Vsevolod Alekseevich Solonnikov is known as one of the outstanding mathema- ciansfromtheSt.PetersburgMathematicalSchool.Hisremarkableresultsonexact estimates of solutions to boundary and initial-boundary value problems for linear elliptic, parabolic, and Stokes systems, his methods and contributions to the - vestigation of free boundary problems, in particular in ?uid mechanics, are well known to specialists all over the world. The International Conference on "Trends in Partial Di?erential Equations of th ' Mathematical Physics" was held on the occasion of his 70 birthday in Obidos (Portugal), from June 7 to 10, 2003. It was an organization of the "Centro de Matem' atica e Aplica, c" oes Fundamentais da Universidade Lisboa", in collaboration with the "Centro de Matem' atica da Universidade de Coimbra", the "Centro de Matem' atica Aplicada do IST/Universidade T' ecnica de Lisboa", the "Centro de Matem' atica da Universidade da Beira Interior",from Portugal,and with the L- oratory of Mathematical Physics of the St.Petersburg Department of the Steklov Institute of Mathematics from Russia. The conference consisted of thirty eight invited and contributed lectures and ' gathered,inthecharminganduniquemedievaltownofObidos,aboutsixtypart- ipants from ?fteen countries, namely USA, Switzerland, Spain, Russia, Portugal, Poland, Lithuania, Korea, Japan, Italy, Germany, France, Canada, Australia and Argentina.Severalcolleaguesgaveusahelpinghandintheorganizationofthec- ference. We are thankful to all of them, and in particular to Stanislav Antontsev, Anvarbek Meirmanov and Ad' elia Sequeira, that integrated also the Organizing Committee. A special acknowledgement is due to Elena Frolova that helped us in compiling the short and necessarily incomplete bio-bibliographical notes below.

This book represents an extended and substantially revised version of my earlierbook, Optimal Control in Problems ofMathematical Physics, originally published in Russian in 1975. About 60% of the text has been completely revised and major additions have been included which have produced a practically new text. My aim was to modernize the presentation but also to preserve the original results, some of which are little known to a Western reader. The idea of composites, which is the core of the modern theory of optimization, was initiated in the early seventies. The reader will find here its implementation in the problem of optimal conductivity distribution in an MHD-generatorchannel flow.Sincethen it has emergedinto an extensive theory which is undergoing a continuous development. The book does not pretend to be a textbook, neither does it offer a systematic presentation of the theory. Rather, it reflects a concept which I consider as fundamental in the modern approach to optimization of dis tributed systems. Bibliographical notes, though extensive, do not pretend to be exhaustive as well. My thanks are due to ProfessorJean-Louis Armand and ProfessorWolf Stadler whose friendly assistance in translating and polishing the text was so valuable. I am indebted to Mrs. Kathleen Durand and Mrs. Colleen Lewis for the hard job of typing large portions of the manuscript."

This book is intended as an introductory lecture in material physics, in which the modern computational group theory and the electronic structure calculation are in collaboration. The first part explains how to use computer algebra for applications in solid-state simulation, based on the GAP computer algebra package. Computer algebra enables us to easily obtain various group theoretical properties, such as the representations, character tables, and subgroups. Furthermore it offers a new perspective on material design, which could be executed in a mathematically rigorous and systematic way. The second part then analyzes the relation between the structural symmetry and the electronic structure in C60 (as an example of a system without periodicity). The principal object of the study was to illustrate the hierarchical change in the quantum-physical properties of the molecule, which correlates to the reduction in the symmetry (as it descends down in the ladder of subgroups). The book also presents the computation of the vibrational modes of the C60 by means of the computer algebra. In order to serve the common interests of researchers, the details of the computations (the required initial data and the small programs developed for the purpose) are explained in as much detail as possible.

The first part of this volume gathers the lecture notes of the courses of the "XVII Escuela Hispano-Francesa", held in Gijon, Spain, in June 2016. Each chapter is devoted to an advanced topic and presents state-of-the-art research in a didactic and self-contained way. Young researchers will find a complete guide to beginning advanced work in fields such as High Performance Computing, Numerical Linear Algebra, Optimal Control of Partial Differential Equations and Quantum Mechanics Simulation, while experts in these areas will find a comprehensive reference guide, including some previously unpublished results, and teachers may find these chapters useful as textbooks in graduate courses. The second part features the extended abstracts of selected research work presented by the students during the School. It highlights new results and applications in Computational Algebra, Fluid Mechanics, Chemical Kinetics and Biomedicine, among others, offering interested researchers a convenient reference guide to these latest advances.

