Books

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Dyadic Walsh Analysis from 1924 Onwards Walsh-Gibbs-Butzer Dyadic Differentiation in Science. Volume 1: Foundations.

R.S. Stankovic, P.L. Butzer, F. Schipp, W.R. Wade, W. Su, Y. Endow, S. Fridli, B.I. Golubov, F. Pichlers

Book Description

Dyadic (Walsh) analysis emerged as a new research area in applied mathematics and engineering in early seventies within attempts to provide answers to demands from practice related to application of spectral analysis of different classes of signals, including audio, video, sonar, and radar signals. In the meantime, it evolved in a mature mathematical discipline with fundamental results and important features providing basis for various applications. The book will provide fundamentals of the area through reprinting carefully selected earlier publications followed by overview of recent results concerning particular subjects in the area written by experts, most of them being founders of the field, and some of their followers. In this way, this first volume of the two volume book offers a rather complete coverage of the development of dyadic Walsh analysis, and provides a deep insight into its mathematical foundations necessary for consideration of generalizations and applications that are the subject of the second volume. The presented theory is quite sufficient to be a basis for further research in the subject area as well as to be applied in solving certain new problems or improving existing solutions for tasks in the areas which motivated development of the dyadic analysis..

More information on this book can be found by the following link:

http://www.springer.com/kr/book/9789462391598.

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Dyadic Walsh Analysis from 1924 Onwards Walsh-Gibbs-Butzer Dyadic Differentiation in Science. Volume 2: Extensions and Generalizations.

R.S. Stankovic, P.L. Butzer, F. Schipp, W.R. Wade, W. Su, Y. Endow, S. Fridli, B.I. Golubov, F. Pichler

Book Description

The second volume of the two volumes book is dedicated to various extensions and generalizations of Dyadic (Walsh) analysis and related applications. Considered are dyadic derivatives on Vilenkin groups and various other Abelian and finite non-Abelian groups. Since some important results were developed in former Soviet Union and China, we provide overviews of former work in these countries. Further, we present translations of three papers that were initially published in Chinese. The presentation continues with chapters written by experts in the area presenting discussions of applications of these results in specific tasks in the area of signal processing and system theory. Efficient computing of related differential operators on contemporary hardware, including graphics processing units, is also considered, which makes the methods and techniques of dyadic analysis and generalizations computationally feasible. The Volume 2 of the book ends with a chapter presenting open problems pointed out by several experts in the area.

More information on this book can be found by the following link:

http://www.springer.com/kr/book/9789462391598.

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Nonlocal Diffusion and Applications

Claudia Bucur, Enrico Valdinoci

Book Description

Working in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation, some applications to water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schr¨odinger equations are given. Furthermore, an example of an s-harmonic function, its harmonic extension and some insight into a fractional version of a classical conjecture due to De Giorgi are presented. Although the aim is primarily to gather some introductory material concerning applications of the fractional Laplacian, some of the proofs and results are new. The work is entirely self-contained, and readers who wish to pursue related subjects of interest are invited to consult the rich bibliography for guidance.

More information on this book can be found by the following link:

http://www.springer.com/us/book/9783319287386.

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Enlightened by the Caputo type of fractional derivative, here we bring forth a concept of “memory-dependent derivative”, which is simply defined in an integral form of a common derivative with a kernel function on a slipping interval. In case the time delay tends to zero it tends to the common derivative. High order derivatives also accord with the first order one. Comparatively, the form of kernel function for the fractional type is fixed, yet that of the memory-dependent type can be chosen freely according to the necessity of applications. So this kind of definition is better than the fractional one for reflecting the memory effect (instantaneous change rate depends on the past state). Its definition is more intuitionistic for understanding the physical meaning and the corresponding memory-dependent differential equation has more expressive force.