On Functions and Functional Equations


Author: Smital
Publisher: CRC Press
ISBN: 9780852744185
Category: Mathematics
Page: 164
View: 9577

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On Functions and Functional Equations introduces the main topics in iteration theory and the theory of functional equations with emphasis on applications in the fields of mathematics, physics, biology, chemistry, and electronics and mechanical engineering. The book contains many classical results as well as important, more recent results. It also includes numerous exercise and some problems that have yet to be resolved. The book is accessible to readers having a secondary level mathematical education.

Lectures on Functional Equations and Their Applications


Author: J. Aczel,Hansjorg Oser
Publisher: Courier Corporation
ISBN: 0486445232
Category: Mathematics
Page: 510
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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 functions and methods of reduction to differential and integral equations. Also includes composite equations, equations with several unknown functions of several variables, vector and matrix equations, more. 1966 edition.

Introduction to Functional Equations

Theory and Problem-solving Strategies for Mathematical Competitions and Beyond
Author: Costas Efthimiou
Publisher: American Mathematical Soc.
ISBN: 0821853147
Category: Mathematics
Page: 363
View: 9333

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Functions and their properties have been part of the rigorous precollege curriculum for decades. And functional equations have been a favorite topic of the leading national and international mathematical competitions. Yet the subject has not received equal attention by authors at an introductory level. The majority of the books on the topic remain unreachable to the curious and intelligent precollege student. The present book is an attempt to eliminate this disparity. The book opens with a review chapter on functions, which collects the relevant foundational information on functions, plus some material potentially new to the reader. The next chapter presents a working definition of functional equations and explains the difficulties in trying to systematize the theory. With each new chapter, the author presents methods for the solution of a particular group of equations. Each chapter is complemented with many solved examples, the majority of which are taken from mathematical competitions and professional journals. The book ends with a chapter of unsolved problems and some other auxiliary material. The book is an invaluable resource for precollege and college students who want to deepen their knowledge of functions and their properties, for teachers and instructors who wish to enrich their curricula, and for any lover of mathematical problem-solving techniques. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession.

Functional Equations and Modelling in Science and Engineering


Author: Enrique Castillo
Publisher: CRC Press
ISBN: 9780824787172
Category: Mathematics
Page: 352
View: 1416

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Provides engineers and applied scientists with some selected results of functional equations and their applications, with the intention of changing the way they think about mathematical modelling. Many of the proofs are simplified or omitted, so as not to bore or confuse engineers. Functional equati

Functional Equations in Applied Sciences


Author: Enrique Castillo,Andres Iglesias,Reyes Ruiz-Cobo
Publisher: Elsevier
ISBN: 9780080477916
Category: Mathematics
Page: 408
View: 6806

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The book provides the reader with the different types of functional equations that s/he can find in practice, showing, step by step, how they can be solved. A general methodology for solving functional equations is provided in Chapter 2. The different types of functional equations are described and solved in Chapters 3 to 8. Many examples, coming from different fields, as geometry, science, engineering, economics, probability, statistics, etc, help the reader to change his/her mind in order to state problems as functional equations as an alternative to differential equations, and to state new problems in terms of functional equations or systems. An interesting feature of the book is that it deals with functional networks, a powerful generalization of neural networks that allows solving many practical problems. The second part of the book, Chapters 9 to 13, is devoted to the applications of this important paradigm. The book contains many examples and end of chapter exercises, that facilitates the understanding of the concepts and applications. · A general methodology for solving functional equations is provided in Chapter 2. · It deals with functional networks, a powerful generalization of neural networks. · Many examples, coming from different fields, as geometry, science, engineering, economics, probability, statistics, etc, illustrate the concept of functional equation. · Functional equations are presented as a powerful alternative to differential equations. · The book contains end of chapter exercises.

Functional Equations and How to Solve Them


Author: Christopher G. Small
Publisher: Springer Science & Business Media
ISBN: 0387489010
Category: Mathematics
Page: 131
View: 4531

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Many books have been written on the theory of functional equations, but very few help readers solve functional equations in mathematics competitions and mathematical problem solving. This book fills that gap. Each chapter includes a list of problems associated with the covered material. These vary in difficulty, with the easiest being accessible to any high school student who has read the chapter carefully. The most difficult will challenge students studying for the International Mathematical Olympiad or the Putnam Competition. An appendix provides a springboard for further investigation of the concepts of limits, infinite series and continuity.

