Search Results (93)

The reliability of mechanical systems hinges on analyzing the actual surface-to-surface contact area, which critically influences dynamic behavior, friction, material performance, and thermal dissipation. Uneven surfaces lead to incomplete contact, where only a fraction of asperities touch, creating a nominal contact area. This study proposes a novel fractal contact model for various mechanical behaviors between mechanical contact surfaces, integrating the Weierstrass–Mandelbrot fractal function and nonuniform rational B-spline interpolation (NURBS) to model material-dependent actual contact conditions. Furthermore, this research delved into the changes in thermal conductivity across the surfaces of metal materials within a simulated setting. It maintained a contact ratio ranging from 0.038% to 15.2%, a factor that remained unaffected by contact pressure. Both experimental and simulated findings unveiled an actual contact rate spanning from 0.44% to 1.06%, thereby underscoring the distinctive interface behaviors specific to different materials. The proposed approach provides fresh perspectives for investigating material–contact interactions and tackling associated engineering hurdles. Full article

In this paper, we consider the family of parameterized Mandelbrot-like sets generated as any point $c\in \mathbb{C}\setminus \left\{0\right\}$ of the complex plane belongs to any member of this family for a real parameter $t\ge 1$, provided that its corresponding orbit of 0 does not escape to infinity under iteration $f_c^n\left(0\right)={\left[f_c^{n-1}\left(0\right)\right]}^2+log{c}^t$; 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. Full article

In this paper, a new class of fifth-order Chebyshev–Halley-type methods with a single parameter is proposed by using the polynomial interpolation method. The convergence order of the new method is proved. The dynamic behavior of the new method on quadratic polynomials $P\left(x\right)=(x-a)(x-b)$ is analyzed, the strange fixed points and the critical points of the operator are obtained, the corresponding parameter planes and dynamic planes are drawn, the stability and convergence of the iterative method are visualized, and some parameter values with good properties are selected. The fractal results of the new method corresponding to different parameters about polynomial $G\left(x\right)$ are plotted. Numerical results show that the new method has less computing and higher computational accuracy than the existing Chebyshev–Halley-type methods. The fractal results show the new method has good stability and convergence. The numerical results of different iteration methods are compared and agree with the results of dynamic analysis. Full article

To investigate the contact performances and meshing characteristics of gears systematically, an improved comprehensive meshing stiffness model of spur gears with consideration of the tooth surface morphology, lubrication, friction, and thermal effects is presented based on the thermal elastohydrodynamic lubrication (TEHL) theory. The fractal feature of the tooth surface morphology is verified experimentally and characterized by the Weierstrass–Mandelbrot fractal function. Based on this, the rough contact stiffness, oil film stiffness, and thermal stiffness are introduced into the stiffness model. Comparisons between smooth and rough models are carried out, and the maximum temperature rise is increased by 24.7%. Subsequently, the influences of the torque, rotational speed, and fractal parameters on the oil film pressure and thickness, friction and temperature rise, and contact stiffness and comprehensive meshing stiffness are investigated. The results show that the oil film pressure and the maximum temperature rise increase by 125.18% and 69.08%, respectively, with an increasing torque from 20 N·m to 300 N·m. As the rotational speed is increased, the oil film thickness sharply increases, the rough peak contact area and friction reduce, and the stiffness fluctuation weakens. For fractal parameters, the oil film pressure and film thickness, friction, and temperature rise are nonlinear with changes in the fractal dimension D and fractal scale characteristic G. The results reveal that this work provides a more reasonable analysis for understanding the meshing characteristics and the design and processing of gears. Full article

Beginning with Mandelbrot’s insight that Fisher information may admit a thermodynamic interpretation, a growing body of work has connected this information-theoretic measure to fluctuation–dissipation relations, thermodynamic geometry, and phase transitions. Yet, these connections have largely remained at the level of formal analogies. In this work, we provide what is, to our knowledge, the first explicit realization of the epistemic-to-physical transition of Fisher information within a finite interacting quantum system. Specifically, we analyze a model of N fermions occupying two degenerate levels and coupled by a spin-flip interaction of strength V, treated in the grand canonical ensemble at inverse temperature $\beta$. We compute the Fisher information $F_N\left(V\right)$ associated with the sensitivity of the thermal state to changes in V, and show that it becomes an observer-independent, experimentally meaningful quantity: it encodes fluctuations, tracks entropy variations, and reveals structural transitions induced by interactions. Our findings thus demonstrate that Fisher information, originally conceived as an inferential and epistemic measure, can operate as a bona fide thermodynamic observable in quantum many-body physics, bridging the gap between information-theoretic foundations and measurable physical law. Full article