This volume contains the invited contributions to the Spring 2012 seminar series at Virginia State University on Mathematical Sciences and Applications. It is a thematic continuation of work presented in Volume 24 of the Springer Proceedings in Mathematics & Statistics series. Contributors present their own work as leading researchers to advance their specific fields and induce a genuine interdisciplinary interaction. Thus all articles therein are selective, self-contained, and are pedagogically exposed to foster student interest in science, technology, engineering and mathematics, stimulate graduate and undergraduate research, as well as collaboration between researchers from different areas. The volume features new advances in mathematical research and its applications: anti-periodicity; almost stochastic difference equations; absolute and conditional stability in delayed equations; gamma-convergence and applications to block copolymer morphology; the dynamics of collision and near-collision in celestial mechanics; almost and pseudo-almost limit cycles; rainbows in spheres and connections to ray, wave and potential scattering theory; null-controllability of the heat equation with constraints; optimal control for systems subjected to null-controllability; the Galerkin method for heat transfer in closed channels; wavelet transforms for real-time noise cancellation; signal, image processing and machine learning in medicine and biology; methodology for research on durability, reliability, damage tolerance of aerospace materials and structures at NASA Langley Research Center. The volume is suitable and valuable for mathematicians, scientists and research students in a variety of interdisciplinary fields, namely physical and life sciences, engineering and technology including structures and materials sciences, computer science for signal, image processing and machine learning in medicine.

There has been a steady advance of the atomic and molecular many-body methodology over the last few years, with a concomitant development of versatile computer codes. Understanding and interpretation of electronic structural features and the associated spectroscopic properties via many-body techniques are becoming competitive with those obtained with the traditional formalisms. Since the many-body techniques are not yet a part of the repertoire of the "black-box tools" of electronic structure and spectroscopy, it seems worthwhile to take stock now of the recent progress in certain selected areas. The present volume is more in the nature of proceedings of a "Paper Symposium," rather than of one which actually took place. We did organize in Calcutta, between December 10 and 12, 1990, a small meeting on Applied Many-Body Methods to Spectroscopy and Electronic Structure, jointly organized by the Indian Association for the Cultivation of Science and the S.N. Bose National Centre for Basic Sciences. Several leading practitioners were invited, among which some could not come for various reasons.

Electromagnetic wave theory is based on Maxwell's equations, and electromagnetic boundary-value problems must be solved to understand electromagnetic scattering, propagation, and radiation. Electromagnetic theory finds practical applications in wireless telecommunications and microwave engineering. This book is written as a text for a two-semester graduate course on electromagnetic wave theory. As such, Electromagnetic Wave Theory for Boundary-Value Problems is intended to help students enhance analytic skills by solving pertinent boundary-value problems. In particular, the techniques of Fourier transform, mode matching, and residue calculus are utilized to solve some canonical scattering and radiation problems.

In three volumes, a distinguished group of scholars from a variety of disciplines in the natural and social sciences, the humanities and the arts contribute essays in honor of Robert S. Cohen, on the occasion of his 70th birthday. The range of the essays, as well as their originality, and their critical and historical depth, pay tribute to the extraordinary scope of Professor Cohen's intellectual interests, as a scientist-philosopher and a humanist, and also to his engagement in the world of social and political practice. The essays presented in Physics, Philosophy, and the Scientific Community (Volume I of Essays in Honor of Robert S. Cohen) focus on philosophical and historical issues in contemporary physics: on the origins and conceptual foundations of quantum mechanics, on the reception and understanding of Bohr's and Einstein's work, on the emergence of quantum electrodynamics, and on some of the sharp philosophical and scientific issues that arise in current scientific practice (e.g. in superconductivity research). In addition, several essays deal with critical issues within the philosophy of science, both historical and contemporary: e.g. with Cartesian notions of mechanism in the philosophy of biology; with the language and logic of science - e.g. with new insights concerning the issue of a physicalistic' language in the arguments of Neurath, Carnap and Wittgenstein; with the notion of elementary logic'; and with rational and non-rational elements in the history of science. Two original contributions to the history of mathematics and some studies in the comparative sociology of science round off this outstanding collection.

