Convergence Methods for Double Sequences and Applications

Convergence Methods for Double Sequences and Applications

Author: M. Mursaleen

Publisher: Springer Science & Business Media

Published: 2013-10-17

Total Pages: 174

ISBN-13: 8132216113

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This book exclusively deals with the study of almost convergence and statistical convergence of double sequences. The notion of “almost convergence” is perhaps the most useful notion in order to obtain a weak limit of a bounded non-convergent sequence. There is another notion of convergence known as the “statistical convergence”, introduced by H. Fast, which is an extension of the usual concept of sequential limits. This concept arises as an example of “convergence in density” which is also studied as a summability method. Even unbounded sequences can be dealt with by using this method. The book also discusses the applications of these non-matrix methods in approximation theory. Written in a self-contained style, the book discusses in detail the methods of almost convergence and statistical convergence for double sequences along with applications and suitable examples. The last chapter is devoted to the study convergence of double series and describes various convergence tests analogous to those of single sequences. In addition to applications in approximation theory, the results are expected to find application in many other areas of pure and applied mathematics such as mathematical analysis, probability, fixed point theory and statistics.


Soft Computing Techniques in Engineering, Health, Mathematical and Social Sciences

Soft Computing Techniques in Engineering, Health, Mathematical and Social Sciences

Author: Pradip Debnath

Publisher: CRC Press

Published: 2021-07-15

Total Pages: 232

ISBN-13: 1000409813

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Soft computing techniques are no longer limited to the arena of computer science. The discipline has an exponentially growing demand in other branches of science and engineering and even into health and social science. This book contains theory and applications of soft computing in engineering, health, and social and applied sciences. Different soft computing techniques such as artificial neural networks, fuzzy systems, evolutionary algorithms and hybrid systems are discussed. It also contains important chapters in machine learning and clustering. This book presents a survey of the existing knowledge and also the current state of art development through original new contributions from the researchers. This book may be used as a one-stop reference book for a broad range of readers worldwide interested in soft computing. In each chapter, the preliminaries have been presented first and then the advanced discussion takes place. Learners and researchers from a wide variety of backgrounds will find several useful tools and techniques to develop their soft computing skills. This book is meant for graduate students, faculty and researchers willing to expand their knowledge in any branch of soft computing. The readers of this book will require minimum prerequisites of undergraduate studies in computation and mathematics.


Current Trends in Mathematical Analysis and Its Interdisciplinary Applications

Current Trends in Mathematical Analysis and Its Interdisciplinary Applications

Author: Hemen Dutta

Publisher: Springer Nature

Published: 2019-08-23

Total Pages: 912

ISBN-13: 3030152421

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This book explores several important aspects of recent developments in the interdisciplinary applications of mathematical analysis (MA), and highlights how MA is now being employed in many areas of scientific research. Each of the 23 carefully reviewed chapters was written by experienced expert(s) in respective field, and will enrich readers’ understanding of the respective research problems, providing them with sufficient background to understand the theories, methods and applications discussed. The book’s main goal is to highlight the latest trends and advances, equipping interested readers to pursue further research of their own. Given its scope, the book will especially benefit graduate and PhD students, researchers in the applied sciences, educators, and engineers with an interest in recent developments in the interdisciplinary applications of mathematical analysis.


Applied Mathematical Analysis: Theory, Methods, and Applications

Applied Mathematical Analysis: Theory, Methods, and Applications

Author: Hemen Dutta

Publisher: Springer

Published: 2019-02-21

Total Pages: 809

ISBN-13: 3319999184

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This book addresses key aspects of recent developments in applied mathematical analysis and its use. It also highlights a broad range of applications from science, engineering, technology and social perspectives. Each chapter investigates selected research problems and presents a balanced mix of theory, methods and applications for the chosen topics. Special emphasis is placed on presenting basic developments in applied mathematical analysis, and on highlighting the latest advances in this research area. The book is presented in a self-contained manner as far as possible, and includes sufficient references to allow the interested reader to pursue further research in this still-developing field. The primary audience for this book includes graduate students, researchers and educators; however, it will also be useful for general readers with an interest in recent developments in applied mathematical analysis and applications.


Advances in Summability and Approximation Theory

Advances in Summability and Approximation Theory

Author: S. A. Mohiuddine

Publisher: Springer

Published: 2018-12-30

Total Pages: 248

ISBN-13: 9811330778

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This book discusses the Tauberian conditions under which convergence follows from statistical summability, various linear positive operators, Urysohn-type nonlinear Bernstein operators and also presents the use of Banach sequence spaces in the theory of infinite systems of differential equations. It also includes the generalization of linear positive operators in post-quantum calculus, which is one of the currently active areas of research in approximation theory. Presenting original papers by internationally recognized authors, the book is of interest to a wide range of mathematicians whose research areas include summability and approximation theory. One of the most active areas of research in summability theory is the concept of statistical convergence, which is a generalization of the familiar and widely investigated concept of convergence of real and complex sequences, and it has been used in Fourier analysis, probability theory, approximation theory and in other branches of mathematics. The theory of approximation deals with how functions can best be approximated with simpler functions. In the study of approximation of functions by linear positive operators, Bernstein polynomials play a highly significant role due to their simple and useful structure. And, during the last few decades, different types of research have been dedicated to improving the rate of convergence and decreasing the error of approximation.


