Fractals are geometric shapes and patterns that may repeat their geometry at smaller or larger scales. It is well established that fractals can describe shapes and surfaces that cannot be represented by the classical Euclidean geometry. An eclectic s...
Fractals are geometric shapes and patterns that can describe the roughness (or irregularity) present in almost every object in nature. Many fractals may repeat their geometry at smaller or larger scales. This paper is the second (and last) part of a...
Fractals with different levels of self-similarity and magnification are defined as reduced fractals. It is shown that spectra of these reduced fractals can be constructed and used to describe levels of complexity of natural phenomena. Specific applic...
The relatively high abundance of fractal properties of complex systems on Earth and in space is considered an argument in support of the general relativity of the geometric theory of gravity. The fractality may be called the fractal symmetry of physi...
Two measures are commonly used to describe scale-invariant complexity in images: fractal dimension (D) and power spectrum decay rate (β). Although a relationship between these measures has been derived mathematically, empirical validation across meas...
Complex fractal dimensions, defined as poles of appropriate fractal zeta functions, describe the geometric oscillations in fractal sets. In this work, we show that the same possible complex dimensions in the geometric setting also govern the asymptot...
In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of the...
How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the ex...
The structure of fractals at nano and micro scales is decisive for their physical properties. Generally, statistically self-similar (random) fractals occur in natural systems, and exactly self-similar (deterministic) fractals are artificially created...
In this research article, a modified algorithm for the generation of a fractal pattern resulting from the iteration of an algebraic function using the numerical iteration method is presented. This fractal pattern shows the dynamical behavior of the n...
The engineering properties of particulate materials are the collective manifestation of interactions among their constituent particles and are structures within which particles adopt their spatial arrangement. For the first time in the literature, th...
This paper explores the generalization of the fixed-point theorem for Fisher contraction on controlled metric space. The controlled metric space and Fisher contractions are playing a very crucial role in this research. The Fisher contraction on the c...
This paper analyzes the extreme limit of iterated function systems (IFSs) when the number of contractions drops to one and the resulting attractors reduce to a single point. While classical fractals have a strictly positive fractal dimension, the deg...
Fractures caused by mining are the main form of water inrush disaster. However, the temporal and spatial development characteristics of fractures of the rock mass due to mining are not clearly understood at present. In this paper, two geometric param...
Porous structures exhibiting randomly sized and distributed pores are required in biomedical applications (producing implants), materials science (developing cermet-based materials with desired properties), engineering applications (objects having co...
Small-angle scattering (SAS; X-rays, neutrons, light) is being increasingly used to better understand the structure of fractal-based materials and to describe their interaction at nano- and micro-scales. To this aim, several minimalist yet specific t...
In 2018, Erdal Karapinar introduced the concept of interpolative Kannan operators, a novel adaptation of the Kannan mapping originally defined in 1969 by Kannan. This new mapping condition is more lenient than the basic contraction condition. In this...
We propose a novel approach to fractals by leveraging the approximation of fixed points, emphasizing the deep connections between fractal theory and fixed-point theory. We include a condition of isomorphism, which not only generates traditional fract...
This study introduces a new metric structure called the Graphical Fuzzy Cone Metric Space (GFCMS) and explores its essential properties in detail. We examine its topological aspects in detail and introduce the notion of Hausdorff distance within this...
We explore the relation between the local fractal dimension and the development of the built-up area of the Northern Margin of the Metropolitan Area of Lisbon (NMAL), for the period between 1960 and 2004. To this end we make use of a Generalized Loca...
We introduce a mathematical framework to characterize the hierarchical complexity of AI-generated fractals within the finite resolution constraints of digital images. Our method analyzes images produced by text-to-image models at multiple intensity t...
Anomaly detection is crucial in large-scale industrial manufacturing as it helps to detect and localize defective parts. Pre-training feature extractors on large-scale datasets is a popular approach for this task. Stringent data security, privacy reg...
If X is a Hilbert space, one can consider the space
Homogeneous random fractals form a probabilistic generalisation of self-similar sets with more dependencies than in random recursive constructions. Under the Uniform Strong Open Set Condition we show that the mean D-dimensional (average) Minkowski co...
