*Cauchy's Equation and Jensen's Inequality*

**Author**: Marek Kuczma

**Publisher:**Springer Science & Business Media

**ISBN:**3764387491

**Category:**Mathematics

**Page:**595

**View:**7384

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# Search Results for: an-introduction-to-the-theory-of-functional-equations-and-inequalities-second-edition-cauchy-s-equation-and-jensen-s-inequality

*Cauchy's Equation and Jensen's Inequality*

**Author**: Marek Kuczma

**Publisher:** Springer Science & Business Media

**ISBN:** 3764387491

**Category:** Mathematics

**Page:** 595

**View:** 7384

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)
*Cauchy's Equation and Jensen's Inequality*

**Author**: Marek Kuczma

**Publisher:** Springer Science & Business Media

**ISBN:** 3764387491

**Category:** Mathematics

**Page:** 595

**View:** 2573

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)

**Author**: Miklós Laczkovich

**Publisher:** Cambridge University Press

**ISBN:** 9780883857229

**Category:** Mathematics

**Page:** 118

**View:** 6576

How to prove interesting and deep mathematical results from first principles, with exercises.
*Functional Inequalities*

**Author**: Themistocles M. Rassias

**Publisher:** Springer

**ISBN:** 1493912461

**Category:** Mathematics

**Page:** 555

**View:** 5702

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, theoretical, and aesthetic.” On the aesthetic aspects, he said, “As has been pointed out, beauty is in the eye of the beholder. However, it is generally agreed that certain pieces of music, art, or mathematics are beautiful. There is an elegance to inequalities that makes them very attractive.” The content of the Handbook focuses mainly on both old and recent developments on approximate homomorphisms, on a relation between the Hardy–Hilbert and the Gabriel inequality, generalized Hardy–Hilbert type inequalities on multiple weighted Orlicz spaces, half-discrete Hilbert-type inequalities, on affine mappings, on contractive operators, on multiplicative Ostrowski and trapezoid inequalities, Ostrowski type inequalities for the Riemann–Stieltjes integral, means and related functional inequalities, Weighted Gini means, controlled additive relations, Szasz–Mirakyan operators, extremal problems in polynomials and entire functions, applications of functional equations to Dirichlet problem for doubly connected domains, nonlinear elliptic problems depending on parameters, on strongly convex functions, as well as applications to some new algorithms for solving general equilibrium problems, inequalities for the Fisher’s information measures, financial networks, mathematical models of mechanical fields in media with inclusions and holes.

**Author**: Anthony Louis Almudevar

**Publisher:** CRC Press

**ISBN:** 0203503414

**Category:** Computers

**Page:** 372

**View:** 3804

Iterative algorithms often rely on approximate evaluation techniques, which may include statistical estimation, computer simulation or functional approximation. This volume presents methods for the study of approximate iterative algorithms, providing tools for the derivation of error bounds and convergence rates, and for the optimal design of such algorithms. Techniques of functional analysis are used to derive analytical relationships between approximation methods and convergence properties for general classes of algorithms. This work provides the necessary background in functional analysis and probability theory. Extensive applications to Markov decision processes are presented. This volume is intended for mathematicians, engineers and computer scientists, who work on learning processes in numerical analysis and are involved with optimization, optimal control, decision analysis and machine learning.
*An Introduction With Applications*

**Author**: Martin Bohner,Allan Peterson

**Publisher:** Springer Science & Business Media

**ISBN:** 9780817642259

**Category:** Computers

**Page:** 358

**View:** 6007

The study of dynamic equations on a measure chain (time scale) goes back to its founder S. Hilger (1988), and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on measure chains can build bridges between continuous and discrete mathematics. Further, the study of measure chain theory has led to several important applications, e.g., in the study of insect population models, neural networks, heat transfer, and epidemic models. Key features of the book: * Introduction to measure chain theory; discussion of its usefulness in allowing for the simultaneous development of differential equations and difference equations without having to repeat analogous proofs * Many classical formulas or procedures for differential and difference equations cast in a new light * New analogues of many of the "special functions" studied * Examination of the properties of the "exponential function" on time scales, which can be defined and investigated using a particularly simple linear equation * Additional topics covered: self-adjoint equations, linear systems, higher order equations, dynamic inequalities, and symplectic dynamic systems * Clear, motivated exposition, beginning with preliminaries and progressing to more sophisticated text * Ample examples and exercises throughout the book * Solutions to selected problems Requiring only a first semester of calculus and linear algebra, Dynamic Equations on Time Scales may be considered as an interesting approach to differential equations via exposure to continuous and discrete analysis. This approach provides an early encounter with many applications in such areas as biology, physics, and engineering. Parts of the book may be used in a special topics seminar at the senior undergraduate or beginning graduate levels. Finally, the work may serve as a reference to stimulate the development of new kinds of equations with potentially new applications.
*An Introduction to the Art of Mathematical Inequalities*

