Sorin G Gal's Books

Global Smoothness and Shape Preserving Interpolation by Classical Operators

By: Sorin G. Gal

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Global Smoothness and Shape Preserving Interpolation by Classical Operators

By: Sorin G. Gal

This monograph examines in detail two aspects in the field of interpolation of functions -the Global Smoothness Preservation Property (GSPP) and the Shape Preservation Property (SPP). By considering well-known classical interpolation operators such as Lagrange, Grünwald, Hermite-Fejér and Shepard type, the study is mainly developed for the univariate and bivariate cases. One of the first books on the subject, it presents to the reader, recent work featuring many new interesting results in this field, including an excellent survey of past research. Accompanied by numerous open problems, an updated set of references, and an appendix featuring illustrations of nine types of Shepard surfaces, this unique text is best suited to graduate students and researchers in mathematical analysis, interpolation of functions, pure and applied mathematicians in numerical analysis, approximation theory, data fitting, computer aided geometric design, fluid mechanics, and engineering researchers.

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Page Count

155

Publisher

Springer Science & Business Media

Published Date

ISBN 10

0817644016

ISBN 13

9780817644017

Evolution Equations With A Complex Spatial Variable

By: Ciprian G Gal,Sorin G Gal,Jerome A Goldstein

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Evolution Equations With A Complex Spatial Variable

By: Ciprian G Gal,Sorin G Gal,Jerome A Goldstein

This book investigates several classes of partial differential equations of real time variable and complex spatial variables, including the heat, Laplace, wave, telegraph, Burgers, Black-Merton-Scholes, Schrödinger and Korteweg-de Vries equations.The complexification of the spatial variable is done by two different methods. The first method is that of complexifying the spatial variable in the corresponding semigroups of operators. In this case, the solutions are studied within the context of the theory of semigroups of linear operators. It is also interesting to observe that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness. The second method is that of complexifying the spatial variable directly in the corresponding evolution equation from the real case. More precisely, the real spatial variable is replaced by a complex spatial variable in the corresponding evolution equation and then analytic and non-analytic solutions are sought.For the first time in the book literature, we aim to give a comprehensive study of the most important evolution equations of real time variable and complex spatial variables. In some cases, potential physical interpretations are presented. The generality of the methods used allows the study of evolution equations of spatial variables in general domains of the complex plane.

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202

Publisher

World Scientific

Published Date

ISBN 10

9814590614

ISBN 13

9789814590617

Approximation by Max-Product Type Operators

By: Barnabás Bede,Lucian Coroianu,Sorin G. Gal

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Approximation by Max-Product Type Operators

By: Barnabás Bede,Lucian Coroianu,Sorin G. Gal

This monograph presents a broad treatment of developments in an area of constructive approximation involving the so-called "max-product" type operators. The exposition highlights the max-product operators as those which allow one to obtain, in many cases, more valuable estimates than those obtained by classical approaches. The text considers a wide variety of operators which are studied for a number of interesting problems such as quantitative estimates, convergence, saturation results, localization, to name several. Additionally, the book discusses the perfect analogies between the probabilistic approaches of the classical Bernstein type operators and of the classical convolution operators (non-periodic and periodic cases), and the possibilistic approaches of the max-product variants of these operators. These approaches allow for two natural interpretations of the max-product Bernstein type operators and convolution type operators: firstly, as possibilistic expectations of some fuzzy variables, and secondly, as bases for the Feller type scheme in terms of the possibilistic integral. These approaches also offer new proofs for the uniform convergence based on a Chebyshev type inequality in the theory of possibility. Researchers in the fields of approximation of functions, signal theory, approximation of fuzzy numbers, image processing, and numerical analysis will find this book most beneficial. This book is also a good reference for graduates and postgraduates taking courses in approximation theory.

