Using examples from finance and modern warfare to the flocking of birds and the swarming of bacteria, the collected research in this volume demonstrates the common methodological approaches and tools for modeling and simulating collective behavior.

Thetopics presented point toward new and challenging frontiers of applied mathematics, making the volume a useful referencetext forapplied mathematicians, physicists, biologists, and economists involved in the modeling of socio-economic systems."

0. 1. The Scope of the Paper. This article is mainly devoted to the oper ators indicated in the title. More specifically, we consider elliptic differential and pseudodifferential operators with infinitely smooth symbols on infinitely smooth closed manifolds, i. e. compact manifolds without boundary. We also touch upon some variants of the theory of elliptic operators in Rn. A separate article (Agranovich 1993) will be devoted to elliptic boundary problems for elliptic partial differential equations and systems. We now list the main topics discussed in the article. First of all, we ex pound theorems on Fredholm property of elliptic operators, on smoothness of solutions of elliptic equations, and, in the case of ellipticity with a parame ter, on their unique solvability. A parametrix for an elliptic operator A (and A-). . J) is constructed by means of the calculus of pseudodifferential also for operators in Rn, which is first outlined in a simple case with uniform in x estimates of the symbols. As functional spaces we mainly use Sobolev - 2 spaces. We consider functions of elliptic operators and in more detail some simple functions and the properties of their kernels. This forms a foundation to discuss spectral properties of elliptic operators which we try to do in maxi mal generality, i. e., in general, without assuming selfadjointness. This requires presenting some notions and theorems of the theory of nonselfadjoint linear operators in abstract Hilbert space."

Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates. The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early results. He shows the fascination of the difficult Hardy-Littlewood theorems and of an unexpected simple proof, and extolls Wiener's breakthrough based on Fourier theory. There are the spectacular "high-indices" theorems and Karamata's "regular variation," which permeates probability theory. The author presents Gelfand's elegant algebraic treatment of Wiener theory and his own distributional approach. There is also a new unified theory for Borel and "circle" methods. The text describes many Tauberian ways to the prime number theorem. A large bibliography and a substantial index round out the book.

Maximum Principles are central to the theory and applications of second-order partial differential equations and systems. This self-contained text establishes the fundamental principles and provides a variety of applications.

Fundamentals of Convex Analysis offers an in-depth look at some of the fundamental themes covered within an area of mathematical analysis called convex analysis. In particular, it explores the topics of duality, separation, representation, and resolution. The work is intended for students of economics, management science, engineering, and mathematics who need exposure to the mathematical foundations of matrix games, optimization, and general equilibrium analysis. It is written at the advanced undergraduate to beginning graduate level and the only formal preparation required is some familiarity with set operations and with linear algebra and matrix theory. Fundamentals of Convex Analysis is self-contained in that a brief review of the essentials of these tool areas is provided in Chapter 1. Chapter exercises are also provided. Topics covered include: convex sets and their properties; separation and support theorems; theorems of the alternative; convex cones; dual homogeneous systems; basic solutions and complementary slackness; extreme points and directions; resolution and representation of polyhedra; simplicial topology; and fixed point theorems, among others. A strength of this work is how these topics are developed in a fully integrated fashion.

Hans Duistermaat, an influential geometer-analyst, made substantial contributions to the theory of ordinary and partial differential equations, symplectic, differential, and algebraic geometry, minimal surfaces, semisimple Lie groups, mechanics, mathematical physics, and related fields. Written in his honor, the invited and refereed articles in this volume contain important new results as well as surveys in some of these areas, clearly demonstrating the impact of Duistermaat's research and, in addition, exhibiting interrelationships among many of the topics.

This text contains a series of self-contained reviews on the state of the art in different areas of partial differential equations, presented by French mathematicians. Topics include qualitative properties of reaction-diffusion equations, multiscale methods coupling atomistic and continuum mechanics, adaptive semi-Lagrangian schemes for the Vlasov-Poisson equation, and coupling of scalar conservation laws.

