Coherent states (CS) were originally introduced in 1926 by Schroedinger and rediscovered in the early 1960s in the context of laser physics. Since then, they have evolved into an extremely rich domain that pervades virtually every corner of physics, and have also given rise to a range of research topics in mathematics. The purpose of the 2016 CIRM conference was to bring together leading experts in the field with scientists interested in related topics, to jointly investigate their applications in physics, their various mathematical properties, and their generalizations in many directions. Instead of traditional proceedings, this book presents sixteen longer review-type contributions, which are the outcome of a collaborative effort by many conference participants, subsequently reviewed by independent experts. The book aptly illustrates the diversity of CS aspects, from purely mathematical topics to physical applications, including quantum gravity.

As is well known, The Great Divide (a.k.a. The Continental Divide) is formed by the Rocky Mountains stretching from north to south across North America. It creates a virtual "stone wall" so high that wind, rain, snow, etc. cannot cross it. This keeps the weather distinct on both sides. Since railroad trains cannot climb steep grades and tunnels through these mountains are almost formidable, the Canadian Pacific Railroad searched for a mountain pass providing the lowest grade for its tracks. Employees discovered a suitable mountain pass, called the Kicking Horse Pass, el. 5404 ft., near Banff, Alberta. (One can speculate as to the reason for the name.) This pass is also used by the Trans-Canada Highway. At the highest point of the pass the railroad tracks are horizontal with mountains rising on both sides. A mountain stream divides into two branches, one flowing into the Atlantic Ocean and the other into the Pacific. One can literally stand (as the author did) with one foot in the Atlantic Ocean and the other in the Pacific. The author has observed many mountain passes in the Rocky Mountains and Alps. What connections do mountain passes have with nonlinear partial dif ferential equations? To find out, read on ..."

The second edition of this textbook presents the basic mathematical knowledge and skills that are needed for courses on modern theoretical physics, such as those on quantum mechanics, classical and quantum field theory, and related areas. The authors stress that learning mathematical physics is not a passive process and include numerous detailed proofs, examples, and over 200 exercises, as well as hints linking mathematical concepts and results to the relevant physical concepts and theories. All of the material from the first edition has been updated, and five new chapters have been added on such topics as distributions, Hilbert space operators, and variational methods. The text is divided into three parts: - Part I: A brief introduction to (Schwartz) distribution theory. Elements from the theories of ultra distributions and (Fourier) hyperfunctions are given in addition to some deeper results for Schwartz distributions, thus providing a rather comprehensive introduction to the theory of generalized functions. Basic properties and methods for distributions are developed with applications to constant coefficient ODEs and PDEs. The relation between distributions and holomorphic functions is considered, as well as basic properties of Sobolev spaces. - Part II: Fundamental facts about Hilbert spaces. The basic theory of linear (bounded and unbounded) operators in Hilbert spaces and special classes of linear operators - compact, Hilbert-Schmidt, trace class, and Schroedinger operators, as needed in quantum physics and quantum information theory - are explored. This section also contains a detailed spectral analysis of all major classes of linear operators, including completeness of generalized eigenfunctions, as well as of (completely) positive mappings, in particular quantum operations. - Part III: Direct methods of the calculus of variations and their applications to boundary- and eigenvalue-problems for linear and nonlinear partial differential operators. The authors conclude with a discussion of the Hohenberg-Kohn variational principle. The appendices contain proofs of more general and deeper results, including completions, basic facts about metrizable Hausdorff locally convex topological vector spaces, Baire's fundamental results and their main consequences, and bilinear functionals. Mathematical Methods in Physics is aimed at a broad community of graduate students in mathematics, mathematical physics, quantum information theory, physics and engineering, as well as researchers in these disciplines. Expanded content and relevant updates will make this new edition a valuable resource for those working in these disciplines.

The 3rd International ISAAC Congress took place from August 20 to 25, 2001 in Berlin, Germany, supported by the German Research Foundation (DFG), the city of Berlin through Investitionsbank Berlin and the Freie Universitiit Berlin. 10 ISAAC Awards were presented to young researchers in analysis its applications and computation from all over the world on the basis of financial support from Siemens, Daimler Crysler, Motorola and the Berlin Mathematical Society and book gifts from Birkhauser Verlag, Elsevier, Kluwer Academic Publisher, Springer Verlag and World Scientific. The ISAAC is grateful to all these institutions, firms and publishers for their support. Due to the support from DFG and from Investitions bank Berlin many of the 362 registrated participants could be financially supported. Unfortunately the financial supports were granted too late to reach more people from former SU as the procedere for visa is still more than cumbersome and embassies are not at all flexible. Hence, a big part of the financial support could not be used and had to be returned. The 10 plenary lectures were 1. Antoniou, 1. Prigogine (Intern. Solvay Inst. Phys. Chem., Brussels): Irreversibility and the probabilistic description of unstable evolutions beyond the Hilbert space framework (read by 1. Antoniou), N.S. Bakhvalov, M.E. Eglit (Math. Mech. Dept., Lomonosov State Univ."

