This Festschrift contains five research surveys and thirty-four shorter contributions by participants of the conference ''Stochastic Partial Differential Equations and Related Fields'' hosted by the Faculty of Mathematics at Bielefeld University, October 10-14, 2016. The conference, attended by more than 140 participants, including PostDocs and PhD students, was held both to honor Michael Roeckner's contributions to the field on the occasion of his 60th birthday and to bring together leading scientists and young researchers to present the current state of the art and promising future developments. Each article introduces a well-described field related to Stochastic Partial Differential Equations and Stochastic Analysis in general. In particular, the longer surveys focus on Dirichlet forms and Potential theory, the analysis of Kolmogorov operators, Fokker-Planck equations in Hilbert spaces, the theory of variational solutions to stochastic partial differential equations, singular stochastic partial differential equations and their applications in mathematical physics, as well as on the theory of regularity structures and paracontrolled distributions. The numerous research surveys make the volume especially useful for graduate students and researchers who wish to start work in the above-mentioned areas, or who want to be informed about the current state of the art.

This book introduces and analyzes the multigrid approach for the numerical solution of large sparse linear systems arising from the discretization of elliptic partial differential equations. Special attention is given to the powerful matrix-based-multigrid approach, which is particularly useful for problems with variable coefficients and nonsymmetric and indefinite problems. This approach applies not only to model problems on rectangular grids but also to more realistic applications with complicated grids and domains and discontinuous coefficients. Matrix-Based Multigrid can be used as a textbook in courses in numerical analysis, numerical linear algebra, and numerical PDEs at the advanced undergraduate and graduate levels in computer science, math, and applied math departments. The theory is written in simple algebraic terms and therefore requires preliminary knowledge in basic linear algebra and calculus only. Because it is self contained and includes useful exercises, the book is also suitable for self study by research students, researchers, engineers, and others interested in the numerical solution of partial differential equations.

This book collects recent research papers by respected specialists in the field. It presents advances in the field of geometric properties for parabolic and elliptic partial differential equations, an area that has always attracted great attention. It settles the basic issues (existence, uniqueness, stability and regularity of solutions of initial/boundary value problems) before focusing on the topological and/or geometric aspects. These topics interact with many other areas of research and rely on a wide range of mathematical tools and techniques, both analytic and geometric. The Italian and Japanese mathematical schools have a long history of research on PDEs and have numerous active groups collaborating in the study of the geometric properties of their solutions.

This comprehensive book presents a rigorous and state-of-the-art treatment of variational inequalities and complementarity problems in finite dimensions. This class of mathematical programming problems provides a powerful framework for the unified analysis and development of efficient solution algorithms for a wide range of equilibrium problems in economics, engineering, finance, and applied sciences. New research material and recent results, not otherwise easily accessible, are presented in a self-contained and consistent manner. The book is published in two volumes, with the first volume concentrating on the basic theory and the second on iterative algorithms. Both volumes contain abundant exercises and feature extensive bibliographies. Written with a wide range of readers in mind, including graduate students and researchers in applied mathematics, optimization, and operations research as well as computational economists and engineers, this book will be an enduring reference on the subject and provide the foundation for its sustained growth.

Pedagogical insights gained through 30 years of teaching applied mathematics led the author to write this set of student-oriented books. Topics such as complex analysis, matrix theory, vector and tensor analysis, Fourier analysis, integral transforms, ordinary and partial differential equations 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.

The Complex Variable Boundary Element Method (CVBEM) has an important role to play in a number of technical engineering situations and can be a tremendous help to scholars and practitioners preoccupied with solving problems in areas such as heat transport, structural mechanics and river hydraulics. As well as describing the extremely useful applications of this method, the authors explain the mathematical background to the CVBEM, which is vital to understanding the subject as a whole. Advances in the Complex Variable Boundary Element Method is the most comprehensive of books on this subject, bringing together ten years of work and boasting the latest news in CVBEM technology. It will be of particular interest to those concerned with solving technical engineering problems - scientists, graduate students, computer programmers and those working in industry may all find the book helpful.