The Handbook of Micrometeorology is the most up-to-date reference for micrometeorological issues and methods related to the eddy covariance technique for estimating mass and energy exchange between the terrestrial biosphere and the atmosphere. It is intended to provide micrometeorologists, ecosystem scientists, boundary-layer meteorologists, and students involved in micrometeorology with the state of science on measurement and analysis.

The IUTAM Symposium on Evolutionary Methods in Mechanics was held in Cracow, Poland, September 24-27, 2002. The site of the S- posium was Cracow University of Technology. The Symposium was - tended by 50 persons from 18 countries. In addition, several Polish students, Ph. D. students and research associates participated in the meeting. The Symposium provided an excellent opportunity for scholars of - chanics, computer sciences and arti?cial intelligence to interact and - change their points of view on the advanced computational and appli- tion aspects of the evolutionary methods in analysis and design of - chanical systems. Recently evolutionary methods have become the most e?ective tools for solving speci?c kinds of problems in mechanics, es- cially in structural and multidisciplinary optimization. The meeting was devotedtoboththeoreticalandpracticaldevelopmentsofcomputational mechanics methods drawing their inspiration from nature with part- ular emphasis on evolutionary models of computation such as genetic algorithms, evolutionary strategies, classi?er systems, evolutionary p- gramming and other evolutionary computation techniques in mechanics. The objective of the Symposium was to provide an international forum forfacilitatingtheexchangeofinformationamongresearchersinvolvedin computational intelligence methods based on evolutionary nature. The Symposium put special emphasis on evolutionary optimization in va- ous?eldsof mechanics. Thesubject of evolutionary optimization has- cently experienced a remarkable growth. New concepts, approaches and applicationsarebeing continually developed and exploited to provide- ?cient tools for solving a variety of optimization problems in mechanics.

Centered around the natural phenomena of relaxations and fluctuations, this monograph provides readers with a solid foundation in the linear and nonlinear Fokker-Planck equations that describe the evolution of distribution functions.

It emphasizes principles and notions of the theory (e.g. self-organization, stochastic feedback, free energy, and Markov processes), while also illustrating the wide applicability (e.g. collective behavior, multistability, front dynamics, and quantum particle distribution).

The focus is on relaxation processes in homogeneous many-body systems describable by nonlinear Fokker-Planck equations. Also treated are Langevin equations and correlation functions.

Since these phenomena are exhibited by a diverse spectrum of systems, examples and applications span the fields of physics, biology and neurophysics, mathematics, psychology, and biomechanics.

This volume features selected and peer-reviewed articles from the Pan-American Advanced Studies Institute (PASI). The chapters are written by international specialists who participated in the conference. Topics include developments based on breakthroughs in the mathematical understanding of phenomena describing systems in highly inhomogeneous and disordered media, including the KPZ universality class (describing the evolution of interfaces in two dimensions), random walks in random environment and percolative systems. PASI fosters a collaboration between North American and Latin American researchers and students. The conference that inspired this volume took place in January 2012 in both Santiago de Chile and Buenos Aires. Researchers and graduate students will find timely research in probability theory, statistical physics and related disciplines.

The proceedings of the conference is devoted mainly to the mathematically rigorous approaches to the problems of quantum mechanics. The spectral properties of Schroedinger operators, including those on regions with a boundary and their generalizations, scattering theory and resonances, time-dependent Hamiltonians and quantum chaos, problems of statistical physics like spin systems, and others are discussed.

Physics: Imagination and Reality introduces the reader to major ideas and the conceptual structure of modern physics, by tracing its development from the introduction of fields into physics by Faraday and Maxwell in the last century. Because the approach is historical, the book provides a comprehensive overview of the subjects. It should appeal to anyone interested in a basic understanding of the contemporary physicists view of the physical world. It avoids all but the simplest mathematics and presents ideas and concepts in everyday language.Physics: Imagination and Reality attempts to provide educated citizens with an understanding of contemporary physics and, at the same time, shows that its ideas have a grandeur, a challenge to the imagination and an aesthetic appeal which merit its recognition as an integral part of our culture.

Physics: Imagination and Reality introduces the reader to major ideas and the conceptual structure of modern physics, by tracing its development from the introduction of fields into physics by Faraday and Maxwell in the last century. Because the approach is historical, the book provides a comprehensive overview of the subjects. It should appeal to anyone interested in a basic understanding of the contemporary physicists view of the physical world. It avoids all but the simplest mathematics and presents ideas and concepts in everyday language.Physics: Imagination and Reality attempts to provide educated citizens with an understanding of contemporary physics and, at the same time, shows that its ideas have a grandeur, a challenge to the imagination and an aesthetic appeal which merit its recognition as an integral part of our culture.