The book presents a systematic and compact treatment of the qualitative theory of half-linear
differential equations. It contains the most updated and comprehensive material and represents the first attempt to present the results of the rapidly developing theory of half-linear differential equations in a unified form. The main topics covered by the book are oscillation and asymptotic theory and the theory of boundary value problems associated with half-linear equations, but the book also contains a treatment of related topics like PDE s with p-Laplacian, half-linear difference equations and various more general nonlinear differential equations.
- The first complete treatment of the qualitative theory of half-linear differential equations.
- Comparison of linear and half-linear theory.
- Systematic approach to half-linear oscillation and asymptotic theory.
- Comprehensive bibliography and index.
- Useful as a reference book in the topic.

The monograph is devoted to the study of functional equations with the transformed argument on the real line and on the unit circle. Such equations systematically arise in dynamical systems, differential equations, probabilities, singularities of smooth mappings, and other areas. The purpose of the book is to present modern methods and new results in the subject, with an emphasis on a connection between local and global solvability. The general concepts developed in the book are applicable to multidimensional functional equations. Some of the methods are presented for the first time in the monograph literature.
The book is addressed to graduates and researchers interested in dynamical systems, differential equations, operator theory, or the theory of functions and their applications.

The topics of this set of student-oriented books are presented in a discursive style that is readable and easy to follow. Numerous clearly stated, completely worked out examples together with carefully selected problem sets with answers are used to enhance students' understanding and manipulative skill. The goal is to help students feel comfortable and confident in using advanced mathematical tools in junior, senior, and beginning graduate courses.

This work is a continuation of the first volume published by Springer in 2011, entitled "A Cp-Theory Problem Book: Topological and Function Spaces." The first volume provided an introduction from scratch to Cp-theory and general topology, preparing the reader for a professional understanding of Cp-theory in the last section of its main text.This present volume covers a wide variety of topics in Cp-theory and general topology at the professional level bringing the reader to the frontiers of modern research. The volume contains 500 problems and exercises with complete solutions. It can also be used as an introduction to advanced set theory and descriptive set theory. The book presents diverse topics of the theory of function spaces with the topology of pointwise convergence, or Cp-theory which exists at the intersection of topological algebra, functional analysis and general topology. Cp-theory has an important role in the classification and unification of heterogeneous results from these areas of research. Moreover, this book gives a reasonably complete coverage of Cp-theory through 500 carefully selected problems and exercises. By systematically introducing each of the major topics of Cp-theory the book is intended to bring a dedicated reader from basic topological principles to the frontiers of modern research."

Graduate students in mathematics, who want to travel light, will find this book invaluable; impatient young researchers in other fields will enjoy it as an instant reference to the highlights of modern analysis. Starting with general topology, it moves on to normed and seminormed linear spaces. From there it gives an introduction to the general theory of operators on Hilbert space, followed by a detailed exposition of the various forms the spectral theorem may take; from Gelfand theory, via spectral measures, to maximal commutative von Neumann algebras. The book concludes with two supplementary chapters: a concise account of unbounded operators and their spectral theory, and a complete course in measure and integration theory from an advanced point of view.

The Virasoro algebra is an infinite dimensional Lie algebra that plays an increasingly important role in mathematics and theoretical physics. This book describes some fundamental facts about the representation theory of the Virasoro algebra in a self-contained manner. Topics include the structure of Verma modules and Fock modules, the classification of (unitarizable) Harish-Chandra modules, tilting equivalence, and the rational vertex operator algebras associated to the so-called minimal series representations.

Covering a wide range of material, this book has three appendices which provide background information required for some of the chapters. The authors organize fundamental results in a unified way and refine existing proofs. For instance in chapter three, a generalization of Jantzen filtration is reformulated in an algebraic manner, and geometric interpretation is provided. Statements, widely believed to be true, are collated, and results which are known but not verified are proven, such as the corrected structure theorem of Fock modules in chapter eight.

This book will be of interest to a wide range of mathematicians and physicists from the level of graduate students to researchers.

Evolution equations of hyperbolic or more general p-evolution type form an active field of current research. This volume aims to collect some recent advances in the area in order to allow a quick overview of ongoing research. The contributors are first rate mathematicians. This collection of research papers is centred around parametrix constructions and microlocal analysis; asymptotic constructions of solutions; energy and dispersive estimates; and associated spectral transforms. Applications concerning elasticity and general relativity complement the volume. The book gives an overview of a variety of ongoing current research in the field and, therefore, allows researchers as well as students to grasp new aspects and broaden their understanding of the area. "

In inverse problems, the aim is to obtain, via a mathematical model, information on quantities that are not directly observable but rather depend on other observable quantities. Inverse problems are encountered in such diverse areas of application as medical imaging, remote sensing, material testing, geosciences and financing. It has become evident that new ideas coming from differential geometry and modern analysis are needed to tackle even some of the most classical inverse problems. This book contains a collection of presentations, written by leading specialists, aiming to give the reader up-to-date tools for understanding the current developments in the field.