to the English Translation This is a concise guide to basic sections of modern functional analysis. Included are such topics as the principles of Banach and Hilbert spaces, the theory of multinormed and uniform spaces, the Riesz-Dunford holomorphic functional calculus, the Fredholm index theory, convex analysis and duality theory for locally convex spaces. With standard provisos the presentation is self-contained, exposing about a h- dred famous "named" theorems furnished with complete proofs and culminating in the Gelfand-Nalmark-Segal construction for C*-algebras. The first Russian edition was printed by the Siberian Division of "Nauka" P- lishers in 1983. Since then the monograph has served as the standard textbook on functional analysis at the University of Novosibirsk. This volume is translated from the second Russian edition printed by the Sobolev Institute of Mathematics of the Siberian Division of the Russian Academy of Sciences. in 1995. It incorporates new sections on Radon measures, the Schwartz spaces of distributions, and a supplementary list of theoretical exercises and problems. This edition was typeset using AMS-'lEX, the American Mathematical Society's 'lEX system. To clear my conscience completely, I also confess that: = stands for the definor, the assignment operator, signifies the end of the proof."

This monograph, addressing researchers as well as engineers, is devoted to nonclassical thermoelastic modelling of the nonlinear dynamics of shells. Differential equations of different dimensionality and different type have to be combined and nonlinearities of different geometrical, physical or elasto-plastic categories are addressed. Special emphasis is given to the Bubnov--Galerkin method. It can be applied to many problems in the theory of plates and shells, even those with very complex geometries, holes and various boundary conditions. The authors made every effort to keep the text intelligible for both practitioners and graduate students, although they offer a rigorous treatment of both purely mathematical and numerical approaches presented so that the reader can understand, analyse and track the nonlinear dynamics of spatial systems (shells) with thermomechanical behaviours.

The present book is the second of the two volume Proceedings of the Mark Krein International Conference on Operator Theory and Applications. This conference, which was dedicated to the 90th Anniversary of the prominent mathematician Mark Krein, was held in Odessa, Ukraine from 18-22 August, 1997. The conference focused on the main ideas, methods, results, and achievements of M. G. Krein. This second volume is devoted to operator theory and related topics. It opens with the bibliography of M. G. Krein and a number of survey papers about his work. The main part of the book consists of original research papers presenting the state of the art in operator theory and its applications. The first volume of these proceedings, entitled Differential Operators and related Topics, concerns the other aspects of the conference. The two volumes will be of interest to a wide-range of readership in pure and applied mathematics, physics and engineering sciences. Table of Contents Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Bibliography of Mark Grigorevich Krein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Review papers: M. G. Krein's Contributions to Prediction Theory H. Dym M. G. Krein's Contribution to the Moment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 AA Nudelman Research Papers: Solution of the Truncated Matrix Hamburger Moment Problem according to M. G. Krein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Y. M. Adamyan and I. M. Tkachenko Extreme Points of a Positive Operator Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 T. Ando M-accretive Extensions of Sectorial Operators and Krein Spaces . . . . . . . . . 67 Y. M. Arlinskii A Simple Proof of the Continuous Commutant Lifting Theorem . . . . . . . . . . 83 R. Bruzual and M.

Functional Equations, Inequalities and Applications provides an extensive study of several important equations and inequalities, useful in a number of problems in mathematical analysis. Subjects dealt with include the generalized Cauchy functional equation, the Ulam stability theory in the geometry of partial differential equations, stability of a quadratic functional equation in Banach modules, functional equations and mean value theorems, isometric mappings, functional inequalities of iterative type, related to a Cauchy functional equation, the median principle for inequalities and applications, Hadamard and Dragomir-Agarwal inequalities, the Euler formulae and convex functions and approximate algebra homomorphisms. Also included are applications to some problems of pure and applied mathematics.

This book will be of particular interest to mathematicians and graduate students whose work involves functional equations, inequalities and applications.

