Functional Equations on Hypergroups

This volume provides an accessible and coherent introduction to some of the scientific progress on functional equations on groups in the last two decades. It presents the latest methods of treating th

The theory of hypergroups is a rapidly developing area of mathematics due to its diverse applications in different areas like probability, harmonic analysis, etc. This book exhibits the use of functio

Numerous detailed proofs highlight this treatment of functional equations. Starting with equations that can be solved by simple substitutions, the book then moves to equations with several unknown fun

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, different

Builds on an earlier work by J. Hale, Theory of Functional Differential Equations , 1977. Approximately one-third of the material is left intact. A completely new presentation of linear systems for r

Presents a unified treatment for a branch of the theory of functional equations, the main idea being to use the fundamental results about spectral synthesis on discrete abelian groups. Acidic paper. A

Some of the most recent and significant results on homomorphisms and derivations in Banach algebras, quasi-Banach algebras, C*-algebras, C*-ternary algebras, non-Archimedean Banach algebras and multi-

This book provides a crash course on various methods from the bifurcationtheory of Functional Differential Equations (FDEs). FDEs arise very naturally in economics, life sciences and engineering and t

This book consists of five introductory contributions by leading mathematicians on the functional analytic treatment of evolution equations. In particular the contributions deal with Markov semigroups

The last, previously unpublished paper by French mathematician Grisvard joins 16 others from a November 1994 conference on functional as applied to partially differential equations, a particular inter

This book focuses on a conjectural class of zeta integrals which arose from a program born in the work of Braverman and Kazhdan around the year 2000, the eventual goal being to prove the analytic cont

Collocation based on piecewise polynomial approximation represents a powerful class of methods for the numerical solution of initial-value problems for functional differential and integral equations arising in a wide spectrum of applications, including biological and physical phenomena. The present book introduces the reader to the general principles underlying these methods and then describes in detail their convergence properties when applied to ordinary differential equations, functional equations with (Volterra type) memory terms, delay equations, and differential-algebraic and integral-algebraic equations. Each chapter starts with a self-contained introduction to the relevant theory of the class of equations under consideration. Numerous exercises and examples are supplied, along with extensive historical and bibliographical notes utilising the vast annotated reference list of over 1300 items. In sum, Hermann Brunner has written a treatise that can serve as an introduction for stu

This book provides cutting-edge results on the existence of multiple positive periodic solutions of first-order functional differential equations. It demonstrates how the Leggett-Williams fixed-point

As Richard Bellman has so elegantly stated at the Second International Conference on General Inequalities (Oberwolfach, 1978), “There are three reasons for the study of inequalities: practical, theore

The purpose of this book is threefold: to be used for graduate courses on integral equations; to be a reference for researchers; and to describe methods of application of the theory. The author emphasizes the role of Volterra equations as a unifying tool in the study of functional equations, and investigates the relation between abstract Volterra equations and other types of functional-differential equations.

The second edition of this successful textbook, first published in 1970, retains the aims of the first, namely to provide a truly introductory course in functional analysis, but the opportunity has been taken to add more detail and worked examples. The main changes are complete revisons of the work on convex sets, metric and topological linear spaces, reflexivity and weak convergence. Additional material on the Weiner algebra of absolutely convergent Fourier series and on weak topologies is included. A final chapter includes elementary applications of functional analysis to differential and integral equations.

This volume collects longer articles on the analysis and numerics of Maxwell’s equations. The topics include functional analytic and Hilbert space methods, compact embeddings, solution theories and as

This book presents applications of Newton-like and other similar methods to solve abstract functional equations involving fractional derivatives. It focuses on Banach space-valued functions of a real

This text focuses on the use of smoothing methods for developing and estimating differential equations following recent developments in functional data analysis and building on techniques described in

This book offers a comprehensive introduction to the theory of linear and nonlinear Volterra integral equations (VIEs), ranging from Volterra's fundamental contributions and the resulting classical theory to more recent developments that include Volterra functional integral equations with various kinds of delays, VIEs with highly oscillatory kernels, and VIEs with non-compact operators. It will act as a 'stepping stone' to the literature on the advanced theory of VIEs, bringing the reader to the current state of the art in the theory. Each chapter contains a large number of exercises, extending from routine problems illustrating or complementing the theory to challenging open research problems. The increasingly important role of VIEs in the mathematical modelling of phenomena where memory effects play a key role is illustrated with some 30 concrete examples, and the notes at the end of each chapter feature complementary references as a guide to further reading.