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468

Publisher

Springer

Published Date

ISBN 10

3319341898

ISBN 13

9783319341897

Approximation Theory

Moduli of Continuity and Global Smoothness Preservation

By: George A. Anastassiou,Sorin Gal

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Approximation Theory

Moduli of Continuity and Global Smoothness Preservation

By: George A. Anastassiou,Sorin Gal

We study in Part I of this monograph the computational aspect of almost all moduli of continuity over wide classes of functions exploiting some of their convexity properties. To our knowledge it is the first time the entire calculus of moduli of smoothness has been included in a book. We then present numerous applications of Approximation Theory, giving exact val ues of errors in explicit forms. The K-functional method is systematically avoided since it produces nonexplicit constants. All other related books so far have allocated very little space to the computational aspect of moduli of smoothness. In Part II, we study/examine the Global Smoothness Preservation Prop erty (GSPP) for almost all known linear approximation operators of ap proximation theory including: trigonometric operators and algebraic in terpolation operators of Lagrange, Hermite-Fejer and Shepard type, also operators of stochastic type, convolution type, wavelet type integral opera tors and singular integral operators, etc. We present also a sufficient general theory for GSPP to hold true. We provide a great variety of applications of GSPP to Approximation Theory and many other fields of mathemat ics such as Functional analysis, and outside of mathematics, fields such as computer-aided geometric design (CAGD). Most of the time GSPP meth ods are optimal. Various moduli of smoothness are intensively involved in Part II. Therefore, methods from Part I can be used to calculate exactly the error of global smoothness preservation. It is the first time in the literature that a book has studied GSPP.

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554

Publisher

Springer Science & Business Media

Published Date

ISBN 10

0817641513

ISBN 13

9780817641511

Approximation By Complex Bernstein And Convolution Type Operators

By: Sorin G Gal

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Approximation By Complex Bernstein And Convolution Type Operators

By: Sorin G Gal

The monograph, as its first main goal, aims to study the overconvergence phenomenon of important classes of Bernstein-type operators of one or several complex variables, that is, to extend their quantitative convergence properties to larger sets in the complex plane rather than the real intervals. The operators studied are of the following types: Bernstein, Bernstein—Faber, Bernstein-Butzer, q-Bernstein, Bernstein-Stancu, Bernstein-Kantorovich, Favard-Szász-Mirakjan, Baskakov and Balázs-Szabados.The second main objective is to provide a study of the approximation and geometric properties of several types of complex convolutions: the de la Vallée Poussin, Fejér, Riesz-Zygmund, Jackson, Rogosinski, Picard, Poisson-Cauchy, Gauss-Weierstrass, q-Picard, q-Gauss-Weierstrass, Post-Widder, rotation-invariant, Sikkema and nonlinear. Several applications to partial differential equations (PDEs) are also presented.Many of the open problems encountered in the studies are proposed at the end of each chapter. For further research, the monograph suggests and advocates similar studies for other complex Bernstein-type operators, and for other linear and nonlinear convolutions.

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Page Count

350

Publisher

World Scientific

Published Date

ISBN 10

9814466972

ISBN 13

9789814466974

Overconvergence in Complex Approximation

By: Sorin G. Gal

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Overconvergence in Complex Approximation

By: Sorin G. Gal

This monograph deals with the quantitative overconvergence phenomenon in complex approximation by various operators. The book is divided into three chapters. First, the results for the Schurer-Faber operator, Beta operators of first kind, Bernstein-Durrmeyer-type operators and Lorentz operator are presented. The main focus is on results for several q-Bernstein kind of operators with q > 1, when the geometric order of approximation 1/qn is obtained not only in complex compact disks but also in quaternion compact disks and in other compact subsets of the complex plane. The focus then shifts to quantitative overconvergence and convolution overconvergence results for the complex potentials generated by the Beta and Gamma Euler's functions. Finally quantitative overconvergence results for the most classical orthogonal expansions (of Chebyshev, Legendre, Hermite, Laguerre and Gegenbauer kinds) attached to vector-valued functions are presented. Each chapter concludes with a notes and open problems section, thus providing stimulation for further research. An extensive bibliography and index complete the text. This book is suitable for researchers and graduate students working in complex approximation and its applications, mathematical analysis and numerical analysis.