**Author**: J. Michael Steele

**Publisher:** Cambridge University Press

**ISBN:** 9780521546775

**Category:** Mathematics

**Page:** 306

**View:** 7941

This lively, problem-oriented text, first published in 2004, is designed to coach readers toward mastery of the most fundamental mathematical inequalities. With the Cauchy-Schwarz inequality as the initial guide, the reader is led through a sequence of fascinating problems whose solutions are presented as they might have been discovered - either by one of history's famous mathematicians or by the reader. The problems emphasize beauty and surprise, but along the way readers will find systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and the inequalities of Hölder, Hilbert, and Hardy. The text is accessible to anyone who knows calculus and who cares about solving problems. It is well suited to self-study, directed study, or as a supplement to courses in analysis, probability, and combinatorics.

**Author**: John B. Conway

**Publisher:** Springer Science & Business Media

**ISBN:** 1461263131

**Category:** Mathematics

**Page:** 322

**View:** 3389

"This book presents a basic introduction to complex analysis in both an interesting and a rigorous manner. It contains enough material for a full year's course, and the choice of material treated is reasonably standard and should be satisfactory for most first courses in complex analysis. The approach to each topic appears to be carefully thought out both as to mathematical treatment and pedagogical presentation, and the end result is a very satisfactory book." --MATHSCINET

**Author**: Alexander Schrijver

**Publisher:** John Wiley & Sons

**ISBN:** 9780471982326

**Category:** Mathematics

**Page:** 484

**View:** 707

Theory of Linear and Integer Programming Alexander Schrijver Centrum voor Wiskunde en Informatica, Amsterdam, The Netherlands This book describes the theory of linear and integer programming and surveys the algorithms for linear and integer programming problems, focusing on complexity analysis. It aims at complementing the more practically oriented books in this field. A special feature is the author's coverage of important recent developments in linear and integer programming. Applications to combinatorial optimization are given, and the author also includes extensive historical surveys and bibliographies. The book is intended for graduate students and researchers in operations research, mathematics and computer science. It will also be of interest to mathematical historians. Contents 1 Introduction and preliminaries; 2 Problems, algorithms, and complexity; 3 Linear algebra and complexity; 4 Theory of lattices and linear diophantine equations; 5 Algorithms for linear diophantine equations; 6 Diophantine approximation and basis reduction; 7 Fundamental concepts and results on polyhedra, linear inequalities, and linear programming; 8 The structure of polyhedra; 9 Polarity, and blocking and anti-blocking polyhedra; 10 Sizes and the theoretical complexity of linear inequalities and linear programming; 11 The simplex method; 12 Primal-dual, elimination, and relaxation methods; 13 Khachiyan's method for linear programming; 14 The ellipsoid method for polyhedra more generally; 15 Further polynomiality results in linear programming; 16 Introduction to integer linear programming; 17 Estimates in integer linear programming; 18 The complexity of integer linear programming; 19 Totally unimodular matrices: fundamental properties and examples; 20 Recognizing total unimodularity; 21 Further theory related to total unimodularity; 22 Integral polyhedra and total dual integrality; 23 Cutting planes; 24 Further methods in integer linear programming; Historical and further notes on integer linear programming; References; Notation index; Author index; Subject index

**Author**: Morris Marden

**Publisher:** American Mathematical Soc.

**ISBN:** 0821815032

**Category:** Electronic books

**Page:** 243

**View:** 369

During the years since the first edition of this well-known monograph appeared, the subject (the geometry of the zeros of a complex polynomial) has continued to display the same outstanding vitality as it did in the first 150 years of its history, beginning with the contributions of Cauchy and Gauss. Thus, the number of entries in the bibliography of this edition had to be increased from about 300 to about 600 and the book enlarged by one third. It now includes a more extensive treatment of Hurwitz polynomials and other topics. The new material on infrapolynomials, abstract polynomials, and matrix methods is of particular interest.
*From Theory to Algorithms*