This monograph presents a collection of results, observations, and examples related to dynamical systems described by linear and nonlinear ordinary differential and difference equations. In particular, dynamical systems that are susceptible to analysis by the Liapunov approach are considered. The naive observation that certain "diagonal-type" Liapunov functions are ubiquitous in the literature attracted the attention of the authors and led to some natural questions. Why does this happen so often? What are the spe cial virtues of these functions in this context? Do they occur so frequently merely because they belong to the simplest class of Liapunov functions and are thus more convenient, or are there any more specific reasons? This monograph constitutes the authors' synthesis of the work on this subject that has been jointly developed by them, among others, producing and compiling results, properties, and examples for many years, aiming to answer these questions and also to formalize some of the folklore or "cul ture" that has grown around diagonal stability and diagonal-type Liapunov functions. A natural answer to these questions would be that the use of diagonal type Liapunov functions is frequent because of their simplicity within the class of all possible Liapunov functions. This monograph shows that, although this obvious interpretation is often adequate, there are many in stances in which the Liapunov approach is best taken advantage of using diagonal-type Liapunov functions. In fact, they yield necessary and suffi cient stability conditions for some classes of nonlinear dynamical systems."

The main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.

This volume consists of eight papers on new advances in interpolation theory for matrix functions and completion theory for matrices and operators. Much emphasis is placed on different interpolation and completion problems when the interpolant is estimated in two different norms. The book also focusses on the study of the spectra of different completions of 2 x 2 block matrices when originally all entries are specified except the lower left corner. A third theme concerns two-sided tangential interpolation problems for real rational matrix functions, and also for the time varying case. A tangential moment problem is also analyzed. All papers deal with related problems of modern matrix analysis, operator theory, complex analysis and system theory and will appeal to a wide group of mathematicians and engineers. The material can be used for advance courses and seminars. Contents: Editorial Introduction ? D. Alpay/P. Loubaton: The tangential trigonometric moment problem on an interval and related topics ? M. Bakonyi/V.G. Kaftal/G. Weiss/H.J. Woerdeman: Maximum entropy and joint norm bounds for operator extensions ? J.A. Ball/I. Gohberg/M.A. Kaashoek: Bitangential interpolation for input-output operators of time varying systems: the discrete time case ? J.A. Ball/I. Gohberg/L. Rodman: Two-sided tangential interpolation of real rational matrix functions ? H. Du/C. Gu: On the spectra of operator completion problems ? C. Foias/A.E. Frazho/W.S. Li: The exact H2 estimate for the central H interpolant ? A.E. Frazho/s.M. Kherat: On mixed H2 - H tangential interpolation ? I. Gohberg/C.Gu: On a completion problem for matrices

This introduction to applied nonlinear dynamics and chaos places emphasis on teaching the techniques and ideas that will enable students to take specific dynamical systems and obtain some quantitative information about their behavior. The new edition has been updated and extended throughout, and contains a detailed glossary of terms.

From the reviews:

"Will serve as one of the most eminent introductions to the geometric theory of dynamical systems." --Monatshefte f r Mathematik

JeanVaillant L'oeuvre de Jean Leray est originale et profonde; ses theoremes et ses theories sont au coeur des recherches mathematiques actuelles: la beaute de chacun de ses travaux ne se divise pas. Son cours de Princeton, sous forme de notes en anglais (et d'une traduction en russe) en est une belle illustration: ce cours presente les equations aux derivees partielles a partir de la transformation de Laplace et du theoreme de Cauchy-Kowaleska et contient l'essentiel de nombreusesrecherchesmodernes. Lerayavaitpourbutderesoudreunprobleme, souvent d'origine mecanique ou physique - qui se pose, et non qu'on se pose -, de demontrer un theoreme; il construit alors son oeuvre de facon complete et essentiellement intrinseque. En fait, Leray construit une theorie dont l'extension tient a son origine naturelle, l'acuite, la perfection, la profondeur d'esprit de son auteur;enmemetempsildominelescalculs,qu'ilmeneavecplaisiretelegance: "Il n'y a pas de mathematiques sans calculs" disait-il. La science etait au centre de la vie de Jean Leray. Il s'inquietait de sa sauvegarde. Rappelons quelques phrases de ses textes de 1974: "D'ailleurs la science ne s'apprend pas: elle se comprend. Elle n'est pas lettre morte et les livres n'assurent pas sa perennite; elle est une pensee vivante. Pour la maitriser notre esprit doit, habilement guide, la redecouvrir de meme que notre corps a du revivre dans le sein mat- nel, toute l'evolution qui crea notre espece. Aussi n'y a-t-il qu'une facon ef?cace d'enseigner les sciences et les techniques: transmettre l'esprit de recherche.