Content delivery networks (CDNs) face steadily rising, uneven demand, straining heuristic cache replacement. Reinforcement learning (RL) is promising, but most work assumes a fully observable Markov Decision Process (MDP), unrealistic under delayed, partial, and noisy signals. We model cache replacement as a Partially Observable MDP (POMDP) and present the Miss-Triggered Cache Transformer (MTCT), a Transformer-decoder Q-learning agent that encodes recent histories with self-attention. MTCT invokes its policy only on cache misses to align compute with informative events and uses a delayed-hit reward to propagate information from hits. A compact, rank-based action set (12 actions by default) captures popularity–recency trade-offs with complexity independent of cache capacity. We evaluate MTCT on a real trace (MovieLens) and two synthetic workloads (Mandelbrot–Zipf, Pareto) against Adaptive Replacement Cache (ARC), Windowed TinyLFU (W-TinyLFU), classical heuristics, and Double Deep Q-Network (DDQN). MTCT achieves the best or statistically comparable cache-hit rates on most cache sizes; e.g., on MovieLens at $M=600$, it reaches $0.4703$ (DDQN $0.4436$, ARC $0.4513$). Miss-triggered inference also lowers mean wall-clock time per episode; Transformer inference is well suited to modern hardware acceleration. Ablations support $CL=50$ and show that finer action grids improve stability and final accuracy. Full article

In the present work, we consider three branching random walk $S_n\mathsf{Z}\left(\mathsf{t}\right),$$\mathsf{Z}\in \{\mathsf{X},\mathsf{Y},\Phi \}$ on a supercritical random Galton–Watson tree $\partial \mathsf{T}$. We compute the Hausdorff and packing dimensions of the level set ${\mathrm{E}}_{\chi }(\alpha ,\beta )=\left\{\mathsf{t}\in \partial \mathsf{T}:{\lim }_{n\to \infty }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}=\alpha \mathrm{and}{\lim }_{n\to \infty }{\displaystyle \frac{S_n\mathsf{Y}\left(\mathsf{t}\right)}{n}}=\beta \right\},$ where $\partial \mathsf{T}$ is endowed with random metric using $S_n\Phi \left(\mathsf{t}\right)$. This is achieved by constructing a suitable Mandelbrot measure supported on $\mathrm{E}(\alpha ,\beta )$. In the case where $\Phi =1$, we develop a formalism that parallels Olsen’s framework (for measures) and Peyrière’s framework (for the vectorial case) within our setting. Full article

In this paper, a pattern for visualizing fractals, namely Julia and Mandelbrot sets for complex functions of the form $T\left(u\right)=u^a-\xi u^2+ru+sin{\rho }^{\sigma }\mathrm{for}\mathrm{all}u\in \mathbb{C}$ and $a\in \mathbb{N}\setminus \left\{1\right\}$, $\xi \in \mathbb{C}$, $r,\rho \in \mathbb{C}\setminus \left\{0\right\}$ are created using novel fast convergent iterative techniques. The new iteration scheme discussed in this study uses s-convexity and improves earlier approaches, including the Mann and Picard–Mann schemes. Further, the proposed approach is amplified by unique escape conditions that regulate the convergence behavior and generate Julia and Mandelbrot sets. This new technique allows greater versatility in fractal design, influencing the shape, size, and aesthetic structure of the designs created. By modifying various parameters in the suggested scheme, a significant number of visually interesting fractals can be generated and evaluated. Furthermore, we provide numerical examples and graphic demonstrations to demonstrate the efficiency of this novel technique. Full article

The principles of fractal geometry have revolutionized the characterization of complex geometric objects since Benoit Mandelbrot’s groundbreaking work. Richardson’s method for determining the fractal dimension of boundaries laid the groundwork for Mandelbrot’s later developments in fractal theory. Despite extensive research, challenges remain in accurately calculating fractal dimensions, particularly when dealing with digital images and their inherent limitations. This study examines the numerical artifacts introduced by Richardson’s method when applied to the Von Koch Island, a classic fractal curve, and proposes a novel approach for computing fractal dimensions in image analysis. The Koch snowflake serves as a key example in this analysis; it serves to assess the algorithm of fractal dimension calculation as his theoretical one is known. However, there is a fundamental difference between the theoretical calculation of fractal dimension and the actual calculation of the fractal dimension from digital images with a given resolution undergoing discretization. We propose eight different calculation methods based on Richardson’s area–perimeter relationship: the Self-Convolution Patterns Research (SCPR) method accurately estimates the fractal dimension, as the 95% confidence interval includes the theoretical dimension. Full article