Functional Equations and Inequalities in Several Variables


Author: Stefan Czerwik
Publisher: World Scientific
ISBN: 9789810248376
Category: Mathematics
Page: 410
View: 3679

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This book outlines the modern theory of functional equations and inequalities in several variables. It consists of three parts. The first is devoted to additive and convex functions defined on linear spaces with semilinear topologies. In the second part, the problems of stability of functional equations in the sense of Ulam-Hyers-Rassias and in some function spaces are considered. In the last part, the functional equations in set-valued functions are dealt with ? for the first time in the mathematical literature. The book contains many fresh results concerning those problems.

An Introduction to the Theory of Functional Equations and Inequalities

Cauchy's Equation and Jensen's Inequality
Author: Marek Kuczma
Publisher: Springer Science & Business Media
ISBN: 3764387491
Category: Mathematics
Page: 595
View: 8169

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Marek Kuczma was born in 1935 in Katowice, Poland, and died there in 1991. After finishing high school in his home town, he studied at the Jagiellonian University in Kraków. He defended his doctoral dissertation under the supervision of Stanislaw Golab. In the year of his habilitation, in 1963, he obtained a position at the Katowice branch of the Jagiellonian University (now University of Silesia, Katowice), and worked there till his death. Besides his several administrative positions and his outstanding teaching activity, he accomplished excellent and rich scientific work publishing three monographs and 180 scientific papers. He is considered to be the founder of the celebrated Polish school of functional equations and inequalities. "The second half of the title of this book describes its contents adequately. Probably even the most devoted specialist would not have thought that about 300 pages can be written just about the Cauchy equation (and on some closely related equations and inequalities). And the book is by no means chatty, and does not even claim completeness. Part I lists the required preliminary knowledge in set and measure theory, topology and algebra. Part II gives details on solutions of the Cauchy equation and of the Jensen inequality [...], in particular on continuous convex functions, Hamel bases, on inequalities following from the Jensen inequality [...]. Part III deals with related equations and inequalities (in particular, Pexider, Hosszú, and conditional equations, derivations, convex functions of higher order, subadditive functions and stability theorems). It concludes with an excursion into the field of extensions of homomorphisms in general." (Janos Aczel, Mathematical Reviews) "This book is a real holiday for all the mathematicians independently of their strict speciality. One can imagine what deliciousness represents this book for functional equationists." (B. Crstici, Zentralblatt für Mathematik)

Optimal Control of Differential and Functional Equations


Author: J. Warga
Publisher: Academic Press
ISBN: 1483259196
Category: Technology & Engineering
Page: 546
View: 3379

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Optimal Control of Differential and Functional Equations presents a mathematical theory of deterministic optimal control, with emphasis on problems involving functional-integral equations and functional restrictions. The book reviews analytical foundations, and discusses deterministic optimal control problems requiring original, approximate, or relaxed solutions. Original solutions involve mathematicians, and approximate solutions concern engineers. Relaxed solutions yield a complete theory that encompasses both existence theorems and necessary conditions. The text also presents general optimal control problems, optimal control of ordinary differential equations, and different types of functional-integral equations. The book discusses control problems defined by equations in Banach spaces, the convex cost functionals, and the weak necessary conditions for an original minimum. The text illustrates a class of ordinary differential problems with examples, and explains some conflicting control problems with relaxed adverse controls, as well as conflicting control problems with hyper-relaxed adverse controls. The book is intended for mature mathematicians, graduate students in analysis, and practitioners of optimal control whose primary interests and training are in science or engineering.

Introduction to Functional Equations


Author: Prasanna K. Sahoo,Palaniappan Kannappan
Publisher: CRC Press
ISBN: 1439841160
Category: Mathematics
Page: 465
View: 3807

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Introduction to Functional Equations grew out of a set of class notes from an introductory graduate level course at the University of Louisville. This introductory text communicates an elementary exposition of valued functional equations where the unknown functions take on real or complex values. In order to make the presentation as manageable as possible for students from a variety of disciplines, the book chooses not to focus on functional equations where the unknown functions take on values on algebraic structures such as groups, rings, or fields. However, each chapter includes sections highlighting various developments of the main equations treated in that chapter. For advanced students, the book introduces functional equations in abstract domains like semigroups, groups, and Banach spaces. Functional equations covered include: Cauchy Functional Equations and Applications The Jensen Functional Equation Pexider's Functional Equation Quadratic Functional Equation D'Alembert Functional Equation Trigonometric Functional Equations Pompeiu Functional Equation Hosszu Functional Equation Davison Functional Equation Abel Functional Equation Mean Value Type Functional Equations Functional Equations for Distance Measures The innovation of solving functional equations lies in finding the right tricks for a particular equation. Accessible and rooted in current theory, methods, and research, this book sharpens mathematical competency and prepares students of mathematics and engineering for further work in advanced functional equations.

