Fractal Functions, Fractal Surfaces, and Wavelets
Fractal Functions, Fractal Surfaces, and Wavelets

Author: Peter R. Massopust

Publisher: Academic Press

Published: 2016-09-02

Total Pages: 428

ISBN-13: 0128044705

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Fractal Functions, Fractal Surfaces, and Wavelets, Second Edition, is the first systematic exposition of the theory of local iterated function systems, local fractal functions and fractal surfaces, and their connections to wavelets and wavelet sets. The book is based on Massopust's work on and contributions to the theory of fractal interpolation, and the author uses a number of tools—including analysis, topology, algebra, and probability theory—to introduce readers to this exciting subject. Though much of the material presented in this book is relatively current (developed in the past decades by the author and his colleagues) and fairly specialized, an informative background is provided for those entering the field. With its coherent and comprehensive presentation of the theory of univariate and multivariate fractal interpolation, this book will appeal to mathematicians as well as to applied scientists in the fields of physics, engineering, biomathematics, and computer science. In this second edition, Massopust includes pertinent application examples, further discusses local IFS and new fractal interpolation or fractal data, further develops the connections to wavelets and wavelet sets, and deepens and extends the pedagogical content. - Offers a comprehensive presentation of fractal functions and fractal surfaces - Includes latest developments in fractal interpolation - Connects fractal geometry with wavelet theory - Includes pertinent application examples, further discusses local IFS and new fractal interpolation or fractal data, and further develops the connections to wavelets and wavelet sets - Deepens and extends the pedagogical content

Fractals, Wavelets, and their Applications
Fractals, Wavelets, and their Applications

Author: Christoph Bandt

Publisher: Springer

Published: 2014-09-27

Total Pages: 499

ISBN-13: 3319081055

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Fractals and wavelets are emerging areas of mathematics with many common factors which can be used to develop new technologies. This volume contains the selected contributions from the lectures and plenary and invited talks given at the International Workshop and Conference on Fractals and Wavelets held at Rajagiri School of Engineering and Technology, India from November 9-12, 2013. Written by experts, the contributions hope to inspire and motivate researchers working in this area. They provide more insight into the areas of fractals, self similarity, iterated function systems, wavelets and the applications of both fractals and wavelets. This volume will be useful for the beginners as well as experts in the fields of fractals and wavelets.

Fractal Functions, Dimensions and Signal Analysis
Fractal Functions, Dimensions and Signal Analysis

Author: Santo Banerjee

Publisher: Springer Nature

Published: 2020-12-14

Total Pages: 132

ISBN-13: 3030626725

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This book introduces the fractal interpolation functions (FIFs) in approximation theory to the readers and the concerned researchers in advanced level. FIFs can be used to precisely reconstruct the naturally occurring functions when compared with the classical interpolants. The book focuses on the construction of fractals in metric space through various iterated function systems. It begins by providing the Mathematical background behind the fractal interpolation functions with its graphical representations and then introduces the fractional integral and fractional derivative on fractal functions in various scenarios. Further, the existence of the fractal interpolation function with the countable iterated function system is demonstrated by taking suitable monotone and bounded sequences. It also covers the dimension of fractal functions and investigates the relationship between the fractal dimension and the fractional order of fractal interpolation functions. Moreover, this book explores the idea of fractal interpolation in the reconstruction scheme of illustrative waveforms and discusses the problems of identification of the characterizing parameters. In the application section, this research compendium addresses the signal processing and its Mathematical methodologies. A wavelet-based denoising method for the recovery of electroencephalogram (EEG) signals contaminated by nonstationary noises is presented, and the author investigates the recognition of healthy, epileptic EEG and cardiac ECG signals using multifractal measures. This book is intended for professionals in the field of Mathematics, Physics and Computer Science, helping them broaden their understanding of fractal functions and dimensions, while also providing the illustrative experimental applications for researchers in biomedicine and neuroscience.