This volume contains several surveys focused on the ideas of approximate solutions, well-posedness and stability of problems in scalar and vector optimization, game theory and calculus of variations. These concepts are of particular interest in many fields of mathematics. The idea of stability goes back at least to J. Hadamard who introduced it in the setting of differential equations; the concept of well-posedness for minimum problems is more recent (the mid-sixties) and originates with A.N. Tykhonov. It turns out that there are connections between the two properties in the sense that a well-posed problem which, at least in principle, is "easy to solve," has a solution set that does not vary too much under perturbation of the data of the problem, i.e. it is "stable." These themes have been studied in depth for minimum problems and now we have a general picture of the related phenomena in this case. But, of course, the same concepts can be studied in other more complicated situations as, e.g. vector optimization, game theory and variational inequalities. Let us mention that in several of these new areas there is not even a unique idea of what should be called approximate solution, and the latter is at the basis of the definition of well posed problem."

Introduction to Large Truncated Toeplitz Matrices is a text on the application of functional analysis and operator theory to some concrete asymptotic problems of linear algebra. The book contains results on the stability of projection methods, deals with asymptotic inverses and Moore-Penrose inversion of large Toeplitz matrices, and embarks on the asymptotic behavoir of the norms of inverses, the pseudospectra, the singular values, and the eigenvalues of large Toeplitz matrices. The approach is heavily based on Banach algebra techniques and nicely demonstrates the usefulness of C*-algebras and local principles in numerical analysis. The book includes classical topics as well as results obtained and methods developed only in the last few years. Though employing modern tools, the exposition is elementary and aims at pointing out the mathematical background behind some interesting phenomena one encounters when working with large Toeplitz matrices. The text is accessible to readers with basic knowledge in functional analysis. It is addressed to graduate students, teachers, and researchers with some inclination to concrete operator theory and should be of interest to everyone who has to deal with infinite matrices (Toeplitz or not) and their large truncations.

By discussing topics such as shape representations, relaxation theory and optimal transport, trends and synergies of mathematical tools required for optimization of geometry and topology of shapes are explored. Furthermore, applications in science and engineering, including economics, social sciences, biology, physics and image processing are covered. Contents Part I Geometric issues in PDE problems related to the infinity Laplace operator Solution of free boundary problems in the presence of geometric uncertainties Distributed and boundary control problems for the semidiscrete Cahn-Hilliard/Navier-Stokes system with nonsmooth Ginzburg-Landau energies High-order topological expansions for Helmholtz problems in 2D On a new phase field model for the approximation of interfacial energies of multiphase systems Optimization of eigenvalues and eigenmodes by using the adjoint method Discrete varifolds and surface approximation Part II Weak Monge-Ampere solutions of the semi-discrete optimal transportation problem Optimal transportation theory with repulsive costs Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations On the Lagrangian branched transport model and the equivalence with its Eulerian formulation On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows Pressureless Euler equations with maximal density constraint: a time-splitting scheme Convergence of a fully discrete variational scheme for a thin-film equatio Interpretation of finite volume discretization schemes for the Fokker-Planck equation as gradient flows for the discrete Wasserstein distance

This EMS volume contains a survey of the principles and advanced techniques of the spectral theory of linear differential and pseudodifferential operators in finite-dimensional spaces. Also including a special section of Sunada's recent solution of Kac's celebrated problem of whether or not "one can hear the shape of a drum."

It was noted in the preface of the book "Inequalities Involving Functions and Their Integrals and Derivatives," Kluwer Academic Publishers, 1991, by D.S. Mitrinovic, J.E. Pecaric and A.M. Fink; since the writing of the classical book by Hardy, Littlewood and Polya (1934), the subject of differential and integral inequalities has grown by about 800%. Ten years on, we can confidently assert that this growth will increase even more significantly. Twenty pages of Chapter XV in the above mentioned book are devoted to integral inequalities involving functions with bounded derivatives, or, Ostrowski type inequalities. This is now itself a special domain of the Theory of Inequalities with many powerful results and a large number of applications in Numerical Integration, Probability Theory and Statistics, Information Theory and Integral Operator Theory. The main aim of the present book, jointly written by the members of the Vic toria University node of RGMIA (Research Group in Mathematical Inequali ties and Applications, http: I /rgmia. vu. edu. au) and Th. M. Rassias, is to present a selected number of results on Ostrowski type inequalities. Results for univariate and multivariate real functions and their natural applications in the error analysis of numerical quadrature for both simple and multiple integrals as well as for the Riemann-Stieltjes integral are given."