Functional Equations, Inequalities and Applications


Author: Themistocles Rassias
Publisher: Springer Science & Business Media
ISBN: 940170225X
Category: Mathematics
Page: 224
View: 8353

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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.

Functional Equations on Groups


Author: Henrik Stetkær
Publisher: World Scientific
ISBN: 9814513148
Category: Mathematics
Page: 396
View: 1082

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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 the topic and contains new and transparent proofs. Its scope extends from the classical functional equations on the real line to those on groups, in particular, non-abelian groups. This volume presents, in careful detail, a number of illustrative examples like the cosine equation on the Heisenberg group and on the group SL(2, ℝ). Some of the examples are not even seen in existing monographs. Thus, it is an essential source of reference for further investigations. Contents:IntroductionAround the Additive Cauchy EquationThe Multiplicative Cauchy EquationAddition and Subtraction FormulasLevi–Civita's Functional EquationThe Symmetrized Sine Addition FormulaEquations with Symmetric Right Hand SideThe Pre-d'Alembert Functional EquationD'Alembert's Functional EquationD'Alembert's Long Functional EquationWilson's Functional EquationJensen's Functional EquationThe Quadratic Functional EquationK-Spherical FunctionsThe Sine Functional EquationThe Cocycle EquationAppendices:Basic Terminology and ResultsSubstitutes for CommutativityThe Casorati DeterminantRegularityMatrix-Coefficients of RepresentationsThe Small Dimension LemmaGroup Cohomology Readership: Advanced undergraduates, graduates and professional mathematicians interested in harmonic analysis and/or functional equations. Keywords:Functional Equation;Group;Harmonic AnalysisKey Features:Most of the material of the book can be found only in research papers, so it is a good source of referenceThe book is self-contained and provides the necessary background material needed to go further into the subject and to explore the research literatureThe book may be used as a textbook for graduate students and even ambitious undergraduate in mathematics, because it presents the material in an accessible way, originating from a course for students at master's levelExercises at the end of each chapter, some with answers, help to provide more examples to enable the student to grasp the topic betterReviews: “It is an excellent, well composed and self-contained monograph, written in good and clear English. It can serve as a complete and independent introduction to the field of trigonometric functional equations and as an excellent source of suitable references for further study. The scope of solutions considered extends from real functions to those acting between groups, also non-abelian.” Prof Janusz Brzdęk Pedagogical University Kraków, Poland “The book is written as an accessible introduction to trigonometric functional equations for graduate students and working mathematicians. It gives a very readable account of recent research in the area, as well as more than 200 references for further study. It would make an excellent textbook for a graduate course on the topic. This monograph is a valuable contribution to the literature on functional equations.” Zentralblatt MATH

Iterative Functional Equations


Author: Marek Kuczma,Bogdan Choczewski,Roman Ger
Publisher: Cambridge University Press
ISBN: 9780521355612
Category: Mathematics
Page: 552
View: 7996

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A cohesive and comprehensive account of the modern theory of iterative functional equations. Many of the results included have appeared before only in research literature, making this an essential volume for all those working in functional equations and in such areas as dynamical systems and chaos, to which the theory is closely related. The authors introduce the reader to the theory and then explore the most recent developments and general results. Fundamental notions such as the existence and uniqueness of solutions to the equations are stressed throughout, as are applications of the theory to such areas as branching processes, differential equations, ergodic theory, functional analysis and geometry. Other topics covered include systems of linear and nonlinear equations of finite and infinite ORD various function classes, conjugate and commutable functions, linearization, iterative roots of functions, and special functional equations.