The present work introduces a new scheme of data cryptography in the context of emerging trends due to the challenge of defending critical network infrastructure against new exploit systems based on artificial intelligence or defending against quantu...
For a complex parameter c outside the unit disk and an integer
Fractals are essential in representing the natural environment due to their important characteristic of self similarity. The dynamical behavior of fractals mostly depends on escape criteria using different iterative techniques. In this article, we es...
In this paper, we examine a sequence of uncountable iterated function systems (U.I.F.S.), where each term in the sequence is constructed from an uncountable collection of contraction mappings along with a linear and continuous operator. Each U.I.F.S....
In this study, we delve into the general theory of operator kernel functions (OKFs) in operational calculus (OC). We established the rigorous mapping relation between the kernel function and the corresponding operator through the primary translation...
It is well established that the introduction of additive manufacturing in various domains has produced significant technological leaps due to the advantages over other manufacturing techniques. Furthermore, additive manufacturing allows the design of...
Winding numbers are key indices in the depiction, modelling, and testing of dynamical processes. They capture phase progression on closed curves and are robust for quasiperiodic dynamics, but their status for chaotic Poincaré sections is uncle...
The relationship between soil structure and salt accumulation is unclear; thus, experiments on salt accumulation under different soil structures were conducted in cotton fields in arid areas of northwest China. Thirty-nine sets of soil samples were c...
The seismo-electromagnetic theory describes the growth of fractally distributed cracks within the lithosphere that generate the emission of magnetic anomalies prior to large earthquakes. One of the main physical properties of this theory is their con...
In this paper, we obtain multifractals (attractors) in the framework of Hausdorff b-metric spaces. Fractals and multifractals are defined to be the fixed points of associated fractal operators, which are known as attractors in the literature of fract...
In this manuscript, we explore stunning fractals as Julia and Mandelbrot sets of complexvalued cosine functions by establishing the escape radii via a four-step iteration scheme extended with s-convexity. We furnish some illustrations to determine th...
For the simplest colored branching process, we prove an analog to the McMillan theorem and calculate the Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this s...
Research interest in iterative multipoint schemes to solve nonlinear problems has increased recently because of the drawbacks of point-to-point methods, which need high-order derivatives to increase the order of convergence. However, this order is no...
The complexity of geological structures significantly impacts both mining production efficiency and operational safety, making its quantitative assessment a core issue in ensuring coal’s safe production and coalbed methane development. Focusing...
The skid resistance of asphalt pavement is vital for traffic safety and reducing accidents. Existing research using only wavelet transforms or fractal theory to study the pavement surface texture-skid resistance relationship has limitations. This pap...
The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a scale-dependence measure, which indicates the scale-free distribution of urban patterns. Thus, th...
In this manuscript, we introduce the concept of fuzzy S-metric spaces and study some of their characteristics. We prove a fixed-point theorem for a self-mapping on a complete fuzzy S-metric space. To illustrate the versatility of our new ideas and re...
In this paper, fractal calculus, which is called
The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefo...
One of the main limitations of the technique surface-enhanced Raman scattering (SERS) for chemical detection relies on the homogeneity, reproducibility and reusability of the substrates. In this work, SERS active platforms based on 3D-fractal microst...
At present, the selection of lunar landing areas is mostly determined by experts’ argumentation and experience. Generally, it is artificially limited to a small zone, and there are few effective quantitative models for landing areas. Under the...
We reveal the analytic relations between a matrix permanent and major nature’s complexities manifested in critical phenomena, fractal structures and chaos, quantum information processes in many-body physics, number-theoretic complexity in mathe...
Rock is a widely used construction material; its mechanical properties change due to the influence of different load speed. In this study, the split Hopkinson pressure bar (SHPB) was used to test the dynamic properties of rock samples by loading four...
In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for sol...
This paper explores the extension of classical transforms to fractal spaces, focusing on the development and application of the Fractal Hankel Transform. We begin with a concise review of fractal calculus to set the theoretical groundwork. The Fracta...
In the present research work, we construct and examine the self-similarity of optical solitons by employing the Riccati Modified Extended Simple Equation Method (RMESEM) within the framework of non-integrable Coupled Nonlinear Helmholtz Equations (CN...
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