Book Description

We often think of our natural environment as being composed ofvery many interacting particles, undergoing individual chaoticmotions, of which only very coarse averages are perceptible atscales natural to us. However, we could as well think of theworld as being made out of individual waves. This is so not justbecause the distinction between waves and particles becomesrather blurred at the atomic level, but also because evenphenomena at much larger scales are better described in terms ofwaves rather than of particles: It is rare in both fluids andsolids to observe energy being carried from one region of spaceto another by a given set of material particles; much moreoften, this transfer occurs through chains of particles, neitherof them moving much, but each communicating with the next, andhence creating these immaterial objects we call waves.

Waves occur at many spatial and temporal scales. Many of thesewaves have small enough amplitude that they can be approximatelydescribed by linear theory. However, the joint effect of largesets of waves is governed by nonlinear interactions which areresponsible for huge cascades of energy among very disparatescales. Understanding these energy transfers is crucial in orderto determine the response of large systems, such as theatmosphere and the ocean, to external forcings and dissipationmechanisms which act on scales decades apart.

The field of wave turbulence attempts to understand the averagebehavior of large ensembles of waves, subjected to forcing anddissipation at opposite ends of their spectrum. It does so bystudying individual mechanisms for energy transfer, such asresonant triads and quartets, and attempting to draw from themeffects that should not survive averaging.

This book presents the proceedings of the AMS-IMS-SIAM JointSummer Research Conference on Dispersive Wave Turbulence held atMt. Holyoke College (MA). It drew together a group ofresearchers from many corners of the world, in the context of aperceived renaissance of the field, driven by heated debateabout the fundamental mechanism of energy transfer among largesets of waves, as well as by novel applications--and old onesrevisited--to the understanding of the natural world. Theseproceedings reflect the spirit that permeated the conference,that of friendly scientific disagreement and genuine wonder atthe rich phenomenology of waves. ... Read more

Book Description

One of the approaches to the study of functions of severalcomplex variables is to use methods originating in realanalysis. In this concise book, the author gives a lucidpresentation of how these methods produce a variety of globalexistence theorems in the theory of functions (based on thecharacterization of holomorphic functions as weak solutions ofthe Cauchy-Riemann equations).

Emphasis is on recent results, including an $L^2$ extensiontheorem for holomorphic functions, that have brought a deeperunderstanding of pseudoconvexity and plurisubharmonic functions.Based on Oka's theorems and his schema for the grouping ofproblems, topics covered in the book are at the intersection ofthe theory of analytic functions of several variables andmathematical analysis.

It is assumed that the reader has a basic knowledge of complexanalysis at the undergraduate level. The book would make a finesupplementary text for a graduate-level course on complexanalysis. ... Read more

Book Description

Questions regarding the interplay of nonlinearity and the creation and propagation of singularities arise in a variety of fields-including nonlinear partial differential equations, noise-driven stochastic partial differential equations, general relativity, and geometry with singularities. A workshop held at the Erwin-Schrödinger International Institute for Mathematical Physics in Vienna investigated these questions and culminated in this volume of invited papers from experts in the fields of nonlinear partial differential equations, structure theory of generalized functions, geometry and general relativity, stochastic partial differential equations, and nonstandard analysis.The authors provide the latest research relevant to work in partial differential equations, mathematical physics, and nonlinear analysis. With a focus on applications, this books provides a compilation of recent approaches to the problem of singularities in nonlinear models. The theory of differential algebras of generalized functions serves as the central theme of the project, along with its interrelations with classical methods. ... Read more

Book Description

A renowned mathematician who considers himself both applied and theoretical in his approach, Peter Lax has spent most of his professional career at NYU, making significant contributions to both mathematics and computing. He has written several important published works and has received numerous honors including the National Medal of Science, the Lester R. Ford Award, the Chauvenet Prize, the Semmelweis Medal, the Wiener Prize, and the Wolf Prize. Several students he has mentored have become leaders in their fields. These two volumes span the years from 1952 up until 1999, and cover many varying topics, from functional analysis, partial differential equations, and numerical methods to conservation laws, integrable systems and scattering theory. After each paper, or collection of papers, is a commentary placing the paper in context and where relevant discussing more recent developments. Many of the papers in these volumes have become classics and should be ready by any serious student of these topics. In terms of insight, depth, and breadth, Las has few equals. The reader of this selecta will quickly appreciate his brilliance as well as his masterful touch. Having this collection of papers in one place allows one to follow the evolution of his ideas and mathematical interests and to appreciate how many of these papers initiated topics that developed lives of their own. ... Read more