**Author**: Shai Shalev-Shwartz,Shai Ben-David

**Publisher:** Cambridge University Press

**ISBN:** 1107057132

**Category:** Computers

**Page:** 409

**View:** 9472

Introduces machine learning and its algorithmic paradigms, explaining the principles behind automated learning approaches and the considerations underlying their usage.
*An Introduction to Random Fields*

**Author**: Davar Khoshnevisan

**Publisher:** Springer Science & Business Media

**ISBN:** 0387216316

**Category:** Mathematics

**Page:** 584

**View:** 8773

Self-contained presentation: from elementary material to state-of-the-art research; Much of the theory in book-form for the first time; Connections are made between probability and other areas of mathematics, engineering and mathematical physics

**Author**: David J. C. MacKay

**Publisher:** Cambridge University Press

**ISBN:** 9780521642989

**Category:** Computers

**Page:** 628

**View:** 9235

Fun and exciting textbook on the mathematics underpinning the most dynamic areas of modern science and engineering.
*Theory and Examples*

**Author**: Rick Durrett

**Publisher:** Cambridge University Press

**ISBN:** 113949113X

**Category:** Mathematics

**Page:** N.A

**View:** 4266

This classic introduction to probability theory for beginning graduate students covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems. The fourth edition begins with a short chapter on measure theory to orient readers new to the subject.

**Author**: Peter Bullen

**Publisher:** CRC Press

**ISBN:** 1482237628

**Category:** Mathematics

**Page:** 405

**View:** 1318

Adding new results that have appeared in the last 15 years, Dictionary of Inequalities, Second Edition provides an easy way for researchers to locate an inequality by name or subject. This edition offers an up-to-date, alphabetical listing of each inequality with a short statement of the result, some comments, references to related inequalities, and sources of information on proofs and other details. The book does not include proofs and uses basic mathematical terminology as much as possible, enabling readers to access a result or inequality effortlessly. New to the Second Edition More than 100 new inequalities, including recently discovered ones Updated inequalities according to the most recent research Inclusion of a name index Updated bibliography that contains URLs for important references The book mainly presents the most common version of the inequality and later gives more general results as extensions or variants. Inequalities that exist at various levels of generality are presented in the simplest form with the other forms as extensions or under a different heading. The author also clarifies any non-standard notations and includes cross-references for transliterations.

**Author**: Dragoslav S. Mitrinovic,J. Pecaric,A.M Fink

**Publisher:** Springer Science & Business Media

**ISBN:** 9401710430

**Category:** Mathematics

**Page:** 740

**View:** 3784

**Author**: Marek Kuczma,Bogdan Choczewski,Roman Ger

**Publisher:** Cambridge University Press

**ISBN:** 9780521355612

**Category:** Mathematics

**Page:** 552

**View:** 9298

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.

**Author**: Ronald G. Douglas

**Publisher:** Springer Science & Business Media

**ISBN:** 1461216567

**Category:** Mathematics

**Page:** 198

**View:** 3948

A discussion of certain advanced topics in operator theory, providing the necessary background while assuming only standard senior-first year graduate courses in general topology, measure theory, and algebra. Each chapter ends with source notes which suggest additional reading along with comments on who proved what and when, followed by a large number of problems of varying difficulty. This new edition will appeal to a whole new generation of students seeking an introduction to this topic.
*Revised Edition*

**Author**: Yousef Saad

**Publisher:** SIAM

**ISBN:** 9781611970739

**Category:** Eigenvalues

**Page:** 276

**View:** 9665

This revised edition discusses numerical methods for computing eigenvalues and eigenvectors of large sparse matrices. It provides an in-depth view of the numerical methods that are applicable for solving matrix eigenvalue problems that arise in various engineering and scientific applications. Each chapter was updated by shortening or deleting outdated topics, adding topics of more recent interest, and adapting the Notes and References section. Significant changes have been made to Chapters 6 through 8, which describe algorithms and their implementations and now include topics such as the implicit restart techniques, the Jacobi-Davidson method, and automatic multilevel substructuring.

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