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.

The book Complex Analysis through Examples and Exercises has come out from the lectures and exercises that the author held mostly for mathematician and physists . The book is an attempt to present the rat her involved subject of complex analysis through an active approach by the reader. Thus this book is a complex combination of theory and examples. Complex analysis is involved in all branches of mathematics. It often happens that the complex analysis is the shortest path for solving a problem in real circum stances. We are using the (Cauchy) integral approach and the (Weierstrass) power se ries approach . In the theory of complex analysis, on the hand one has an interplay of several mathematical disciplines, while on the other various methods, tools, and approaches. In view of that, the exposition of new notions and methods in our book is taken step by step. A minimal amount of expository theory is included at the beinning of each section, the Preliminaries, with maximum effort placed on weil selected examples and exercises capturing the essence of the material. Actually, I have divided the problems into two classes called Examples and Exercises (some of them often also contain proofs of the statements from the Preliminaries). The examples contain complete solutions and serve as a model for solving similar problems given in the exercises. The readers are left to find the solution in the exercisesj the answers, and, occasionally, some hints, are still given."

This book presents an operator theoretic approach to robust control analysis for linear time-varying systems. It emphasizes the conceptual similarity with the H control theory for time-invariant systems and at the same time clarifies the major difficulties confronted in the time varying case. The necessary operator theory is developed from first principles and the book is as self-contained as possible. After presenting the necessary results from the theories of Toeplitz operators and nest algebras, linear systems are defined as input- output operators and the relationship between stabilization and the existance of co-prime factorizations is described. Uniform optimal control problems are formulated as model-matching problems and are reduced to four block problems. Robustness is considered both from the point of view of fractional representations and the "time varying gap" metric, and the relationship between these types of uncertainties is clarified. The book closes with the solution of the orthogonal embedding problem for time varying contractive systems. This book will be useful to both mathematicians interested in the potential applications of operator theory in control and control engineers who wish to deal with some of the more mathematically sophisticated extension of their work.

This and the next volume of the OT series contain the proceedings of the Work shop on Operator Theory and its Applications, IWOTA 95, which was held at the University of Regensburg, Germany, July 31 to August 4, 1995. It was the eigth workshop of this kind. Following is a list of the seven previous workshops with reference to their proceedings: 1981 Operator Theory (Santa Monica, California, USA) 1983 Applications of Linear Operator Theory to Systems and Networks (Rehovot, Israel), OT 12 1985 Operator Theory and its Applications (Amsterdam, The Netherlands), OT 19 1987 Operator Theory and Functional Analysis (Mesa, Arizona, USA), OT 35 1989 Matrix and Operator Theory (Rotterdam, The Netherlands), OT 50 1991 Operator Theory and Complex Analysis (Sapporo, Japan), OT 59 1993 Operator Theory and Boundary Eigenvalue Problems (Vienna, Austria), OT 80 IWOTA 95 offered a rich programme on a wide range of latest developments in operator theory and its applications. The programme consisted of 6 invited plenary lectures, 54 invited special topic lectures and more than 100 invited session talks. About 180 participants from 25 countries attended the workshop, more than a third came from Eastern Europe. The conference covered different aspects of linear and nonlinear spectral prob lems, starting with problems for abstract operators up to spectral theory of ordi nary and partial differential operators, pseudodifferential operators, and integral operators. The workshop was also focussed on operator theory in spaces with indefinite metric, operator functions, interpolation and extension problems.

Recent years have witnessed an increasingly close relationship growing between potential theory, probability and degenerate partial differential operators. The theory of Dirichlet (Markovian) forms on an abstract finite or infinite-dimensional space is common to all three disciplines. This is a fascinating and important subject, central to many of the contributions to the conference on Potential Theory and Degenerate Partial Differential Operators', held in Parma, Italy, February 1994.

The aim of this proceeding is addressed to present recent developments of the mathematical research on the Navier-Stokes equations, the Euler equations and other related equations. In particular, we are interested in such problems as: 1) existence, uniqueness and regularity of weak solutions2) stability and its asymptotic behavior of the rest motion and the steady state3) singularity and blow-up of weak and strong solutions4) vorticity and energy conservation5) fluid motions around the rotating axis or outside of the rotating body6) free boundary problems7) maximal regularity theorem and other abstract theorems for mathematical fluid mechanics.