The recent appearance of wavelets as a new computational tool in applied mathematics has given a new impetus to the field of numerical analysis of Fredholm integral equations. This book gives an account of the state of the art in the study of fast multiscale methods for solving these equations based on wavelets. The authors begin by introducing essential concepts and describing conventional numerical methods. They then develop fast algorithms and apply these to solving linear, nonlinear Fredholm integral equations of the second kind, ill-posed integral equations of the first kind and eigen-problems of compact integral operators. Theorems of functional analysis used throughout the book are summarised in the appendix. The book is an essential reference for practitioners wishing to use the new techniques. It may also be used as a text, with the first five chapters forming the basis of a one-semester course for advanced undergraduates or beginning graduates.

Delay Differential Equations: Recent Advances and New Directions cohesively presents contributions from leading experts on the theory and applications of functional and delay differential equations (

The Navier–Stokes equations are a set of nonlinear partial differential equations comprising the fundamental dynamical description of fluid motion. They are applied routinely to problems in engineering, geophysics, astrophysics, and atmospheric science. This book is an introductory physical and mathematical presentation of the Navier–Stokes equations, focusing on unresolved questions of the regularity of solutions in three spatial dimensions, and the relation of these issues to the physical phenomenon of turbulent fluid motion. Intended for graduate students and researchers in applied mathematics and theoretical physics, results and techniques from nonlinear functional analysis are introduced as needed with an eye toward communicating the essential ideas behind the rigorous analyses.

The basis of this graduate-level textbook is a careful survey of a wide range of problems affecting the solution of linear partial differential equations. The book begins with a fairly elementary introduction to the theory of Fourier series of continuous functions and goes on to describe the fundamental theory of linear partial differential equations of elliptic and hyperbolic types, equations of evolution, semi-linear hyperbolic equations and selected topics on Green's functions and spectra of some special operators. The book is intended for use by pure mathematicians in functional analysis. The selection of material is interesting and differs from existing literature in European languages. This paperback edition will make it particularly attractive to graduate students in pure and applied mathematics.

This comprehensive introduction to the calculus of variations and its main principles also presents their real-life applications in various contexts: mathematical physics, differential geometry, and optimization in economics. Based on the authors' original work, it provides an overview of the field, with examples and exercises suitable for graduate students entering research. The method of presentation will appeal to readers with diverse backgrounds in functional analysis, differential geometry and partial differential equations. Each chapter includes detailed heuristic arguments, providing thorough motivation for the material developed later in the text. Since much of the material has a strong geometric flavor, the authors have supplemented the text with figures to illustrate the abstract concepts. Its extensive reference list and index also make this a valuable resource for researchers working in a variety of fields who are interested in partial differential equations and functional

This exposition is devoted to a consistent treatment of quantization problems, based on appealing to some nontrivial items of functional analysis concerning the theory of linear operators in Hilbert s

The study of electronic structure of materials is at a momentous stage, with new computational methods and advances in basic theory. Many properties of materials can be determined from the fundamental equations, and electronic structure theory is now an integral part of research in physics, chemistry, materials science and other fields. This book provides a unified exposition of the theory and methods, with emphasis on understanding each essential component. New in the second edition are recent advances in density functional theory, an introduction to Berry phases and topological insulators explained in terms of elementary band theory, and many new examples of applications. Graduate students and research scientists will find careful explanations with references to original papers, pertinent reviews, and accessible books. Each chapter includes a short list of the most relevant works and exercises that reveal salient points and challenge the reader.

The theory of Toeplitz matrices and operators is a vital part of modern analysis, with applications to moment problems, orthogonal polynomials, approximation theory, integral equations, bounded- and vanishing-mean oscillations, and asymptotic methods for large structured determinants, among others. This friendly introduction to Toeplitz theory covers the classical spectral theory of Toeplitz forms and Wiener–Hopf integral operators and their manifestations throughout modern functional analysis. Numerous solved exercises illustrate the results of the main text and introduce subsidiary topics, including recent developments. Each chapter ends with a survey of the present state of the theory, making this a valuable work for the beginning graduate student and established researcher alike. With biographies of the principal creators of the theory and historical context also woven into the text, this book is a complete source on Toeplitz theory.

