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"Mandelbrot, Benoit B."
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The fractalist : memoir of a scientific maverick
A fascinating memoir from the man who revitalized visual geometry, and whose ideas about fractals have changed how we look at both the natural world and the financial world. Benoit Mandelbrot, the creator of fractal geometry, has significantly improved our understanding of, among other things, financial variability and erratic physical phenomena.
Book
Study of the Nature and Dynamics of Processes in Terms of Fractals on the Example of Selected Joint Stock Companies
Originality/value: The originality of the content of the article lies in the combination of theoretical concepts related to the research on the fractal nature of some reality processes with empirical research.
Journal Article
The islands of Benoمit Mandelbrot : fractals, chaos, and the materiality of thinking
\"Over the past few decades, the \"pictorial turn\" in the natural sciences, prompted by the computer's capacity to produce visual representations, has generated considerable theoretical interest. Poised between their materiality and the abstract level they are meant to convey, scientific images are always intersections of form and meaning. Benoمit Mandelbrot (1924-2010), one of the best-known producers of digital images in scientific and industrial research, was particularly curious about the ways in which the materiality of scientific representation was able to influence the development of the ideas and abstractions the images embodied.Using images and objects found in Mandelbrot's office, this book questions the relationship between the visual and scientific reasoning in fractal geometry and chaos theory, among the most popular fields to use digital scientific imagery in the past century. These unpublished materials offer new connections between the material world and that of mathematical ideas. Work by Adrien Douady and Otto Rèossler provides historical depth to the analysis\"-- Provided by publisher.
Book
Multifractality approach of a generalized Shannon index in financial time series
Multifractality is a concept that extends locally the usual ideas of fractality in a system. Nevertheless, the multifractal approaches used lack a multifractal dimension tied to an entropy index like the Shannon index. This paper introduces a generalized Shannon index (GSI) and demonstrates its application in understanding system fluctuations. To this end, traditional multifractality approaches are explained. Then, using the temporal Theil scaling and the diffusive trajectory algorithm, the GSI and its partition function are defined. Next, the multifractal exponent of the GSI is derived from the partition function, establishing a connection between the temporal Theil scaling exponent and the generalized Hurst exponent. Finally, this relationship is verified in a fractional Brownian motion and applied to financial time series. In fact, this leads us to proposing an approximation called local fractional Brownian motion approximation, where multifractal systems are viewed as a local superposition of distinct fractional Brownian motions with varying monofractal exponents. Also, we furnish an algorithm for identifying the optimal q -th moment of the probability distribution associated with an empirical time series to enhance the accuracy of generalized Hurst exponent estimation.
Journal Article
Generation of Julia and Mandelbrot Sets for a Complex Function via Jungck–Noor Iterative Method with s-Convexity
This paper introduces novel, non-classical Julia and Mandelbrot sets using the Jungck–Noor iterative method with s-convexity, and derives an escape criterion for higher-order complex polynomials of the form zn+ℑz3−ℜz+ω, where n≥4 and ℑ,ℜ,ω∈C. The proposed method advances existing algorithms, enabling the visualization of intricate fractal patterns as Julia and Mandelbrot sets with enhanced complexity. Through graphical representations, we illustrate how parameter variations influence the color, size, and shape of the resulting images, producing visually striking and aesthetically appealing fractals. Furthermore, we explore the dynamic behavior of these sets under fixed input parameters while varying the degree n. The presented results, both methodologically and visually, offer new insights into fractal geometry and inspire further research.
Journal Article
New Estimates for Csiszár Divergence and Zipf–Mandelbrot Entropy via Jensen–Mercer’s Inequality
Jensen’s inequality is one of the fundamental inequalities which has several applications in almost every field of science. In 2003, Mercer gave a variant of Jensen’s inequality which is known as Jensen–Mercer’s inequality. The purpose of this article is to propose new bounds for Csiszár and related divergences by means of Jensen–Mercer’s inequality. Also, we investigate several new bounds for Zipf–Mandelbrot entropy. The idea of this article may further stimulate research on information theory with the help of Jensen–Mercer’s inequality.
Journal Article
Principles Entailed by Complexity, Crucial Events, and Multifractal Dimensionality
Complexity is one of those descriptive terms adopted in science that we think we understand until it comes time to form a coherent definition upon which everyone can agree. Suddenly, we are awash in conditions that qualify this or that situation, much like we were in the middle of the last century when it came time to determine the solutions to differential equations that were not linear. Consequently, this tutorial is not an essay on the mathematics of complexity nor is it a rigorous review of the recent growth spurt of complexity science, but is rather an exploration of how physiologic time series (PTS) in the life sciences that have eluded traditional mathematical modeling become less mysterious when certain historical assumptions are discarded and so-called ordinary statistical events in PTS are replaced with crucial events (CEs) using mutifractal dimensionality as the working measure of complexity. The empirical datasets considered include respiration, electrocardiograms (ECGs), and electroencephalograms (EEGs), and as different as these time series appear from one another when recorded, they are in fact shown to be in synchrony when properly processed using the technique of modified diffusion entropy analysis (MDEA). This processing reveals a new synchronization mechanism among the time series which simultaneously measures their complexity by means of the multifractal dimension of each time series and are shown to track one another across time. These results reveal a set of priciples that capture the manner in which information is exchanged among physiologic organ networks.
Journal Article
Family of Fuzzy Mandelblog Sets
In this paper, we consider the family of parameterized Mandelbrot-like sets generated as any point c∈C∖0 of the complex plane belongs to any member of this family for a real parameter t≥1, provided that its corresponding orbit of 0 does not escape to infinity under iteration f[sub.c] [sup.n](0)=fcn−10[sup.2]+logc[sup.t]; otherwise, it is not a member of this set. This classically means there is only a binary membership possibility for all points. Here, we call this type of fractal set a Mandelblog set, and then we introduce a membership function that assigns a degree to each c to be an element of a fuzzy Mandelblog set under the iterations, even if the orbits of the points are not limited. Moreover, we provide numerical examples and gray-scale graphics that illustrate the membership degrees of the points of the fuzzy Mandelblog sets under the effects of iteration parameters. This approach enables the formation of graphs for these fuzzy fractal sets by representing points that belong to the set as white pixels, points that do not belong as black pixels, and other points, based on their membership degrees, as gray-toned pixels. Furthermore, the membership function facilitates the direct proofs of the symmetry criteria for these fractal sets.
Journal Article
Mathematical Modeling of Physical Reality: From Numbers to Fractals, Quantum Mechanics and the Standard Model
In physics, we construct idealized mathematical models in order to explain various phenomena which we observe or create in our laboratories. In this article, I recall how sophisticated mathematical models evolved from the concept of a number created thousands of years ago, and I discuss some challenges and open questions in quantum foundations and in the Standard Model. We liberated nuclear energy, landed on the Moon and built ‘quantum computers’. Encouraged by these successes, many believe that when we reconcile general relativity with quantum theory we will have the correct theory of everything. Perhaps we should be much humbler. Our perceptions of reality are biased by our senses and by our brain, bending them to meet our priors and expectations. Our abstract mathematical models describe only in an approximate way different layers of physical reality. To describe the motion of a meteorite, we can use a concept of a material point, but the point-like approximation breaks completely when the meteorite hits the Earth. Similarly, thermodynamic, chemical, molecular, atomic, nuclear and elementary particle layers of physical reality are described using specific abstract mathematical models and approximations. In my opinion, the theory of everything does not exist.
Journal Article