Calculus and linear algebra are two dominant themes in contemporary mathematics and its applications. The aim of this book is to introduce linear algebra in an intuitive geometric setting as the study of linear maps and to use these simpler linear functions to study more complicated nonlinear functions. In this way, many of the ideas, techniques, and formulas in the calculus of several variables are clarified and understood in a more conceptual way. After using this text a student should be well prepared for subsequent advanced courses in both algebra and linear differential equations as well as the many applications where linearity and its interplay with nonlinearity are significant. This second edition has been revised to clarify the concepts. Many exercises and illustrations have been included to make the text more usable for students.

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 book was undertaken to provide a text and reference on the theory and practice of the FFT and its common usage. This book is organized in only four chapters, and is intended as a tutorial on the use of the FFf and its trade space. The trade space of the FFT is the parameters in its usage and the relationships between them - the sampie rate, the total number of points or the interval over which processing occurs in a single FFf, the selectivity of tuning to a given frequency over signals out-of-band, and the bandwidth over which a signal appears. The examples given in this text are in FORTRAN 9512003. FORTRAN 2003 was frozen as a standard while this work was in progress. The listings given here are intended as an aid in understanding the FFT and associated algorithms such as spectral window weightings, with the goal of making the best of them more accessible to the reader. The code I use here provides a simple bridge between the material in the text and implementation in FORTRAN 2003, C++, Java, MATLAB (c), and other modem languages. The examples are sufficiently simple to be translated into older languages such as C and FORTRAN 77 if desired.

This two-volume work mainly addresses undergraduate and graduate students in the engineering sciences and applied mathematics. Hence it focuses on partial differential equations with a strong emphasis on illustrating important applications in mechanics. The presentation considers the general derivation of partial differential equations and the formulation of consistent boundary and initial conditions required to develop well-posed mathematical statements of problems in mechanics. The worked examples within the text and problem sets at the end of each chapter highlight engineering applications. The mathematical developments include a complete discussion of uniqueness theorems and, where relevant, a discussion of maximum and miniumum principles. The primary aim of these volumes is to guide the student to pose and model engineering problems, in a mathematically correct manner, within the context of the theory of partial differential equations in mechanics.

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.

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.

This edited book aims at presenting current research activities in the field of robust variable-structure systems. The scope equally comprises highlighting novel methodological aspects as well as presenting the use of variable-structure techniques in industrial applications including their efficient implementation on hardware for real-time control. The target audience primarily comprises research experts in the field of control theory and nonlinear dynamics but the book may also be beneficial for graduate students.

Functional Equations andInequalities provides an extensive studyofsome of the most important topics of current interest in functional equations and inequalities. Subjects dealt with include: a Pythagorean functional equation, a functional definition oftrigonometric functions, the functional equation ofthe square root spiral, a conditional Cauchy functional equation, an iterative functional equation, the Hille-type functional equation, the polynomial-like iterative functional equation, distribution ofzeros and inequalities for zeros of algebraic polynomials, a qualitative study ofLobachevsky's complex functional equation, functional inequalities in special classesoffunctions, replicativity and function spaces, normal distributions, some difference equations, finite sums decompositions of functions, harmonic functions, set-valued quasiconvex functions, the problem of expressibility in some extensions of free groups, Aleksandrov problem and mappings which preserve distances, Ulam's problem, stability of some functional equation for generalized trigonometric functions, Hyers-Ulam stability of Hosszil's equation, superstability of a functional equation, and some demand functions in a duopoly market with advertising. It is a pleasureto express my deepest appreciationto all the mathematicians who contributed to this volume. Finally, we wish to acknowledge the superb assistance provided by the staffofKluwer Academic Publishers. June 2000 Themistocles M. Rassias xi ON THE STABILITY OF A FUNCTIONAL EQUATION FOR GENERALIZED TRIGONOMETRIC FUNCTIONS ROMAN BADORA lnstytut Matematyki, Uniwersytet Sli;ski, ul. Bankowa 14, PL-40-007 Katowice, Poland, e-mail: robadora@gate. math. us. edu. pl Abstract. In the present paper the stability result concerning a functional equation for generalized trigonometric functions is presented. Z.