On Applications and Theory of Functional Equations


Author: J. Aczél
Publisher: Academic Press
ISBN: 1483262650
Category: Mathematics
Page: 64
View: 1702

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On Applications and Theory of Functional Equations focuses on the principles and advancement of numerical approaches used in functional equations. The publication first offers information on the history of functional equations, noting that the research on functional equations originated in problems related to applied mathematics. The text also highlights the influence of J. d'Alembert, S. D. Poisson, E. Picard, and A. L. Cauchy in promoting the processes of numerical analyses involving functional equations. The role of vectors in solving functional equations is also noted. The book ponders on the international Fifth Annual Meeting on Functional Equations, held in Waterloo, Ontario, Canada on April 24-30, 1967. The meeting gathered participants from America, Asia, Australia, and Europe. One of the topics presented at the meeting focuses on the survey of materials dealing with the progress of approaches in the processes and methodologies involved in solving problems dealing with functional equations. The influence, works, and contributions of A. L. Cauchy, G. Darboux, and G. S. Young to the field are also underscored. The publication is a valuable reference for readers interested in functional equations.

Mean Value Theorems and Functional Equations


Author: Prasanna Sahoo,Thomas Riedel
Publisher: World Scientific
ISBN: 9789810235444
Category: Mathematics
Page: 245
View: 1516

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This book takes a comprehensive look at mean value theorems and their connection with functional equations. Besides the traditional Lagrange and Cauchy mean value theorems, it covers the Pompeiu and the Flett mean value theorems as well as extension to higher dimensions and the complex plane. Furthermore the reader is introduced to the field of functional equations through equations that arise in connection with the many mean value theorems discussed.

Theory of Approximate Functional Equations

In Banach Algebras, Inner Product Spaces and Amenable Groups
Author: Madjid Eshaghi Gordji,Sadegh Abbaszadeh
Publisher: Academic Press
ISBN: 012803971X
Category: Mathematics
Page: 148
View: 4294

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Presently no other book deals with the stability problem of functional equations in Banach algebras, inner product spaces and amenable groups. Moreover, in most stability theorems for functional equations, the completeness of the target space of the unknown functions contained in the equation is assumed. Recently, the question, whether the stability of a functional equation implies this completeness, has been investigated by several authors. In this book the authors investigate these developments in the theory of approximate functional equations. A useful text for graduate seminars and of interest to a wide audience including mathematicians and applied researchers Presents recent developments in the theory of approximate functional equations Discusses the stability problem of functional equations in Banach algebras, inner product spaces and amenable groups

Functions and Graphs


Author: I. M. Gelfand,E. G. Glagoleva,E. E. Shnol
Publisher: Courier Corporation
ISBN: 0486317137
Category: Mathematics
Page: 112
View: 5813

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This text demonstrates the fundamentals of graph theory. The 1st part employs simple functions to analyze basics; 2nd half deals with linear functions, quadratic trinomials, linear fractional functions, power functions, rational functions. 1969 edition.

Functional Equations in Probability Theory


Author: Ramachandran Balasubrahmanyan,Ka-Sing Lau
Publisher: Elsevier
ISBN: 1483272222
Category: Mathematics
Page: 268
View: 7337

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Functional Equations in Probability Theory deals with functional equations in probability theory and covers topics ranging from the integrated Cauchy functional equation (ICFE) to stable and semistable laws. The problem of identical distribution of two linear forms in independent and identically distributed random variables is also considered, with particular reference to the context of the common distribution of these random variables being normal. Comprised of nine chapters, this volume begins with an introduction to Cauchy functional equations as well as distribution functions and characteristic functions. The discussion then turns to the nonnegative solutions of ICFE on R+; ICFE with a signed measure; and application of ICFE to the characterization of probability distributions. Subsequent chapters focus on stable and semistable laws; ICFE with error terms on R+; independent/identically distributed linear forms and the normal laws; and distribution problems relating to the arc-sine, the normal, and the chi-square laws. The final chapter is devoted to ICFE on semigroups of Rd. This book should be of interest to mathematicians and statisticians.

Functional Equations in Several Variables


Author: J. Aczel,Jean G. Dhombres
Publisher: Cambridge University Press
ISBN: 9780521352765
Category: Mathematics
Page: 462
View: 9713

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Deals with modern theory of functional equations in several variables and their applications to mathematics, information theory, and the natural, behavioral, and social sciences. The authors emphasize applications, although not at the expense of theory, and have kept the prerequisites to a minimum; the reader should be familiar with calculus and some simple structures of algebra and have a basic knowledge of Lebesque integration. For the applications the authors have included references and explained the results used. The book is designed so that the chapters may be read almost independently of each other, enabling a selection of material to be chosen for introductory and advanced courses. Each chapter concludes with exercises and further results, 400 in all, which extend and test the material presented in the text. The history of functional equations is well documented in a final chapter which is complemented by an encyclopedic bibliography of over 1600 items. This volume will be of interest to professionals and graduate students in pure and applied mathematics.