Measure of Noncompactness, Fixed Point Theorems, and Applications

Measure of Noncompactness, Fixed Point Theorems, and Applications

Author: S. A. Mohiuddine

Publisher: CRC Press

Published: 2024-04-24

Total Pages: 205

ISBN-13: 1040013325

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The theory of the measure of noncompactness has proved its significance in various contexts, particularly in the study of fixed point theory, differential equations, functional equations, integral and integrodifferential equations, optimization, and others. This edited volume presents the recent developments in the theory of the measure of noncompactness and its applications in pure and applied mathematics. It discusses important topics such as measures of noncompactness in the space of regulated functions, application in nonlinear infinite systems of fractional differential equations, and coupled fixed point theorem. Key Highlights: Explains numerical solution of functional integral equation through coupled fixed point theorem, measure of noncompactness and iterative algorithm Showcases applications of the measure of noncompactness and Petryshyn’s fixed point theorem functional integral equations in Banach algebra Explores the existence of solutions of the implicit fractional integral equation via extension of the Darbo’s fixed point theorem Discusses best proximity point results using measure of noncompactness and its applications Includes solvability of some fractional differential equations in the holder space and their numerical treatment via measures of noncompactness This reference work is for scholars and academic researchers in pure and applied mathematics.


Applied Summability Methods

Applied Summability Methods

Author: M. Mursaleen

Publisher: Springer Science & Business Media

Published: 2014-03-27

Total Pages: 129

ISBN-13: 3319046098

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This short monograph is the first book to focus exclusively on the study of summability methods, which have become active areas of research in recent years. The book provides basic definitions of sequence spaces, matrix transformations, regular matrices and some special matrices, making the material accessible to mathematicians who are new to the subject. Among the core items covered are the proof of the Prime Number Theorem using Lambert's summability and Wiener's Tauberian theorem, some results on summability tests for singular points of an analytic function, and analytic continuation through Lototski summability. Almost summability is introduced to prove Korovkin-type approximation theorems and the last chapters feature statistical summability, statistical approximation, and some applications of summability methods in fixed point theorems.


Double Sequence Spaces and Four-Dimensional Matrices

Double Sequence Spaces and Four-Dimensional Matrices

Author: Feyzi Başar

Publisher: CRC Press

Published: 2022-04-22

Total Pages: 252

ISBN-13: 1000594475

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Double Sequence Spaces and Four-Dimensional Matrices provides readers with a clear introduction to the spaces of double sequences and series, as well as their properties. The book then goes beyond this to investigate paranormed double sequence spaces and their algebraic and topological properties, triangle matrices and their domains in certain spaces of double sequences, dual spaces of double sequence spaces, and matrix transformations between double sequence spaces and related topics. Each chapter contains a conclusion section highlighting the importance of results and pointing out possible new ideas that can be studied further. Features Suitable for students at graduate or post-graduate level and researchers Investigates different types of summable spaces and computes their duals Characterizes several four-dimensional matrix classes transforming one summable space into other Discusses several algebraic and topological properties of new sequence spaces generated by the domain of triangles.


Summable Series and Convergence Factors

Summable Series and Convergence Factors

Author: Charles Napoleon Moore

Publisher: American Mathematical Soc.

Published: 1938-12-31

Total Pages: 114

ISBN-13: 0821846205

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Fairly early in the development of the theory of summability of divergent series, the concept of convergence factors was recognized as of fundamental importance in the subject. One of the pioneers in this field was C. N. Moore, the author of the book under review.... Moore classifies convergence factors into two types. In type I he places the factors which have only the property that they preserve convergence for a convergent series or produce convergence for a summable series. In type II he places the factors which not only maintain or produce convergence but have the additional property that they may be used to obtain the sum or generalized sum of the series. This book gives a generalized systematic treatment of the theory of convergence factors of both types, for simply infinite series and for multiple series, convergent and summable.... --Bulletin of the American Mathematical Society


Current Topics in Summability Theory and Applications

Current Topics in Summability Theory and Applications

Author: Hemen Dutta

Publisher: Springer

Published: 2016-04-28

Total Pages: 436

ISBN-13: 9811009139

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This book discusses recent developments in and contemporary research on summability theory, including general summability methods, direct theorems on summability, absolute and strong summability, special methods of summability, functional analytic methods in summability, and related topics and applications. All contributing authors are eminent scientists, researchers and scholars in their respective fields, and hail from around the world. The book can be used as a textbook for graduate and senior undergraduate students, and as a valuable reference guide for researchers and practitioners in the fields of summability theory and functional analysis. Summability theory is generally used in analysis and applied mathematics. It plays an important part in the engineering sciences, and various aspects of the theory have long since been studied by researchers all over the world.