Abstract and Applied Analysis
Abstract and Applied Analysis

Author: N. M. Chuong

Publisher: World Scientific

Published: 2004

Total Pages: 579

ISBN-13: 9812702547

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This volume takes up various topics in Mathematical Analysis including boundary and initial value problems for Partial Differential Equations and Functional Analytic methods. Topics include linear elliptic systems for composite material OCo the coefficients may jump from domain to domain; Stochastic Analysis OCo many applied problems involve evolution equations with random terms, leading to the use of stochastic analysis. The proceedings have been selected for coverage in: . OCo Index to Scientific & Technical Proceedings (ISTP CDROM version / ISI Proceedings). OCo CC Proceedings OCo Engineering & Physical Sciences. Contents: Deterministic Analysis: Differentiation of Hypergeometric Functions with Respect to Parameters (Yu A Brychkov & K O Geddes); On the Lagrange Problem About the Strongest Columns (Yu V Egorov); Wavelet Based Fast Solution of Boundary Integral Equations (H Harbrecht & R Schneider); Semi-Classical Methods in GinzburgOCoLandau Theory (B Helffer); Stability of Equilibriums in One-Dimensional Motion of Compressible Viscous Gas Forced by Self-Gravity (Y Iwata & Y Yamamoto); Estimates for Elliptic Systems for Composite Material (L Nirenberg); On Asymptotics for the Mabuchi Energy Functional (D H Phong & J Sturm); Regularity of Solutions of the Initial Boundary Value Problem for Linearized Equations of Ideal Magneto-Hydrodynamics (M Yamamoto); Stochastic Analysis: Impulsive Stochastic Evolution Inclusions with Multi-Valued Diffusion (N U Ahmed); Some of Future Directions of White Noise Analysis (T Hida); Constructing Random Probability Distributions (T P Hill & D E R Sitton); Multiparameter Additive Processes of Mixture Type (K Inoue); The Random Integral Representation Hypothesis Revisited: New Classes of S-Selfdecomposable Laws (Z J Jurek); Semigroups and Processes with Parameter in a Cone (J Pedersen & K-I Sato); and other papers. Readership: Researchers and academics in the fields of analysis and differential equations, approximation theory, probability and statistics."

Abstract And Applied Analysis - Proceedings Of The International Conference
Abstract And Applied Analysis - Proceedings Of The International Conference

Author: Nguyen Minh Chuong

Publisher: World Scientific

Published: 2004-06-01

Total Pages: 579

ISBN-13: 9814482110

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This volume takes up various topics in Mathematical Analysis including boundary and initial value problems for Partial Differential Equations and Functional Analytic methods.Topics include linear elliptic systems for composite material — the coefficients may jump from domain to domain; Stochastic Analysis — many applied problems involve evolution equations with random terms, leading to the use of stochastic analysis.The proceedings have been selected for coverage in:• Index to Scientific & Technical Proceedings (ISTP CDROM version / ISI Proceedings)• CC Proceedings — Engineering & Physical Sciences

Applied Analysis, Optimization and Soft Computing
Applied Analysis, Optimization and Soft Computing

Author: Tanmoy Som

Publisher: Springer Nature

Published: 2023-06-10

Total Pages: 425

ISBN-13: 9819905974

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This book contains select contributions presented at the International Conference on Nonlinear Applied Analysis and Optimization (ICNAAO-2021), held at the Department of Mathematics Sciences, Indian Institute of Technology (BHU) Varanasi, India, from 21–23 December 2021. The book discusses topics in the areas of nonlinear analysis, fixed point theory, dynamical systems, optimization, fractals, applications to differential/integral equations, signal and image processing, and soft computing, and exposes the young talents with the newer dimensions in these areas with their practical approaches and to tackle the real-life problems in engineering, medical and social sciences. Scientists from the U.S.A., Austria, France, Mexico, Romania, and India have contributed their research. All the submissions are peer reviewed by experts in their fields.