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 orig

Ulam Stability of Operators presents a modern, unified, and systematic approach to the field. Focusing on the stability of functional equations across single variable, difference equations, differenti

A careful and accessible exposition of a functional analytic approach to initial boundary value problems for semilinear parabolic differential equations, with a focus on the relationship between analytic semigroups and initial boundary value problems. This semigroup approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of pseudo-differential operators, one of the most influential works in the modern history of analysis. Complete with ample illustrations and additional references, this new edition offers both streamlined analysis and better coverage of important examples and applications. A powerful method for the study of elliptic boundary value problems, capable of further extensive development, is provided for advanced undergraduates or beginning graduate students, as well as mathematicians with an interest in functional analysis and partial differential equations.

In this monograph, the authors present some powerful methods for dealing with singularities in elliptic and parabolic partial differential inequalities. Here, the authors take the unique approach of investigating differential inequalities rather than equations, the reason being that the simplest way to study an equation is often to study a corresponding inequality; for example, using sub and superharmonic functions to study harmonic functions. Another unusual feature of the present book is that it is based on integral representation formulae and nonlinear potentials, which have not been widely investigated so far. This approach can also be used to tackle higher order differential equations. The book will appeal to graduate students interested in analysis, researchers in pure and applied mathematics, and engineers who work with partial differential equations. Readers will require only a basic knowledge of functional analysis, measure theory and Sobolev spaces.

This book is concerned with digital image processing techniques that use partial differential equations (PDEs) for the task of image 'inpainting', an artistic term for virtual image restoration or interpolation, whereby missing or occluded parts in images are completed based on information provided by intact parts. Computer graphic designers, artists and photographers have long used manual inpainting to restore damaged paintings or manipulate photographs. Today, mathematicians apply powerful methods based on PDEs to automate this task. This book introduces the mathematical concept of PDEs for virtual image restoration. It gives the full picture, from the first modelling steps originating in Gestalt theory and arts restoration to the analysis of resulting PDE models, numerical realisation and real-world application. This broad approach also gives insight into functional analysis, variational calculus, optimisation and numerical analysis and will appeal to researchers and graduate studen

The theory of optimal transportation has its origins in the eighteenth century when the problem of transporting resources at a minimal cost was first formalised. Through subsequent developments, particularly in recent decades, it has become a powerful modern theory. This book contains the proceedings of the summer school 'Optimal Transportation: Theory and Applications' held at the Fourier Institute in Grenoble. The event brought together mathematicians from pure and applied mathematics, astrophysics, economics and computer science. Part I of this book is devoted to introductory lecture notes accessible to graduate students, while Part II contains research papers. Together, they represent a valuable resource on both fundamental and advanced aspects of optimal transportation, its applications, and its interactions with analysis, geometry, PDE and probability, urban planning and economics. Topics covered include Ricci flow, the Euler equations, functional inequalities, curvature-dimensio

Written by prominent experts in the field, this monograph provides the first comprehensive, unified presentation of the structural, algorithmic and applied aspects of the theory of Boolean functions. The book focuses on algebraic representations of Boolean functions, especially disjunctive and conjunctive normal form representations. This framework looks at the fundamental elements of the theory (Boolean equations and satisfiability problems, prime implicants and associated short representations, dualization), an in-depth study of special classes of Boolean functions (quadratic, Horn, shellable, regular, threshold, read-once functions and their characterization by functional equations) and two fruitful generalizations of the concept of Boolean functions (partially defined functions and pseudo-Boolean functions). Several topics are presented here in book form for the first time. Because of the depth and breadth and its emphasis on algorithms and applications, this monograph will have sp

This book lays the foundations for a theory on almost periodic stochastic processes and their applications to various stochastic differential equations, functional differential equations with delay, p

A unique introduction to the design, analysis, and presentation of scientific projects, this is an essential textbook for undergraduate majors in science and mathematics. The textbook gives an overview of the main methods used in scientific research, including hypothesis testing, the measurement of functional relationships, and observational research. It describes important features of experimental design, such as the control of errors, instrument calibration, data analysis, laboratory safety, and the treatment of human subjects. Important concepts in statistics are discussed, focusing on standard error, the meaning of p values, and use of elementary statistical tests. The textbook introduces some of the main ideas in mathematical modeling, including order-of-magnitude analysis, function fitting, Fourier transforms, recursion relations, and difference approximations to differential equations. It also provides guidelines on accessing scientific literature, and preparing scientific paper