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.
| Preface |
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1 Some Remarks on Multivariate Fractal Approximation |
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1 | (24) |
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1 | (4) |
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1.2 Dimension Preserving Approximation of Multivariate Functions |
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5 | (12) |
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1.3 Some Multi-valued Mappings |
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17 | (6) |
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23 | (2) |
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2 Fractal Interpolation: From Global to Local, to Nonstationary and Quaternionic |
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25 | (25) |
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25 | (1) |
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2.2 Global Fractal Interpolation |
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26 | (5) |
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2.3 Local Fractal Interpolation |
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31 | (5) |
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2.4 Non-stationary Fractal Interpolation |
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36 | (5) |
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2.5 Quaternionic Fractal Interpolation |
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41 | (5) |
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46 | (4) |
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3 A Study on Fractal Operator Corresponding to Non-stationary Fractal Interpolation Functions |
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50 | (17) |
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50 | (1) |
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51 | (1) |
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3.3 Non-stationary Univariate a-fractal Functions |
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52 | (3) |
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3.4 Associated Non-stationary Fractal Operator on C(I) |
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55 | (5) |
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3.5 Fractal Dimension of Non-stationary a-fractal Functions |
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60 | (1) |
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61 | (4) |
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65 | (2) |
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67 | (16) |
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Alireza Khalili Golmankhaneh |
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67 | (2) |
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4.2 Local Fractal Calculus |
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69 | (2) |
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4.3 Non-local Fractal Calculus |
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71 | (1) |
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4.4 The Local Fractal Fourier Transform |
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72 | (1) |
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4.5 The Local Fractal Laplace Transform |
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73 | (1) |
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4.6 Discrete Scale Invariance |
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74 | (1) |
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4.7 Fractal Calculus and its Applications |
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75 | (4) |
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79 | (4) |
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5 Perspective of Fractal Calculus on Types of Fractal Interpolation Functions |
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83 | (18) |
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83 | (1) |
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5.2 Fractal Interpolation Function |
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84 | (2) |
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5.3 Hidden Variable Fractal Interpolation Function |
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86 | (2) |
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88 | (1) |
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5.5 Fractal Calculus of Fractal Functions |
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89 | (10) |
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99 | (2) |
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6 On the Borel Regularity of the Relative Centered Multifractal Measures |
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101 | (13) |
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101 | (1) |
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6.2 The Relative Multifractal Measures |
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102 | (3) |
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6.3 The Borel Regularity of the Relative Centered Multifractal Measures |
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105 | (6) |
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6.4 Remarks and Discussion |
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111 | (1) |
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111 | (3) |
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7 A Mixed Multifractal Analysis of Vector-valued Measures: Review and Extension to Densities and Regularities of Non-necessary Gibbs Cases |
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114 | (39) |
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114 | (3) |
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7.2 The φ-mixed Multifractal Measures and Dimensions |
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117 | (11) |
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7.3 The φ-mixed Multifractal Densities |
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128 | (8) |
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7.4 Regularities of φ-mixed Multifractal Measures |
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136 | (3) |
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7.5 Some Motivations, Applications, Examples and Discussions |
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139 | (4) |
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143 | (1) |
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144 | (9) |
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8 Multifractal Dimensions and Fractional Differentiation in Automated Edge Detection on Intuitionistic Fuzzy Enhanced Image |
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153 | (20) |
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153 | (2) |
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155 | (7) |
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8.3 Proposed Method of Edge Detection |
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162 | (1) |
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8.4 Experimental Results and Discussion |
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163 | (4) |
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167 | (6) |
| Index |
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Santo Banerjee is associated with the Department of Mathematics, Politecnico di Torino, Italy. Prior to this, he was an Associate Professor in the Institute for Mathematical Research (INSPEM), University Putra Malaysia, Malaysia, until 2020, and also a founder member of the Malaysia-Italy Centre of Excellence in Mathematical Science, UPM, Malaysia. His research is mainly concerned with Nonlinear Dynamics, Chaos, Complexity and Secure Communication. He is a Managing Editor of EPJ Plus (Springer).
A. Gowrisankar has a master's degree and Ph.D in Mathematics from The Gandhigram Rural Institute (Deemed to be University), Gandhigram, Dindigul, India, in 2012 and 2017 respectively. He got an institute postdoctoral fellowship from the Indian Institute of Technology Guwahati (IITG), Guwahati, Assam, India, in 2017. At present, he is an Assistant Professor in the Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, India. His broad areas of research include Fractal Analysis, Image Processing, Fractional Calculus and Fractal Functions.