This volume collects the edited and reviewed contribution presented in the 9th iTi Conference that took place virtually, covering fundamental and applied aspects in turbulence. In the spirit of the iTi conference, the volume is produced after the conference so that the authors had the opportunity to incorporate comments and discussions raised during the meeting. In the present book, the contributions have been structured according to the topics: I Experiments II Simulations and Modelling III Data Processing and Scaling IV Theory V Miscellaneous topics

This monograph explores nonoscillation and existence of positive solutions for functional differential equations and describes their applications to maximum principles, boundary value problems and stability of these equations. In view of this objective the volume considers a wide class of equations including, scalar equations and systems of different types, equations with variable types of delays and equations with variable deviations of the argument. Each chapter includes an introduction and preliminaries, thus making it complete. Appendices at the end of the book cover reference material. Nonoscillation Theory of Functional Differential Equations with Applications is addressed to a wide audience of researchers in mathematics and practitioners.

The theory of set-valued maps and of differential inclusion is developed in recent years both as a field of his own and as an approach to control theory. The book deals with the theory of semilinear differential inclusions in infinite dimensional spaces. In this setting, problems of interest to applications do not suppose neither convexity of the map or compactness of the multi-operators. These assumption implies the development of the theory of measure of noncompactness and the construction of a degree theory for condensing mapping. Of particular interest is the approach to the case when the linear part is a generator of a condensing, strongly continuous semigroup. In this context, the existence of solutions for the Cauchy and periodic problems are proved as well as the topological properties of the solution sets. Examples of applications to the control of transmission line and to hybrid systems are presented.

This volume constitutes the proceedings of a workshop whose main purpose was to exchange information on current topics in complex analysis, differential geometry, mathematical physics and applications, and to group aspects of new mathematics.

In August 1995 an international symposium on "Quasiconformal Mappings and Analysis" was held in Ann Arbor on the occasion of Professor Fred- erick W. Gehring's 70th birthday and his impending retirement from the Mathematics Department at the University of Michigan. The concept of the symposium was to feature broad survey talks on a wide array of topics related to Gehring's basic research contributions in the field of quasicon- formal mappings, emphasizing their relations to other parts of analysis. Principal speakers were Kari Astala, Albert Baernstein, Clifford Earle, Pe- ter Jones, Irwin Kra, OUi Lehto, Gaven Martin, Dennis Sullivan, and Jussi Vaisala. Financial support was provided by the National Science Founda- tion, with additional grants from the University of Michigan and from the Institute for Mathematics and its Applications. The symposium was a great success. The speakers rose to the occasion and presented excellent survey lectures. The present volume was conceived as a means for disseminating those expositions to a wider audience. Ad- ditional mathematicians, some of whom had not been able to attend the symposium, were invited to contribute similar articles. The result is a fit- ting tribute to Fred Gehring's pre-eminent role in developing the theory of quasiconformal mappings, through his own research and writings and lec- tures, and through his supervision of graduate students. The volume begins with descriptions of Gehring's mathematical career and an overview of his research achievements.

This book illustrates the wide range of research subjects developed by the Italian research group in harmonic analysis, originally started by Alessandro Figa-Talamanca, to whom it is dedicated in the occasion of his retirement. In particular, it outlines some of the impressive ramifications of the mathematical developments that began when Figa-Talamanca brought the study of harmonic analysis to Italy; the research group that he nurtured has now expanded to cover many areas. Therefore the book is addressed not only to experts in harmonic analysis, summability of Fourier series and singular integrals, but also in potential theory, symmetric spaces, analysis and partial differential equations on Riemannian manifolds, analysis on graphs, trees, buildings and discrete groups, Lie groups and Lie algebras, and even in far-reaching applications as for instance cellular automata and signal processing (low-discrepancy sampling, Gaussian noise).