This paper addresses the critical gap in the existing literature regarding the combined buoyancy–Marangoni convection of power-law fluids in three-dimensional porous media with complex evaporation surfaces. Previous studies have rarely investigated the convective heat transfer mechanisms in such systems, and there is a lack of effective methods to accurately track fractal evaporation surfaces, which are ubiquitous in natural and engineering porous media (e.g., geological formations, industrial heat exchangers). This research is significant because understanding heat transfer in these complex porous media is essential for optimizing energy systems, enhancing thermal management in industrial processes, and improving the efficiency of phase-change-based technologies. For this scientific issue, a general model is designed. There is a significant temperature difference on the left and right sides of the model, which drives the internal fluid movement through the temperature difference. The upper end of the model is designed as a complex evaporation surface, and there is flowing steam above it, thus forming a coupled flow field. The VOF fractal reconstruction method is adopted to approximate the shape of the complex evaporation surface, which is a major highlight of this study. Different from previous research, this method can more accurately reflect the flow and phase change on the upper surface of the porous medium. Through numerical simulation, the influence of the evaporation coefficient on the flow and heat transfer rate can be determined. Key findings from numerical simulations reveal the following: (1) Heat transfer rates decrease with increasing fractal dimension (surface complexity) and evaporation coefficient; (2) As the thermal Rayleigh number increases, the influence of the Marangoni number on heat transfer diminishes; (3) The coupling of buoyancy and Marangoni effects in porous media with complex evaporation surfaces significantly alters flow and heat transfer patterns compared to smooth-surfaced porous media. This study provides a robust numerical framework for analyzing non-Newtonian fluid convection in complex porous media, offering insights into optimizing thermal systems involving phase changes and irregular surfaces. The findings contribute to advancing heat transfer theory and have practical implications for industries such as energy storage, chemical engineering, and environmental remediation. Full article

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 $z^n+\Im z^3-\Re z+\omega$, where $n\ge 4$ and $\Im ,\Re ,\omega \in \mathbb{C}$. 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. Full article

Bottlenose dolphins’ broadband click vocalisations are well-studied in the literature concerning their echolocation function. Their potential use for communication among conspecifics has long been speculated but has yet to be conclusively established. In this study, we first categorised dolphins’ click production based on their amplitude contour and then analysed the distribution of individual clicks and click sequences against their duration and length. The results show that the repertoire and composition of clicks and click sequences adhere to the three essential linguistic laws of efficient communication: Zipf’s rank–frequency law, the law of brevity, and the Menzerath–Altmann law. Conforming to the rank–frequency law suggests that clicks may form a linguistic code subject to selective pressures for unification, on the one hand, and diversification, on the other. Conforming to the other two laws also implies that dolphins use clicks according to the compression criterion or minimisation of code length without losing information. Such conformity of dolphin clicks might indicate that these linguistic laws are more general, which produces an exciting research perspective on animal communication. Full article

The torsional stiffness of rehabilitation robot joints is a critical performance determinant, significantly affecting motion accuracy, stability, and user comfort. This paper introduces an innovative traction drive mechanism that transmits torque through friction forces, overcoming mechanical impact issues of traditional gear transmissions, though accurately modeling surface roughness effects remains challenging. Based on fractal theory, this study presents a comprehensive torsional stiffness analysis for advanced traction drive joints. Surface topography is characterized using the Weierstrass–Mandelbrot function, and a contact mechanics model accounting for elastic–plastic deformation of micro-asperities is developed to derive the tangential stiffness of individual contact pairs. Static force analysis determines load distribution, and overall joint torsional stiffness is calculated through the integration of individual contact contributions. Parametric analyses reveal that contact stiffness increases with normal load, contact length, and radius, while decreasing with the tangential load and roughness parameter. Stiffness exhibits a non-monotonic relationship with fractal dimension, reaching a maximum at intermediate values. Overall system stiffness demonstrates similar parameter dependencies, with a slight decrease under increasing output load when sufficient preload is applied. This fractal-based model enables more accurate stiffness prediction and offers valuable theoretical guidance for design optimization and performance improvement in rehabilitation robot joints. Full article

In this paper, we introduce a generalised formulation of the logistic map extended to the complex plane and correspondingly redefine the classical Mandelbrot and Julia sets within this broader framework. Central to our approach is the development of an escape criterion based on the Picard orbit, which underpins the escape-time algorithms employed for graphical approximations of these sets. We analyse the structural and dynamical properties of the resulting Mandelbrot and Julia sets, emphasising their inherent symmetries through detailed visualisations. Furthermore, we examine how variations in a key parameter of the generalised map affect two critical numerical metrics: the average escape time and the non-escaping area index. Our computational study reveals that, particularly for Julia sets, these dependencies are characterised by intricate, highly non-linear behaviour—highlighting the profound complexity and sensitivity of the system under this generalised mapping. Full article

Fractal dimension has proven to be a valuable tool in the analysis of geological data. For instance, it can be used for assessing the distribution and connectivity of fractures in rocks, which is important for evaluating hydrocarbon storage potential. However, while calculating a single fractal dimension for an entire geological profile provides a general overview, it can obscure local variations. These localized fluctuations, if analyzed, can offer a more detailed and nuanced understanding of the profile’s characteristics. Hence, this study proposes a fractal characterization procedure using a new strategy based on moving windows applied to the analysis domain, enabling the evaluation of data multifractality through the Dynamical Approach Method. Validations for the proposed methodology were performed using controlled artificial data generated from Weierstrass–Mandelbrot functions. Then, the methodology was applied to real geological profile data measuring permeability and porosity in oil wells, revealing the fractal dimensions of these data along the depth of each analyzed case. The results demonstrate that the proposed methodology effectively captures a wide range of fractal dimensions, from high to low, in artificially generated data. Moreover, when applied to geological datasets, it successfully identifies regions exhibiting distinct fractal characteristics, which may contribute to a deeper understanding of reservoir properties and fluid flow dynamics. Full article