This book discusses the latest advances in algorithms for symbolic summation, factorization, symbolic-numeric linear algebra and linear functional equations. It presents a collection of papers on original research topics from the Waterloo Workshop on Computer Algebra (WWCA-2016), a satellite workshop of the International Symposium on Symbolic and Algebraic Computation (ISSAC'2016), which was held at Wilfrid Laurier University (Waterloo, Ontario, Canada) on July 23-24, 2016. This workshop and the resulting book celebrate the 70th birthday of Sergei Abramov (Dorodnicyn Computing Centre of the Russian Academy of Sciences, Moscow), whose highly regarded and inspirational contributions to symbolic methods have become a crucial benchmark of computer algebra and have been broadly adopted by many Computer Algebra systems.

Spectral theoryis an important part of functional analysis.It has numerousapp- cations in many parts of mathematics and physics including matrix theory, fu- tion theory, complex analysis, di?erential and integral equations, control theory and quantum physics. In recent years, spectral theory has witnessed an explosive development. There are many types of spectra, both for one or several commuting operators, with important applications, for example the approximate point spectrum, Taylor spectrum, local spectrum, essential spectrum, etc. The present monograph is an attempt to organize the available material most of which exists only in the form of research papers scattered throughout the literature. The aim is to present a survey of results concerning various types of spectra in a uni?ed, axiomatic way. The central unifying notion is that of a regularity, which in a Banach algebra isasubsetofelementsthatareconsideredtobe nice .AregularityRinaBanach algebraA de?nes the corresponding spectrum ? (a)={ C: a / ? R} in R the same wayas the ordinaryspectrum is de?ned by means of invertible elements, ?(a)={ C: a / ? Inv(A)}. Axioms of a regularity are chosen in such a way that there are many natural interesting classes satisfying them. At the same time they are strong enough for non-trivial consequences, for example the spectral mapping theorem. Spectra ofn-tuples ofcommuting elements ofa Banachalgebraaredescribed similarly by means of a notion of joint regularity. This notion is closely related to ? the axiomatic spectral theory of Zelazko and S lodkowski."

This two-volume work mainly addresses undergraduate and gra- duate students in the engineering sciences and applied ma- thematics. Hence it focuses on partial differential equati- ons with a strong emphasis on illustrating important appli- cations in mechanics. The presentation considers the general derivation of partial differential equations and the formu- lation of consistent boundary and initial conditions requi- red to develop well-posed mathematical statements of pro- blems in mechanics. The worked examples within the text and problem sets at the end of each chapter highlight enginee- ring applications. The mathematical developments include a complete discussion of uniqueness theorems and, where rele- vant, a discussion of maximum and miniumum principles. The primary aim of these volumes is to guide the student to pose and model engineering problems, in a mathematically correct manner, within the context of the theory of partial differential equations in mechanics.

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.

It is hard to imagine that another elementary analysis book would contain ma terial that in some vision could qualify as being new and needed for a discipline already abundantly endowed with literature. However, to understand analysis, be ginning with the undergraduate calculus student through the sophisticated math ematically maturing graduate student, the need for examples and exercises seems to be a constant ingredient to foster deeper mathematical understanding. To a talented mathematical student, many elementary concepts seem clear on their first encounter. However, it is the belief of the authors, this understanding can be deepened with a guided set of exercises leading from the so called "elementary" to the somewhat more "advanced" form. Insight is instilled into the material which can be drawn upon and implemented in later development. The first year graduate student attempting to enter into a research environment begins to search for some original unsolved area within the mathematical literature. It is hard for the student to imagine that in many circumstances the advanced mathematical formulations of sophisticated problems require attacks that draw upon, what might be termed elementary techniques. However, if a student has been guided through a serious repertoire of examples and exercises, he/she should certainly see connections whenever they are encountered."

Infotext]((Kurztext))These are the proceedings of the 7th International Conference on Hyperbolic Problems, held in Zurich in February 1998. The speakers and contributors have been rigorously selected and present the state of the art in this field. The articles, both theoretical and numerical, encompass a wide range of applications, such as nonlinear waves in solids, various computational fluid dynamics from small-scale combustion to relativistic astrophysical problems, multiphase phenomena and geometrical optics.

((Volltext))These proceedings contain, in two volumes, approximately one hundred papers presented at the conference on hyperbolic problems, which has focused to a large extent on the laws of nonlinear hyperbolic conservation. Two-fifths of the papers are devoted to mathematical aspects such as global existence, uniqueness, asymptotic behavior such as large time stability, stability and instabilities of waves and structures, various limits of the solution, the Riemann problem and so on. Roughly the same number of articles are devoted to numerical analysis, for example stability and convergence of numerical schemes, as well as schemes with special desired properties such as shock capturing, interface fitting and high-order approximations to multidimensional systems. The results in these contributions, both theoretical and numerical, encompass a wide range of applications such as nonlinear waves in solids, various computational fluid dynamics from small-scale combustion to relativistic astrophysical problems, multiphase phenomena and geometrical optics."

