An Introduction to the Theory of Wave Maps and Related Geometric Problems

Author: Dan-Andrei Geba
Publisher: World Scientific Publishing Company
ISBN: 9814713929
Format: PDF, ePub, Mobi
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The wave maps system is one of the most beautiful and challenging nonlinear hyperbolic systems, which has captured the attention of mathematicians for more than thirty years now. In the study of its various issues, such as the well-posedness theory, the formation of singularities, and the stability of the solitons, in order to obtain optimal results, one has to use intricate tools coming not only from analysis, but also from geometry and topology. Moreover, the wave maps system is nothing other than the Euler–Lagrange system for the nonlinear sigma model, which is one of the fundamental problems in classical field theory. One of the goals of our book is to give an up-to-date and almost self-contained overview of the main regularity results proved for wave maps. Another one is to introduce, to a wide mathematical audience, physically motivated generalizations of the wave maps system (e.g., the Skyrme model), which are extremely interesting and difficult in their own right.

Nonlinear Partial Differential Equations in Geometry and Physics

Author: Garth Baker
Publisher: Birkhäuser
ISBN: 3034888953
Format: PDF, ePub, Docs
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This volume presents the proceedings of a series of lectures hosted by the Math ematics Department of The University of Tennessee, Knoxville, March 22-24, 1995, under the title "Nonlinear Partial Differential Equations in Geometry and Physics" . While the relevance of partial differential equations to problems in differen tial geometry has been recognized since the early days of the latter subject, the idea that differential equations of differential-geometric origin can be useful in the formulation of physical theories is a much more recent one. Perhaps the earliest emergence of systems of nonlinear partial differential equations having deep geo metric and physical importance were the Einstein equations of general relativity (1915). Several basic aspects of the initial value problem for the Einstein equa tions, such as existence, regularity and stability of solutions remain prime research areas today. eighty years after Einstein's work. An even more recent development is the realization that structures originally the context of models in theoretical physics may turn out to have introduced in important geometric or topological applications. Perhaps its emergence can be traced back to 1954, with the introduction of a non-abelian version of Maxwell's equations as a model in elementary-particle physics, by the physicists C.N. Yang and R. Mills. The rich geometric structure ofthe Yang-Mills equations was brought to the attention of mathematicians through work of M.F. Atiyah, :"J. Hitchin, I.

New Trends in the Theory of Hyperbolic Equations

Author: Michael Reissig
Publisher: Springer Science & Business Media
ISBN: 9783764372835
Format: PDF, ePub
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This book presents several recent developments in the theory of hyperbolic equations. The carefully selected invited and peer-reviewed contributions deal with questions of low regularity, critical growth, ill-posedness, decay estimates for solutions of different non-linear hyperbolic models, and introduce new approaches based on microlocal methods.

Collection of Papers on Geometry Analysis and Mathematical Physics

Author: T-T Li
Publisher: World Scientific
ISBN: 9814497797
Format: PDF, Docs
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This volume is published in honor of Professor Gu Chaohao, a renowned mathematician and member of the Chinese Academy of Sciences, on the occasion of his 70th birthday and his 50th year of educational work. The subjects covered by this collection are closely related to differential geometry, partial differential equations and mathematical physics — the major areas in which Professor Gu has received notable achievements. Many distinguished mathematicians all over the world contributed their papers to this collection. This collection also consists of “Gu Chaohao and I” written by C N Yang, “The academic career and accomplishment of Professor Gu Chaohao” by T T Li and “List of publications of Professor Gu Chaohao”. Contents:A Global Existence Theorem for Ultra Relativistic Fluids on Minkowski Space Time (Y Choquet-Bruhat)Asymptotic Analysis of Elastic Shells (P G Ciarlet)Generalized Solutions Defined by Lebesque–Stieltjies Integrals (X X Ding)Automorphisms of the Circle — and Teichmüller Theory (J Eells)Nonlinear Relativistic Wave Equations in General Dimensions (M Flato et al.)Remarks on the Domain-Dependence of Convergence Rate of Iterations in a Certain Domain Decomposition Method — Analysis by the Steklov–Poincaré Operator (J Fujita)Compactification of Moduli of Vector Bundles Over Algebraic Surfaces (J Li)Analysis on Singular Spaces (F H Lin)A Generic Result of Approximate Controllability (J L Lions)and other papers Readership: Mathematicians. keywords:Geometry;Analysis;Mathematical Physics;Dedication

An Introduction to the Mathematical Theory of Waves

Author: Roger Knobel
Publisher: American Mathematical Soc.
ISBN: 0821820397
Format: PDF, ePub, Docs
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Linear and nonlinear waves are a central part of the theory of PDEs. This book begins with a description of one-dimensional waves and their visualization through computer-aided techniques. Next, traveling waves are covered, such as solitary waves for the Klein-Gordon and KdV equations. Finally, the author gives a lucid discussion of waves arising from conservation laws, including shock and rarefaction waves. As an application, interesting models of traffic flow are used to illustrate conservation laws and wave phenomena. This book is based on a course given by the author at the IAS/Park City Mathematics Institute. It is suitable for independent study by undergraduate students in mathematics, engineering, and science programs.

Best Approximation by Linear Superpositions approximate Nomography

Author: S. I͡A. Khavinson
Publisher: American Mathematical Soc.
ISBN: 9780821897737
Format: PDF, ePub
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This book deals with problems of approximation of continuous or bounded functions of several variables by linear superposition of functions that are from the same class and have fewer variables. The main topic is the space of linear superpositions D considered as a sub-space of the space of continous functions C(X) on a compact space X. Such properties as density of D in C(X), its closedness, proximality, etc. are studied in great detail. The approach to these and other problems based on duality and the Hahn-Banach theorem is emphasized. Also, considerable attention is given to the discussion of the Diliberto-Straus algorithm for finding the best approximation of a given function by linear superpositions.

Operator Theory Analysis and Mathematical Physics

Author: Jan Janas
Publisher: Springer Science & Business Media
ISBN: 3764381353
Format: PDF, Docs
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This volume contains lectures delivered at the International Conference Operator Theory and its Applications in Mathematical Physics (OTAMP 2004), held at the Mathematical Research and Conference Center in Bedlewo near Poznan, Poland. The idea behind these lectures was to present interesting ramifications of operator methods in current research of mathematical physics.

Hyperbolic Problems Theory Numerics Applications

Author: Heinrich Freistühler
Publisher: Springer Science & Business Media
ISBN: 9783764367107
Format: PDF, ePub
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Hyperbolic partial differential equations describe phenomena of material or wave transport in physics, biology and engineering, especially in the field of fluid mechanics. The mathematical theory of hyperbolic equations has recently made considerable progress. Accurate and efficient numerical schemes for computation have been and are being further developed. This two-volume set of conference proceedings contains about 100 refereed and carefully selected papers. The books are intended for researchers and graduate students in mathematics, science and engineering interested in the most recent results in theory and practice of hyperbolic problems. Applications touched in these proceedings concern one-phase and multiphase fluid flow, phase transitions, shallow water dynamics, elasticity, extended thermodynamics, electromagnetism, classical and relativistic magnetohydrodynamics, cosmology. Contributions to the abstract theory of hyperbolic systems deal with viscous and relaxation approximations, front tracking and wellposedness, stability of shock profiles and multi-shock patterns, traveling fronts for transport equations. Numerically oriented articles study finite difference, finite volume, and finite element schemes, adaptive, multiresolution, and artificial dissipation methods.