This book is dedicated to the Soviet theoretician Yakov Ilich Frenkel (1894 - 1952), whose work in solid and liquid state physics is considered to be the golden foundation of twentieth century physics. Best known are the Frenkel pairs (defects), kinetic theory of liquids, theory of mobile dislocations (Frenkel- Kontorova solitons). Today, the electron theory of solids is inconceivable without excitons - the quasiparticles he introduced in 1930. Frenkel also contributed important concepts to classical electrodynamics (which now go under Feynmana s appellation "Frenkela s Fields") and to nuclear physics (the Bohr-Frenkel drop model). The book surveys the genesis and ramifications of Yakov Frenkela s scientific achievements. Special attention is paid to Frenkela s civic convictions, his fight against official Soviet philosophy for the acceptance and development of the theory of relativity and quantum mechanics in the Soviet Union of the 1920sa "1940s, a crucial thirty-year period in the history of Russian physics following the October Revolution. Much of the book is based on a wealth of archival documents, personal reminiscences and of Frenkela s letters. Thanks to his trenchant observations, a vivid picture emerges of scientists, universities and cultures in Europe, the United States and various cities of the Soviet Union. The book is richly illustrated by unique photos and copies of drawings and portraits from Frenkela s own hand.

The study of phase transitions is among the most fascinating fields in physics. Originally limited to transition phenomena in equilibrium systems, this field has outgrown its classical confines during the last two decades. The behavior of far from equilibrium systems has received more and more attention and has been an extremely active and productive subject of research for physicists, chemists and biologists. Their studies have brought about a more unified vision of the laws which govern self-organization processes of physico-chemical and biological sys tems. A major achievement has been the extension of the notion of phase transi tion to instabilities which occur only in open nonlinear systems. The notion of phase transition has been proven fruitful in apphcation to nonequilibrium ins- bihties known for about eight decades, like certain hydrodynamic instabilities, as well as in the case of the more recently discovered instabilities in quantum optical systems such as the laser, in chemical systems such as the Belousov-Zhabotinskii reaction and in biological systems. Even outside the realm of natural sciences, this notion is now used in economics and sociology. In this monograph we show that the notion of phase transition can be extend ed even further. It apphes also to a new class of transition phenomena which occur only in nonequilibrium systems subjected to a randomly fluctuating en vironment."

This book should be considered as an introduction to a special dass of hierarchical systems of optimal control, where subsystems are described by partial differential equations of various types. Optimization is carried out by means of a two-level scheme, where the center optimizes coordination for the upper level and subsystems find the optimal solutions for independent local problems. The main algorithm is a method of iterative aggregation. The coordinator solves the problern with macrovariables, whose number is less than the number of initial variables. This problern is often very simple. On the lower level, we have the usual optimal control problems of math ematical physics, which are far simpler than the initial statements. Thus, the decomposition (or reduction to problems ofless dimensions) is obtained. The algorithm constructs a sequence of so-called disaggregated solutions that are feasible for the main problern and converge to its optimal solutionunder certain assumptions ( e.g., under strict convexity of the input functions). Thus, we bridge the gap between two disciplines: optimization theory of large-scale systems and mathematical physics. The first motivation was a special model of branch planning, where the final product obeys a preset assortment relation. The ratio coefficient is maximized. Constraints are given in the form of linear inequalities with block diagonal structure of the part of a matrix that corresponds to subsystems. The central coordinator assem bles the final production from the components produced by the subsystems."

Physics has been applied to medical diagnosis for very nearly 400 years, and has now become an essential element of medical practice. This book concentrates on the theoretical basis of the physics which supports diagnostic techniques in modern clinical practice. Arising out of over a decade of teaching a course on medical physics to third year undergraduate students, the book has been structured so that individuals with a non-physics background, such as medical students or practitioners, can also benefit.

Based on several recent courses given to mathematical physics students, this volume is an introduction to bundle theory. It aims to provide newcomers to the field with solid foundations in topological K-theory. A fundamental theme, emphasized in the book, centers around the gluing of local bundle data related to bundles into a global object. One renewed motivation for studying this subject, comes from quantum field theory, where topological invariants play an important role.