Optimization is a rich and thriving mathematical discipline. The theory underlying current computational optimization techniques grows ever more sophisticated. The powerful and elegant language of convex analysis unifies much of this theory. The aim of this book is to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. It can serve as a teaching text, at roughly the level of first year graduate students. While the main body of the text is self-contained, each section concludes with an often extensive set of optional exercises. The new edition adds material on semismooth optimization, as well as several new proofs that will make this book even more self-contained.

The theory of integral and integrodifferential equations has ad vanced rapidly over the last twenty years. Of course the question of existence is an age-old problem of major importance. This mono graph is a collection of some of the most advanced results to date in this field. The book is organized as follows. It is divided into twelve chap ters. Each chapter surveys a major area of research. Specifically, some of the areas considered are Fredholm and Volterra integral and integrodifferential equations, resonant and nonresonant problems, in tegral inclusions, stochastic equations and periodic problems. We note that the selected topics reflect the particular interests of the authors. Donal 0 'Regan Maria Meehan CHAPTER 1 INTRODUCTION AND PRELIMINARIES 1.1. Introduction The aim of this book is firstly to provide a comprehensive existence the ory for integral and integrodifferential equations, and secondly to present some specialised topics in integral equations which we hope will inspire fur ther research in the area. To this end, the first part of the book deals with existence principles and results for nonlinear, Fredholm and Volterra inte gral and integrodifferential equations on compact and half-open intervals, while selected topics (which reflect the particular interests of the authors) such as nonresonance and resonance problems, equations in Banach spaces, inclusions, and stochastic equations are presented in the latter part."

This book is the first comprehensive treatment of Painleve differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, the Nevanlinna theory will be applied to offer a detailed exposition of growth aspects and value distribution of Painleve transcendents. The subsequent main part of the book is devoted to topics of classical background such as representations and expansions of solutions, solutions of special type like rational and special transcendental solutions, Backlund transformations and higher order analogues, treated separately for each of these six equations. The final chapter offers a short overview of applications of Painleve equations, including an introduction to their discrete counterparts. Due to the present important role of Painleve equations in physical applications, this monograph should be of interest to researchers in both mathematics and physics and to graduate students interested in mathematical physics and the theory of differential equations.

The linear theory of oscillations traditionally operates with frequency representa- tions based on the concepts of a transfer function and a frequency response. The universality of the critria of Nyquist and Mikhailov and the simplicity and obvi- ousness of the application of frequency and amplitude - frequency characteristics in analysing forced linear oscillations greatly encouraged the development of practi- cally important nonlinear theories based on various forms of the harmonic balance hypothesis [303]. Therefore mathematically rigorous frequency methods of investi- gating nonlinear systems, which appeared in the 60s, also began to influence many areas of nonlinear theory of oscillations. First in this sphere of influence was a wide range of problems connected with multidimensional analogues of the famous van der Pol equation describing auto- oscillations of generators of various radiotechnical devices. Such analogues have as a rule a unique unstable stationary point in the phase space and are Levinson dis- sipative. One of the pioneering works in this field, which started the investigation of a three-dimensional analogue of the van der Pol equation, was K. O. Friedrichs's paper [123]. The author suggested a scheme for constructing a positively invariant set homeomorphic to a torus, by means of which the existence of non-trivial periodic solutions was established. That scheme was then developed and improved for dif- ferent classes of multidimensional dynamical systems [131, 132, 297, 317, 334, 357, 358]. The method of Poincare mapping [12, 13, 17] in piecewise linear systems was another intensively developed direction.

A combinatorial method is developed in this book to explore the mysteries of chaos, which has became a topic of science since 1975. Using tools from theoretical computer science, formal languages and automata, the complexity of symbolic behaviors of dynamical systems is classified and analysed thoroughly. This book is mainly devoted to explanation of this method and apply it to one-dimensional dynamical systems, including the circle and interval maps, which are typical in exhibiting complex behavior through simple iterated calculations. The knowledge for reading it is self-contained in the book.