Frontiers of Fractal Analysis
Frontiers of Fractal Analysis

Author: Santo Banerjee

Publisher: CRC Press

Published: 2022-07-07

Total Pages: 183

ISBN-13: 1000625877

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The history of describing natural objects using geometry is as old as the advent of science itself, in which traditional shapes are the basis of our intuitive understanding of geometry. However, nature is not restricted to such Euclidean objects which are only characterized typically by integer dimensions. Hence, the conventional geometric approach cannot meet the requirements of solving or analysing nonlinear problems which are related with natural phenomena, therefore, the fractal theory has been born, which aims to understand complexity and provide an innovative way to recognize irregularity and complex systems. Although the concepts of fractal geometry have found wide applications in many forefront areas of science, engineering and societal issues, they also have interesting implications of a more practical nature for the older classical areas of science. Since its discovery, there has been a surge of research activities in using this powerful concept in almost every branch of scientific disciplines to gain deep insights into many unresolved problems. This book includes eight chapters which focus on gathering cutting-edge research and proposing application of fractals features in both traditional scientific disciplines and in applied fields.

Analysis, Probability And Mathematical Physics On Fractals
Analysis, Probability And Mathematical Physics On Fractals

Author: Patricia Alonso Ruiz

Publisher: World Scientific

Published: 2020-02-26

Total Pages: 594

ISBN-13: 9811215545

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In the 50 years since Mandelbrot identified the fractality of coastlines, mathematicians and physicists have developed a rich and beautiful theory describing the interplay between analytic, geometric and probabilistic aspects of the mathematics of fractals. Using classical and abstract analytic tools developed by Cantor, Hausdorff, and Sierpinski, they have sought to address fundamental questions: How can we measure the size of a fractal set? How do waves and heat travel on irregular structures? How are analysis, geometry and stochastic processes related in the absence of Euclidean smooth structure? What new physical phenomena arise in the fractal-like settings that are ubiquitous in nature?This book introduces background and recent progress on these problems, from both established leaders in the field and early career researchers. The book gives a broad introduction to several foundational techniques in fractal mathematics, while also introducing some specific new and significant results of interest to experts, such as that waves have infinite propagation speed on fractals. It contains sufficient introductory material that it can be read by new researchers or researchers from other areas who want to learn about fractal methods and results.

Dynamics with Chaos and Fractals
Dynamics with Chaos and Fractals

Author: Marat Akhmet

Publisher: Springer Nature

Published: 2020-01-01

Total Pages: 233

ISBN-13: 3030358542

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The book is concerned with the concepts of chaos and fractals, which are within the scopes of dynamical systems, geometry, measure theory, topology, and numerical analysis during the last several decades. It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. This is the first time in the literature that the description of chaos is initiated from a single motion. Chaos is now placed on the line of oscillations, and therefore, it is a subject of study in the framework of the theories of dynamical systems and differential equations, as in this book. The techniques introduced in the book make it possible to develop continuous and discrete dynamics which admit fractals as points of trajectories as well as orbits themselves. To provide strong arguments for the genericity of chaos in the real and abstract universe, the concept of abstract similarity is suggested.

Mathematical Theorems
Mathematical Theorems

Author: Lyudmila Alexeyeva

Publisher: BoD – Books on Demand

Published: 2020-12-09

Total Pages: 149

ISBN-13: 1838800719

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The main content of this book is related to construction of analytical solutions of differential equations and systems of mathematical physics, to development of analytical methods for solving boundary value problems for such equations and the study of properties of their solutions. A wide class of equations (elliptic, parabolic, and hyperbolic) is considered here, on the basis of which complex wave processes in biological and physical media can be simulated.The method of generalized functions presented in the book for solving boundary value problems of mathematical physics is universal for constructing solutions of boundary value problems for systems of linear differential equations with constant coefficients of any type. In the last sections of the book, the issues of calculating functions based on Padé approximations, binomial expansions, and fractal representations are considered. The book is intended for specialists in the field of mathematical and theoretical physics, mechanics and biophysics, students of mechanics, mathematics, physics and biology departments of higher educational institutions.