The objective of this book is to present an introduction to the ideas, phenomena, and methods of partial differential equations. This material can be presented in one semester and requires no previous knowledge of differential equations, but assumes the reader to be familiar with advanced calculus, real analysis, the rudiments of complex analysis, and thelanguage of functional analysis. Topics discussed in the text include elliptic, hyperbolic, and parabolic equations, the energy method, maximum principle, and the Fourier Transform. The text features many historical and scientific motivations and applications. Included throughout are exercises, hints, and discussions which form an important and integral part of the course.

This volume collects contributions from the speakers at an INdAM Intensive period held at the University of Bari in 2017. The contributions cover several aspects of partial differential equations whose development in recent years has experienced major breakthroughs in terms of both theory and applications. The topics covered include nonlocal equations, elliptic equations and systems, fully nonlinear equations, nonlinear parabolic equations, overdetermined boundary value problems, maximum principles, geometric analysis, control theory, mean field games, and bio-mathematics. The authors are trailblazers in these topics and present their work in a way that is exhaustive and clearly accessible to PhD students and early career researcher. As such, the book offers an excellent introduction to a variety of fundamental topics of contemporary investigation and inspires novel and high-quality research.

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

In recent years many researchers in material science have focused their attention on the study of composite materials, equilibrium of crystals and crack distribution in continua subject to loads. At the same time several new issues in computer vision and image processing have been studied in depth. The understanding of many of these problems has made significant progress thanks to new methods developed in calculus of variations, geometric measure theory and partial differential equations. In particular, new technical tools have been introduced and successfully applied. For example, in order to describe the geometrical complexity of unknown patterns, a new class of problems in calculus of variations has been introduced together with a suitable functional setting: the free-discontinuity problems and the special BV and BH functions. The conference held at Villa Olmo on Lake Como in September 1994 spawned successful discussion of these topics among mathematicians, experts in computer science and material scientists.

This work is a revised and enlarged edition of a book with the same title published in Romanian by the Publishing House of the Romanian Academy in 1989. It grew out of lecture notes for a graduate course given by the author at the University if Ia i and was initially intended for students and readers primarily interested in applications of optimal control of ordinary differential equations. In this vision the book had to contain an elementary description of the Pontryagin maximum principle and a large number of examples and applications from various fields of science. The evolution of control science in the last decades has shown that its meth ods and tools are drawn from a large spectrum of mathematical results which go beyond the classical theory of ordinary differential equations and real analy ses. Mathematical areas such as functional analysis, topology, partial differential equations and infinite dimensional dynamical systems, geometry, played and will continue to play an increasing role in the development of the control sciences. On the other hand, control problems is a rich source of deep mathematical problems. Any presentation of control theory which for the sake of accessibility ignores these facts is incomplete and unable to attain its goals. This is the reason we considered necessary to widen the initial perspective of the book and to include a rigorous mathematical treatment of optimal control theory of processes governed by ordi nary differential equations and some typical problems from theory of distributed parameter systems."

In this book, ring-theoretical properties of skew Laurent series rings A((x; )) over a ring A, where A is an associative ring with non-zero identity element are described. In addition, we consider Laurent rings and Malcev-Neumann rings, which are proper extensions of skew Laurent series rings.

The book consists of solicited articles from a select group of mathematicians and physicists working at the interface between positivity and the geometry, combinatorics or analysis of polynomials of one or several variables. It is dedicated to the memory of Julius Borcea (1968-2009), a distinguished mathematician, Professor at the University of Stockholm. With his extremely original contributions and broad vision, his impact on the topics of the planned volume cannot be underestimated. All contributors knew or have exchanged ideas with Dr. Borcea, and their articles reflect, at least partially, his heritage.

Presents a discrete in time-space universal map of relative dynamics that is used to unfold an extensive catalogue of dynamic events not previously discussed in mathematical or social science literature. With emphasis on the chaotic dynamics that may ensue, the book describes the evolution on the basis of temporal and locational advantages. It explains nonlinear discrete time dynamic maps primarily through numerical simulations. These very rich qualitative dynamics are linked to evolution processes in socio-spatial systems. Important features include: The analytical properties of the one-stock, two- and three-location map; the numerical results from the one- and two-stock, two- and three-location dynamics; and the demonstration of the map's potential applicability in the social sciences through simulating population dynamics of the U.S. Regions over a two-century period. In addition, this book includes new findings: the Hopf equivalent discrete time dynamics bifurcation; the Feigenbaum slope-sequences; the presence of strange local attractors and containers; switching of extreme states; the presence of different types of turbulence; local and global turbulence. Intended for researchers and advanced graduate students in applied mathematics and an interest in dynamics and chaos. Mathematical social scientists in many other fields will also find this book useful.

