Edited by: Paul A. Milewski, Leslie M. Smith, and Fabian Waleffe, University of Wisconsin, Madison, WI, and Esteban G. Tabak, New York University-Courant Inst. of Math. Sciences, NYC,
Advances in Wave Interaction and Turbulence
Description
We often think of our natural environment as being composed of very many interacting particles, undergoing individual chaotic motions, of which only very coarse averages are perceptible at scales natural to us. However, we could as well think of the world as being made out of individual waves. This is so not just because the distinction between waves and particles becomes rather blurred at the atomic level, but also because even phenomena at much larger scales are better described in terms of waves rather than of particles: It is rare in both fluids and solids to observe energy being carried from one region of space to another by a given set of material particles; much more often, this transfer occurs through chains of particles, neither of them moving much, but each communicating with the next, and hence creating these immaterial objects we call waves.
Waves occur at many spatial and temporal scales. Many of these waves have small enough amplitude that they can be approximately described by linear theory. However, the joint effect of large sets of waves is governed by nonlinear interactions which are responsible for huge cascades of energy among very disparate scales. Understanding these energy transfers is crucial in order to determine the response of large systems, such as the atmosphere and the ocean, to external forcings and dissipation mechanisms which act on scales decades apart.
The field of wave turbulence attempts to understand the average behavior of large ensembles of waves, subjected to forcing and dissipation at opposite ends of their spectrum. It does so by studying individual mechanisms for energy transfer, such as resonant triads and quartets, and attempting to draw from them effects that should not survive averaging.
This book presents the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Dispersive Wave Turbulence held at Mt. Holyoke College (MA). It drew together a group of researchers from many corners of the world, in the context of a perceived renaissance of the field, driven by heated debate about the fundamental mechanism of energy transfer among large sets of waves, as well as by novel applications-and old ones revisited-to the understanding of the natural world. These proceedings reflect the spirit that permeated the conference, that of friendly scientific disagreement and genuine wonder at the rich phenomenology of waves.
Contents
A. Babin, A. Mahalov, and B. Nicolaenko -- Strongly stratified limit of 3D primitive equations in an infinite layer
A. M. Balk -- Anomalous transport by wave turbulence
R. Jordan and B. Turkington -- Statistical equilibrium theories for the nonlinear Schrodinger equation
R. M. Kerr -- Is there a 2D cascade in 3D convection?
F. Menzaque, R. R. Rosales, E. G. Tabak, and C. V. Turner -- The forced inviscid Burgers equation as a model for nonlinear interactions among dispersive waves
P. Panayotaros -- Traveling surface elastic waves in the half-plane
L. M. Smith -- Numerical study of two-dimensional stratified turbulence
V. E. Zakharov, P. Guyenne, A. N. Pushkarev, and F. Dias -- Turbulence of one-dimensional weakly nonlinear dispersive waves
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Contemporary Mathematics, Volume: 283
Publication Year: 2001
ISBN: 0-8218-2714-6
Paging: 116 pp.
Binding: Softcover
Eyal Z. Goren, McGill University, Montreal, PQ, Canada
Lectures on Hilbert Modular Varieties and Modular Forms
Expected publication date is November 25, 2001
Description
This book is devoted to certain aspects of the theory of $p$-adic Hilbert modular forms and moduli spaces of abelian varieties with real multiplication.
The theory of $p$-adic modular forms is presented first in the elliptic case, introducing the reader to key ideas of N. M. Katz and J.-P. Serre. It is re-interpreted from a geometric point of view, which is developed to present the rudiments of a similar theory for Hilbert modular forms.
The theory of moduli spaces of abelian varieties with real multiplication is presented first very explicitly over the complex numbers. Aspects of the general theory are then exposed, in particular, local deformation theory of abelian varieties in positive characteristic.
The arithmetic of $p$-adic Hilbert modular forms and the geometry of moduli spaces of abelian varieties are related. This relation is used to study $q$-expansions of Hilbert modular forms, on the one hand, and stratifications of moduli spaces on the other hand.
The book is addressed to graduate students and non-experts. It attempts to provide the necessary background to all concepts exposed in it. It may serve as a textbook for an advanced graduate course.
Contents
Introduction
Tori and abelian varieties
Complex abelian varieties with real multiplication and Hilbert modular forms
Abelian varieties with real multiplication over general fields
$p$-adic elliptic modular forms
$p$-adic Hilbert modular forms
Deformation theory of abelian varieties
Group schemes
Calculating with cusps
Bibliography
Notation index
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: CRM Monograph Series, Volume: 14
Publication Year: 2002
ISBN: 0-8218-1995-X
Paging: 270 pp.
