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.

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.

The book features original chapters on sequence spaces involving the idea of ideal convergence, modulus function, multiplier sequences, Riesz mean, Fibonacci difference matrix etc., and illustrate their involvement in various applications. The preliminaries have been presented in the beginning of each chapter and then the advanced discussion takes place, so it is useful for both expert and nonexpert on aforesaid topics. The book consists of original thirteen research chapters contributed by the well-recognized researchers in the field of sequence spaces with associated applications. Features Discusses the Fibonacci and vector valued difference sequence spaces Presents the solution of Volterra integral equation in Banach algebra Discusses some sequence spaces involving invariant mean and related to the domain of Jordan totient matrix Presents the Tauberian theorems of double sequences Discusses the paranormed Riesz difference sequence space of fractional order Includes a technique for studying the existence of solutions of infinite system of functional integro-differential equations in Banach sequence spaces The subject of book is an active area of research of present time internationally and would serve as a good source for researcher and educators involved with the topic of sequence spaces.

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.

This book publishes original research chapters on the theory of approximation by positive linear operators as well as theory of sequence spaces and illustrates their applications. Chapters are original and contributed by active researchers in the field of approximation theory and sequence spaces. Each chapter describes the problem of current importance and summarizes ways of their solution and possible applications which improve the current understanding pertaining to sequence spaces and approximation theory. The presentation of the articles is clear and self-contained throughout the book.

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.

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.