Asymptotic methods belong to the, perhaps, most romantic area of modern mathematics. They are widely known and have been used in me chanics, physics and other exact sciences for many, many decades. But more than this, asymptotic ideas are found in all branches of human knowledge, indeed in all areas of life. In this broader context they have not and perhaps cannot be fully formalized. However, they are mar velous, they leave room for fantasy, guesses and intuition; they bring us very near to the border of the realm of art. Many books have been written and published about asymptotic meth ods. Most of them presume a mathematically sophisticated reader. The authors here attempt to describe asymptotic methods on a more accessi ble level, hoping to address a wider range of readers. They have avoided the extreme of banishing formulae entirely, as done in some popular science books that attempt to describe mathematical methods with no mathematics. This is impossible (and not wise). Rather, the authors have tried to keep the mathematics at a moderate level. At the same time, using simple examples, they think they have been able to illustrate all the key ideas of asymptotic methods and approaches, to depict in de tail the results of their application to various branches of knowledg- from astronomy, mechanics, and physics to biology, psychology and art. The book is supplemented by several appendices, one of which con tains the profound ideas of R. G."

Ship optimization design is critical to the preliminary design of a ship. With the rapid development of computer technology, the simulation-based design (SBD) technique has been introduced into the field of ship design. Typical SBD consists of three parts: geometric reconstruction; CFD numerical simulation; and optimization. In the context of ship design, these are used to alter the shape of the ship, evaluate the objective function and to assess the hull form space respectively. As such, the SBD technique opens up new opportunities and paves the way for a new method for optimal ship design. This book discusses the problem of optimizing ship's hulls, highlighting the key technologies of ship optimization design and presenting a series of hull-form optimization platforms. It includes several improved approaches and novel ideas with significant potential in this field

In this book the details of many calculations are provided for access to work in quantum groups, algebraic differential calculus, noncommutative geometry, fuzzy physics, discrete geometry, gauge theory, quantum integrable systems, braiding, finite topological spaces, some aspects of geometry and quantum mechanics and gravity.

This book presents complex analysis in one variable in the context of modern mathematics, with clear connections to several complex variables, de Rham theory, real analysis, and other branches of mathematics. Thus, covering spaces are used explicitly in dealing with Cauchy's theorem, real variable methods are illustrated in the Loman-Menchoff theorem and in the corona theorem, and the algebraic structure of the ring of holomorphic functions is studied.
Using the unique position of complex analysis, a field drawing on many disciplines, the book also illustrates powerful mathematical ideas and tools, and requires minimal background material. Cohomological methods are introduced, both in connection with the existence of primitives and in the study of meromorphic functionas on a compact Riemann surface. The proof of Picard's theorem given here illustrates the strong restrictions on holomorphic mappings imposed by curvature conditions.

This work fills an important gap in the literature by providing an important link between MAPLE and its successful use in solving problems in Operations Research (OR). The symbolic, numerical, and graphical aspects of MAPLE make this software package an ideal tool for treating certain OR problems and providing descriptive and optimization-based analyses of deterministic and stochastic models. Detailed is MAPLE's treatment of some of the mathematical techniques used in OR modeling: e.g., algebra and calculus, ordinary and partial differential equations, linear algebra, transform methods, and probability theory.

A number of examples of OR techniques and applications are presented, such as linear and nonlinear programming, dynamic programming, stochastic processes, inventory models, queueing systems, and simulation. Throughout the text MAPLE statements used in the solutions of problems are clearly explained. At the same time, technical background material is presented in a rigorous mathematical manner to reach the OR novice and professional. Numerous end-of- chapter exercises, a good bibliography and overall index at the end of the book are also included, as well as MAPLE worksheets that are easily downloadable from the author's website at www.business.mcmaster.ca/msis/profs/parlar, or from the Birkhauser website at www.birkhauser.com/cgi-win/ISBN/0-8176-4165-3.

The book is intended for advanced undergraduate and graduate students in operations research, management science departments of business schools, industrial and systems engineering, economics, and mathematics. As a self-study resource, the text can be used by researchers and practitioners who want a quick overview ofMAPLE's usefulness in solving realistic OR problems that would be difficult or impossible to solve with other software packages.