Binding: Hardcover
Edited by: Alan Coley, Dalhousie University, Halifax, NS, Canada, Decio Levi, University of Rome III, Italy, Robert Milson, Dalhousie University, Halifax, NS, Canada, Colin Rogers, University of New South Wales, Sydney, NSW, Australia, and Pavel Winternitz, Universite de Montreal, QC
Backlund and Darboux Transformations. The Geometry of Solitons
Expected publication date is November 25, 2001
Description
This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Backlund and Darboux transformations and their applications in the theory of integrable systems, also known as soliton theory.
The book consists of two parts. The first is a series of introductory pedagogical lectures presented by leading experts in the field. They are devoted respectively to Backlund transformations of Painleve equations, to the dressing method and Backlund and Darboux transformations, and to the classical geometry of Backlund transformations and their applications to soliton theory. The second part contains original contributions that represent new developments in the theory and applications of these transformations.
Both the introductory lectures and the original talks were presented at an International Workshop that took place in Halifax, Nova Scotia (Canada). This volume covers virtually all recent developments in the theory and applications of Backlund and Darboux transformations.
Contents
Introductory lectures
V. I. Gromak -- Backlund transformations of the higher order Painleve equations
D. Levi and O. Ragnisco -- Dressing method and Backlund and Darboux transformations
C. Rogers and W. K. Schief -- The classical geometry of Backlund transformations. Introduction to applications in soliton theory
W. K. Schief -- An introduction to integrable difference and differential geometries: Affine spheres, their natural generalization and discretization
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: CRM Proceedings & Lecture Notes, Volume: 29
Publication Year: 2001
ISBN: 0-8218-2803-7
Paging: 436 pp.
Binding: Softcover
Edited by: Edward L. Green, Virginia Polytechnic Institute and State University, Blacksburg, VA, Serkan Hosten, San Francisco State University, CA, Reinhard C. Laubenbacher, New Mexico State University, Las Cruces, NM, and Victoria Ann Powers, Emory University, Atlanta, GA
Symbolic Computation:
Solving Equations in Algebra, Geometry, and Engineering
Expected publication date is December 28, 2001
Description
This volume presents the proceedings from the research conference, "Symbolic Computation: Solving Equations in Algebra, Analysis, and Engineering," held at Mount Holyoke College (MA). It provides an overview of current research in symbolic computation as it applies to the solution of polynomial systems. The conference brought together pure and applied mathematicians, computer scientists, and engineers, who use symbolic computation to solve systems of equations or who develop the theoretical background and tools needed for this purpose. Within this general framework, the conference focused on several themes: systems of polynomials, systems of differential equations, noncommutative systems, and applications.
Contents
D. A. Cox -- Equations of parametric curves and surfaces via syzygies
G. M. Diaz-Toca and L. Gonzalez-Vega -- An explicit description for the triangular decomposition of a zero-dimensional ideal through trace computations
A. J. Sommese, J. Verschelde, and C. W. Wampler -- Numerical irreducible decomposition using projections from points on the components
K. Gatermann -- Counting stable solutions of sparse polynomial systems in chemistry
I. S. Kotsireas -- Central configurations in the Newtonian N-body problem of celestial mechanics
D. Napoletani -- A power function approach to Kouchnirenko's conjecture
J. M. Rojas -- Finiteness for arithmetic fewnomial systems
D. Grigoriev -- Constructing double-exponential number of vectors of multiplicities of solutions of polynomial systems
C. D'Andrea and I. Z. Emiris -- Computing sparse projection operators
B. Sturmfels -- Grobner bases of abelian matrix groups
G. Boffi and F. Rossi -- Lexicographic Grobner bases of 3-dimensional transportation problems
E. Briales, A. Campillo, P. Pison, and A. Vigneron -- Simplicial complexes and syzygies of lattice ideals
U. Walther -- Algorithmic determination of the rational cohomology of complex varieties via differential forms
M. Saito and W. N. Traves -- Differential algebras on semigroup algebras
M. J. Bardzell -- Noncommutative Grobner bases and Hochschild cohomology
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Contemporary Mathematics,
Publication Year: 2001
ISBN: 0-8218-2679-4
Paging: approximately 248 pp.
Binding: Softcover
Seiki Nishikawa, Mathematical Institute, Tohoku University, Sendai, Japan
Variational Problems in Geometry
Expected publication date is December 15, 2001
Description
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
Contents
Arc-length of curves and geodesics
First and second variation formulas
Energy of maps and harmonic maps
Existence of harmonic maps
Fundamentals of theory of manifolds and functional analysis
Prospects for the contemporary mathematics
Bibliography
Books
Solutions to exercise problems
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Translations of Mathematical Monographs, ISSN: 0065-9282
Subseries: Iwanami Series in Modern Mathematics
Publication Year: 2001
ISBN: 0-8218-1356-0
Paging: approximately 240 pp.
Binding: Softcover