgan,Dusa McDuff,Mohammad Tehrani,Kenji Fukaya,Dominic Joyce
Virtual Fundamental Cycles in Symplectic Topology Book PDF

Author : John W. Morgan,Dusa McDuff,Mohammad Tehrani,Kenji Fukaya,Dominic Joyce
Publisher : American Mathematical Soc.
Page : 300 pages
File Size : 49,7 Mb
Release : 2019-04-12
Category : Geometry, Differential
ISBN : 9781470450144

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Virtual Fundamental Cycles in Symplectic Topology by John W. Morgan,Dusa McDuff,Mohammad Tehrani,Kenji Fukaya,Dominic Joyce Pdf

The method of using the moduli space of pseudo-holomorphic curves on a symplectic manifold was introduced by Mikhail Gromov in 1985. From the appearance of Gromov's original paper until today this approach has been the most important tool in global symplectic geometry. To produce numerical invariants of these manifolds using this method requires constructing a fundamental cycle associated with moduli spaces. This volume brings together three approaches to constructing the “virtual” fundamental cycle for the moduli space of pseudo-holomorphic curves. All approaches are based on the idea of local Kuranishi charts for the moduli space. Workers in the field will get a comprehensive understanding of the details of these constructions and the assumptions under which they can be made. These techniques and results will be essential in further applications of this approach to producing invariants of symplectic manifolds.