Beginning with the works of N.N.Krasovskii [81, 82, 83], which clari fied the functional nature of systems with delays, the functional approach provides a foundation for a complete theory of differential equations with delays. Based on the functional approach, different aspects of time-delay system theory have been developed with almost the same completeness as the corresponding field of ODE (ordinary differential equations) the ory. The term functional differential equations (FDE) is used as a syn onym for systems with delays 1. The systematic presentation of these re sults and further references can be found in a number of excellent books [2, 15, 22, 32, 34, 38, 41, 45, 50, 52, 77, 78, 81, 93, 102, 128]. In this monograph we present basic facts of i-smooth calculus ~ a new differential calculus of nonlinear functionals, based on the notion of the invariant derivative, and some of its applications to the qualitative theory of functional differential equations. Utilization of the new calculus is the main distinction of this book from other books devoted to FDE theory. Two other distinguishing features of the volume are the following: - the central concept that we use is the separation of finite dimensional and infinite dimensional components in the structures of FDE and functionals; - we use the conditional representation of functional differential equa tions, which is convenient for application of methods and constructions of i~smooth calculus to FDE theory.

This book introduces the reader to important concepts in modern applied analysis, such as homogenization, gradient flows on metric spaces, geometric evolution, Gamma-convergence tools, applications of geometric measure theory, properties of interfacial energies, etc. This is done by tackling a prototypical problem of interfacial evolution in heterogeneous media, where these concepts are introduced and elaborated in a natural and constructive way. At the same time, the analysis introduces open issues of a general and fundamental nature, at the core of important applications. The focus on two-dimensional lattices as a prototype of heterogeneous media allows visual descriptions of concepts and methods through a large amount of illustrations.

Written by a distinguished specialist in functional analysis, this book presents a comprehensive treatment of the history of Banach spaces and (abstract bounded) linear operators. Banach space theory is presented as a part of a broad mathematics context, using tools from such areas as set theory, topology, algebra, combinatorics, probability theory, logic, etc. Equal emphasis is given to both spaces and operators. The book may serve as a reference for researchers and as an introduction for graduate students who want to learn Banach space theory with some historical flavor.

This is the first systematic presentation of the capacitory approach and symmetrization in the context of complex analysis. The content of the book is original - the main part has not been covered by existing textbooks and monographs. After an introduction to the theory of condenser capacities in the plane, the monotonicity of the capacity under various special transformations (polarization, Gonchar transformation, averaging transformations and others) is established, followed by various types of symmetrization which are one of the main objects of the book. By using symmetrization principles, some metric properties of compact sets are obtained and some extremal decomposition problems are solved. Moreover, the classical and present facts for univalent and multivalent meromorphic functions are proven. This book will be a valuable source for current and future researchers in various branches of complex analysis and potential theory.

Since the early 1960s, the mathematical theory of variational inequalities has been under rapid development, based on complex analysis and strongly influenced by 'real-life' application. Many, but of course not all, moving free (Le., a priori un known) boundary problems originating from engineering and economic applica tions can directly, or after a transformation, be formulated as variational inequal ities. In this work we investigate an evolutionary variational inequality with a memory term which is, as a fixed domain formulation, the result of the application of such a transformation to a degenerate moving free boundary problem. This study includes mathematical modelling, existence, uniqueness and regularity results, numerical analysis of finite element and finite volume approximations, as well as numerical simulation results for applications in polymer processing. Essential parts of these research notes were developed during my work at the Chair of Applied Mathematics (LAM) of the Technical University Munich. I would like to express my sincerest gratitude to K. -H. Hoffmann, the head of this chair and the present scientific director of the Center of Advanced European Studies and Research (caesar), for his encouragement and support. With this work I am fol lowing a general concept of Applied Mathematics to which he directed my interest and which, based on application problems, comprises mathematical modelling, mathematical and numerical analysis, computational aspects and visualization of simulation results."