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206

Publisher

Springer Science & Business Media

Published Date

ISBN 10

1461470986

ISBN 13

9781461470984

Shape-Preserving Approximation by Real and Complex Polynomials

By: Sorin G. Gal

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Shape-Preserving Approximation by Real and Complex Polynomials

By: Sorin G. Gal

First comprehensive treatment in book form of shape-preserving approximation by real or complex polynomials in one or several variables Of interest to grad students and researchers in approximation theory, mathematical analysis, numerical analysis, Computer Aided Geometric Design, robotics, data fitting, chemistry, fluid mechanics, and engineering Contains many open problems to spur future research Rich and updated bibliography

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Page Count

359

Publisher

Springer Science & Business Media

Published Date

ISBN 10

0817647031

ISBN 13

9780817647032

Introduction to Geometric Function Theory of Hypercomplex Variables

By: Sorin G. Gal

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Introduction to Geometric Function Theory of Hypercomplex Variables

By: Sorin G. Gal

Introduction to Geometric Function Theory of Hypercomplex Variables

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340

Publisher

Nova Publishers

Published Date

ISBN 10

1590333640

ISBN 13

9781590333648

Defects Of Properties In Mathematics: Quantitative Characterizations

By: Adrian I Ban,Sorin G Gal

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Defects Of Properties In Mathematics: Quantitative Characterizations

By: Adrian I Ban,Sorin G Gal

This book introduces a method of research which can be used in various fields of mathematics. It examines, in a systematic way, the quantitative characterizations of the “deviation from a (given) property”, called the “defect of a property”, in: set theory; topology; measure theory; real, complex and functional analysis; algebra; geometry; number theory; fuzzy mathematics.Besides well-known “defects”, the book introduces and studies new ones, such as: measures of noncompactness for fuzzy sets; fuzzy and intuitionistic entropies; the defect of (sub, super)additivity; complementarity; monotonicity for set functions; the defect of convexity; monotonicity; differentiability for real functions; the defect of equality for inequalities; the defect of orthogonality for sets and defects of properties for linear operators in normed spaces; defects of properties (commutativity, associativity, etc.) for binary operations; defects of orthogonality and parallelness in Euclidean and non-Euclidean geometries; defects of integer, perfect, prime and amicable numbers; the defect of tautology in fuzzy logic.

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365

Publisher

World Scientific

Published Date

ISBN 10

9814488771

ISBN 13

9789814488778

Quaternionic Approximation

With Application to Slice Regular Functions

By: Sorin G. Gal,Irene Sabadini

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Quaternionic Approximation

With Application to Slice Regular Functions

By: Sorin G. Gal,Irene Sabadini

This book presents the extensions to the quaternionic setting of some of the main approximation results in complex analysis. It also includes the main inequalities regarding the behavior of the derivatives of polynomials with quaternionic cofficients. With some few exceptions, all the material in this book belongs to recent research of the authors on the approximation of slice regular functions of a quaternionic variable. The book is addressed to researchers in various areas of mathematical analysis, in particular hypercomplex analysis, and approximation theory. It is accessible to graduate students and suitable for graduate courses in the above framework.

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228

Publisher

Springer

Published Date

ISBN 10

3030106667

ISBN 13

9783030106669

Chemical Process Design

Computer-Aided Case Studies

By: Alexandre C. Dimian,Costin Sorin Bildea

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Chemical Process Design

Computer-Aided Case Studies

By: Alexandre C. Dimian,Costin Sorin Bildea

This practical how-to-do book deals with the design of sustainable chemical processes by means of systematic methods aided by computer simulation. Ample case studies illustrate generic creative issues, as well as the efficient use of simulation techniques, with each one standing for an important issue taken from practice. The didactic approach guides readers from basic knowledge to mastering complex flow-sheets, starting with chemistry and thermodynamics, via process synthesis, efficient use of energy and waste minimization, right up to plant-wide control and process dynamics. The simulation results are compared with flow-sheets and performance indices of actual industrial licensed processes, while the complete input data for all the case studies is also provided, allowing readers to reproduce the results with their own simulators. For everyone interested in the design of innovative chemical processes.

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