Book Description

This annual directory provides a handy reference to variousorganizations in the mathematical sciences community. Listed inthe directory are the following: officers and committee membersof over thirty professional mathematical organizations (terms ofoffice and other pertinent information are also provided in somecases); key mathematical sciences personnel of selectedgovernment agencies; academic departments in the mathematicalsciences; mathematical units in nonacademic organizations; andalphabetic listings of colleges and universities. Currentaddresses, telephone numbers, and electronic addresses forindividuals when provided are listed in the directory. ... Read more

Book Description

The synergism between the World Wide Web and fiber optics is a familiar story to researchers of digital communications. Fibers are the enablers of the rates of information flow that make the Internet possible. Currently transoceanic optical fiber cables transmit data at rates that could transfer the contents of a respectable university library in a few minutes. No other medium is capable of this rate of transmission at such distances.

With the maturing of mobile portable telephony and the emerging broadband access market, greater fiber transmission capacity will be essential in the early 21st century. Since the demand for more capacity drives the development of new optics-based technologies, fiber optics therefore remains a vibrant area for research. Knowing that the basic fiber optic technology is mature means that open questions are more sharply focused and permit deeper mathematic content.

Mathematical Principles of Optical Fiber Communications is intended to support and promote interdisciplinary research in optical fiber communications by providing essential background in both the physical and mathematical principles of the discipline. Chapter topics include the basics of fibers and their construction, fiber modes and the criterion of single mode operation, the nonlinear Schrödinger equation, the variational approach to the analysis of pulse propagation, and, finally, solitons and some new results on soliton formation energy thresholds. These chapters are written to be as independent as possible while taking the reader to the frontiers of research on fiber optics communications. ... Read more

Book Description

Variational analysis is a fruitful area in mathematics that, on the one hand, deals with the study of optimization and equilibrium problems and, on the other hand, applies optimization, perturbation, and approximation ideas to the analysis of a broad range of problems that may not be of a variational nature.

This monograph in two volumes contains a comprehensive and state-of-the art study of the basic concepts and principles of variational analysis and generalized differentiation in both finite-dimensional and infinite dimensional spaces and presents numerous applications to problems in the optimization, equilibria, stability and sensitivity, control theory, economics, mechanics, etc. The first volume is mainly devoted to the basic theory of variational analysis and generalized differentiations, while the second volume contains various applications. Both volumes contain abundant bibliographies and extensive commentaries.

Book Description

In an era when increasing numbers of people are conducting research and interacting with one another through the internet, the study of ?Inhabited Information Spaces? is aimed at encouraging a more fruitful exchange between the users, and the digital data they are accessing. Introducing the new and developing field of Inhabited Information Spaces, this book covers all types of collaborative systems including virtual environments and more recent innovations such as hybrid and augmented real-world systems. Divided into separate sections, each covering a different aspect of Inhabited Information Systems, this book includes: · How best to design and construct social work spaces · Analysis of how users interact with existing systems, and the technological and sociological challenges designers face · How Inhabited Information Spaces are likely to evolve in the future and the new communities that they will create ... Read more

Book Description

This is a presentation of the main results of the 2nd International Conference on Multivariate Approximation, held in Haus Bommerholz near Witten, Germany, September 29 - October 4, 1996. There are contributions to the following topics: Interpolation and hyperinterpolation on the sphere, quadrature, discrepancy and spherical designs, multivariate convex spline interpolation, triangular finite element spaces, polar form of splines, refinable spline pairs, local box dimension of fractal functions, symbolic treatment of algebraic systems, harmonic approximation and boundary behaviour, quasi-balayage, Landau-Kolmogorov inequality, trigonometric approximation in multivariate periodic Hilbert spaces, orthonormal polynomial bases of low degree, synthesis and analysis with multivariate wavelets and frames. ... Read more