This monograph is the first to provide a comprehensive, self-contained and rigorous presentation of some of the most powerful preconditioning methods for solving finite element equations in a common block-matrix factorization framework. Topics covered include the classical incomplete block-factorization preconditioners and the most efficient methods such as the multigrid, algebraic multigrid, and domain decomposition. Additionally, the author discusses preconditioning of saddle-point, nonsymmetric and indefinite problems, as well as preconditioning of certain nonlinear and quadratic constrained minimization problems that typically arise in contact mechanics. The book presents analytical as well as algorithmic aspects. This text can serve as an indispensable reference for researchers, graduate students, and practitioners. It can also be used as a supplementary text for a topics course in preconditioning and/or multigrid methods at the graduate level.

The notion of a dominated or rnajorized operator rests on a simple idea that goes as far back as the Cauchy method of majorants. Loosely speaking, the idea can be expressed as follows. If an operator (equation) under study is dominated by another operator (equation), called a dominant or majorant, then the properties of the latter have a substantial influence on the properties of the former . Thus, operators or equations that have "nice" dominants must possess "nice" properties. In other words, an operator with a somehow qualified dominant must be qualified itself. Mathematical tools, putting the idea of domination into a natural and complete form, were suggested by L. V. Kantorovich in 1935-36. He introduced the funda mental notion of a vector space normed by elements of a vector lattice and that of a linear operator between such spaces which is dominated by a positive linear or monotone sublinear operator. He also applied these notions to solving functional equations. In the succeedingyears many authors studied various particular cases of lattice normed spaces and different classes of dominated operators. However, research was performed within and in the spirit of the theory of vector and normed lattices. So, it is not an exaggeration to say that dominated operators, as independent objects of investigation, were beyond the reach of specialists for half a century. As a consequence, the most important structural properties and some interesting applications of dominated operators have become available since recently."

This monograph develops a generalised energy flow theory to investigate non-linear dynamical systems governed by ordinary differential equations in phase space and often met in various science and engineering fields. Important nonlinear phenomena such as, stabilities, periodical orbits, bifurcations and chaos are tack-led and the corresponding energy flow behaviors are revealed using the proposed energy flow approach. As examples, the common interested nonlinear dynamical systems, such as, Duffing's oscillator, Van der Pol's equation, Lorenz attractor, Roessler one and SD oscillator, etc, are discussed. This monograph lights a new energy flow research direction for nonlinear dynamics. A generalised Matlab code with User Manuel is provided for readers to conduct the energy flow analysis of their nonlinear dynamical systems. Throughout the monograph the author continuously returns to some examples in each chapter to illustrate the applications of the discussed theory and approaches. The book can be used as an undergraduate or graduate textbook or a comprehensive source for scientists, researchers and engineers, providing the statement of the art on energy flow or power flow theory and methods.

This reference work deals with important topics in general topology and their role in functional analysis and axiomatic set theory, for graduate students and researchers working in topology, functional analysis, set theory and probability theory. It provides a guide to recent research findings, with three contributions by Arhangel'skii and Choban.

The Centre de recherches mathCmatiques (CRM) was created in 1968 by the Universite de Montreal to promote research in the mathematical sci- ences. It is now a national institute that hosts several groups, holds special theme years, summer schools, workshops, postdoctoral program. The focus of its scientific activities ranges from pure to applied mathematics, and includes satistics, theoretical computer science, mathematical methods in biology and life sciences, and mathematical and theoretical physics. The CRM also promotes collaboration between mathematicians and industry. It is subsidized by the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR od the Province of Quebec, the Canadian Institute for Advanced Research and has private endowments. Current ac- tivities, fellowships, and annual reports can be found on the CRM web page at http://www . CRM. UMontreal. CAl. The CRM Series in Mathematical Physics will publish monographs, lec- ture notes, and proceedings base on research pursued and events held at the Centre de recherches mathematiques. Yvan Saint-Aubin Montreal Preface The subject of this three-week school was the explicit integration, that is, analytical as opposed to numerical, of all kinds of nonlinear differential equations (ordinary differential, partial differential, finite difference). The result of such integration is ideally the "general solution," but there are numerous physical systems for which only a particular solution is accessible, for instance the solitary wave of the equation of Kuramoto and Sivashinsky in turbulence.