The analysis of singular perturbed di?erential equations began early in the twentieth century, when approximate solutions were constructed from asy- totic expansions. (Preliminary attempts appear in the nineteenth century - see[vD94].)Thistechniquehas?ourishedsincethemid-1960sanditsprincipal ideas and methods are described in several textbooks; nevertheless, asy- totic expansions may be impossible to construct or may fail to simplify the given problem and then numerical approximations are often the only option. Thesystematicstudyofnumericalmethodsforsingularperturbationpr- lems started somewhat later - in the 1970s. From this time onwards the - search frontier has steadily expanded, but the exposition of new developments in the analysis of these numerical methods has not received its due attention. The ?rst textbook that concentrated on this analysis was [DMS80], which collected various results for ordinary di?erential equations. But after 1980 no further textbook appeared until 1996, when three books were published: Miller et al. [MOS96], which specializes in upwind ?nite di?erence methods on Shishkin meshes, Morton's book [Mor96], which is a general introduction to numerical methods for convection-di? usion problems with an emphasis on the cell-vertex ?nite volume method, and [RST96], the ?rst edition of the present book. Nevertheless many methods and techniques that are important today, especially for partial di?erential equations, were developed after 1996.

This IMA Volume in Mathematics and its Applications ESSAYS ON MATHEMATICAL ROBOTICS is based on the proceedings of a workshop that was an integral part of the 1992-93 IMA program on "Control Theory." The workshop featured a mathematicalintroductionto kinematics and fine motion planning; dynam- ics and control of kinematically redundant robot arms including snake-like robots, multi-fingered robotic hands; methods of non-holonomic motion planning for space robots, multifingered robot hands and mobile robots; new techniques in analytical mechanics for writing the dynamics of com- plicated multi-body systems subject to constraints on angular momentum or other non-holonomic constraints. In addition to papers representing proceedings of the Workshop, this volume contains several longer papers surveying developments of the intervening years. We thank John Baillieul, Shankar S. Sastry, and Hector J. Sussmann for organizing the workshop and editing the proceedings. We also take this opportunity to thank the National Science Foundation and the Army Research Office, whose financial support made the workshop possible. Avner Friedman Willard Miller, Jr.

Many results, both from semi group theory itself and from the applied sciences, are phrased in discipline-specific languages and hence are hardly known to a broader community. This volume contains a selection of lectures presented at a conference that was organised as a forum for all mathematicians using semi group theory to learn what is happening outside their own field of research. The collection will help to establish a number of new links between various sub-disciplines of semigroup theory, stochastic processes, differential equations and the applied fields. The theory of semigroups of operators is a well-developed branch of functional analysis. Its foundations were laid at the beginning of the 20th century, while the fundamental generation theorem of Hille and Yosida dates back to the forties. The theory was, from the very beginning, designed as a universal language for partial differential equations and stochastic processes, but at the same time it started to live as an independent branch of operator theory. Nowadays, it still has the same distinctive flavour: it develops rapidly by posing new 'internal' questions and in answering them, discovering new methods that can be used in applications. On the other hand, it is influenced by questions from PDEs and stochastic processes as well as from applied sciences such as mathematical biology and optimal control, and thus it continually gathers a new momentum. Researchers and postgraduate students working in operator theory, partial differential equations, probability and stochastic processes, analytical methods in biology and other natural sciences, optimization and optimal control will find this volume useful.

Fifteen years ago, most mathematicians who worked in the intersection of function theory and operator theory thought that progress on the Bergman spaces was unlikely, yet today the situation has completely changed. For several years, research interest and activity have expanded in this area and there are now rich theories describing the Bergman spaces and their operators. This book is a timely treatment of the theory, written by three of the major players in the field.

This book attempts to place the basic ideas of real analysis and numerical analysis together in an applied setting that is both accessible and motivational to young students. The essentials of real analysis are presented in the context of a fundamental problem of applied mathematics, which is to approximate the solution of a physical model. The framework of existence, uniqueness, and methods to approximate solutions of model equations is sufficiently broad to introduce and motivate all the basic ideas of real analysis. The book includes background and review material, numerous examples, visualizations and alternate explanations of some key ideas, and a variety of exercises ranging from simple computations to analysis and estimates to computations on a computer. The book can be used in an honor calculus sequence typically taken by freshmen planning to major in engineering, mathematics, and science, or in an introductory course in rigorous real analysis offered to mathematics majors. Donald Estep is Professor of Mathematics at Colorado State University. He is the author of Computational Differential Equations, with K. Eriksson, P. Hansbo and C. Johnson (Cambridge University Press 1996) and Error of Numerical Solutions of Systems of Nonlinear Reaction-Diffusion Equations with M. Larson and R. Williams (A.M.S. 2000), and recently co-edited Collected Lectures on the Preservation of Stability under Discretization, with Simon Tavener (S.I.A.M., 2002), as well as numerous research articles. His research interests include computational error estimation and adaptive finite element methods, numerical solution of evolutionary problems, and computational investigation of physical models.

This book provides a comprehensive introduction to the theory of ordinary differential equations with a focus on mechanics and dynamical systems as important applications of the theory. The text is written to be used in the traditional way or in a more applied way. In addition to its use in a traditional one or two semester graduate course in mathematics, the book is organized to be used for interdisciplinary courses in applied mathematics, physics, and engineering.