Fractal Research Articles

Multi-scale contact characteristics and leakage prediction of flange seal based on fractal geometry

Multi-scale contact characteristics and leakage prediction of flange seal based on fractal geometry

Polynomial Fourier decay for fractal measures and their pushforwards

Abstract We prove that the pushforwards of a very general class of fractal measures $$\mu $$ μ on $$\mathbb {R}^d$$ R d under a large family of non-linear maps $$F{:}\,{\mathbb {R}}^d \rightarrow {\mathbb {R}}$$ F : R d → R exhibit polynomial Fourier decay: there exist $$C,\eta >0$$ C , η > 0 such that $$|\widehat{F\mu }(\xi )|\le C|\xi |^{-\eta }$$ | F μ ^ ( ξ ) | ≤ C | ξ | - η for all $$\xi \ne 0$$ ξ ≠ 0 . Using this, we prove that if $$\Phi = \{ \varphi _a{:}\,[0,1] \rightarrow [0,1]\}_{a \in \mathcal {A}}$$ Φ = { φ a : [ 0 , 1 ] → [ 0 , 1 ] } a ∈ A is an iterated function system consisting of analytic contractions, and there exists $$a \in \mathcal {A}$$ a ∈ A such that $$\varphi _a$$ φ a is not an affine map, then every non-atomic self-conformal measure for $$\Phi $$ Φ has polynomial Fourier decay; this result was obtained simultaneously by Algom, Rodriguez Hertz, and Wang. We prove applications related to the Fourier uniqueness problem, Fractal Uncertainty Principles, Fourier restriction estimates, and quantitative equidistribution properties of numbers in fractal sets.

System of fractal-fractional differential equations and Bernstein wavelets: a comprehensive study of environmental, epidemiological, and financial applications

Abstract The fractal-fractional derivatives uniquely incorporate memory effects, heterogeneity, and fractal geometry, making them essential for studying dynamic systems compared to integer-order derivatives which cannot capture such intricate dynamics. So, this study aims to analyze the dynamical behaviour of the model of systems of differential equations under Caputo fractal-fractional derivative by designing a numerical framework based on the fractional order Bernstein wavelets. The Caputo fractal-fractional derivative under the power law kernel has been employed to obtain more accurate performances of the considered model as compared to fractional and integer order. A key novelty of this work lies in the implementation of fractional order Bernstein wavelets with collocation grids, which transform the complex differential equations into simpler algebraic equations, ensuring computational efficiency. The validity of the mentioned scheme is demonstrated by examining some famous problems, such as pond pollution problem, SIR epidemic modelling of infectious disease, a model of HIV interactions with CD4+T cells, and a financial model, showcasing its broad applicability across applied science, finance and engineering. For compactness, an error analysis using residual function is performed for fractal-fractional order. The graphs of the solution in integer, fractional and fractal-fractional order show that the achieved solutions are very close to the actual result of the examples and the error progressively decreases as the number of wavelets basis increases. In the case α = β = 1, the obtained approximated wavelet solutions for the suggested model are in good harmony with the exact solutions, the fourth-order Runge-Kutta (RK4) method and existing schemes provided in the literature. This investigation demonstrates that the mentioned scheme is very effective and straightforward for solving such kinds of fractal-fractional models.

Matematiksel Modellemeler Yoluyla Bediüzzaman’ın Yedinci Söz Eserinin Kavram Analizleri

In this article, Bedüzzaman Said Nursi's work The Seventh Word is discussed in terms of mathematical modeling. Thus, it is aimed to analyze a text belonging to human sciences from a different perspective with mathematical modeling. The relationships between the concepts in the text are modeled with mathematical approaches such as network theory, heat map, concept map, Probability and Utility Function, differential equation, Sierpinski triangle, fractals. The concepts and judgment sentences in the researched text are visualized. How each produced graph and shape is produced, which model is used and the obtained result are interpreted. It has become debatable how such an analysis can contribute to future research or how similar studies can be expanded. Document analysis and descriptive analysis are used in the research. Keywords: The Seventh Word, mathematical modeling, network theory, fractal geometry.

Study on particle size distribution of coal particle flocs based on hydrophobic flocculation kinetics combined with fractal theory

ABSTRACT Based on the multi-domain fractal theory, the distribution characteristics of each granularity level of coal particles during hydrophobic flocculation were analyzed. The fractal dimension of fine-grained increased from 1.07 to 1.33, indicating that the particle size distribution range was wide, and the internal difference was significant. The fractal dimension of middle-grained decreased from 2.71 to 2.03, showing that the particle size growth noticeable and the morphology of flocs tended to be regular. The fractal dimension of the coarse-grained changes slightly around 2.90, demonstrating that the flocs had a relatively stable spatial structure under the dynamic balance of fragmentation and reaggregation. By introducing the fractal dimension of each granularity level into the flocculation kinetic equation, the conservation relationship between the median particle size and the standard deviation of the floc was quantified, and the analysis of the particle distribution characteristics of the hydrophobic flocculation system at any time was realized.

LEVERAGING BLOCKCHAIN WITH CHAOTIC OPPOSITIONAL BARNACLES MATING OPTIMIZER-BASED DEEP LEARNING MODEL FOR SECURE IOT ENVIRONMENT IN CONSUMER ELECTRONICS

The security of IoT networks has become a major concern with the ubiquity of Internet of Things (IoT) technology. The concept of intrusion detection system (IDS) is a complex system to discover an intruder in the IoT platform, where the intruder can be a host that tries to gain some other nodes without authorization. Due to the complexity and resource constraints, classical IDS have their limitations in the context of IoT networks. There has been significant research in integrating both IDS and blockchain (BC) to detect existing and emerging cyberattacks and improve data privacy, correspondingly. In these approaches, learning-based ensemble algorithms can concurrently ensure data privacy and facilitate the detection of sophisticated malicious events. This study introduces a BC with fractal chaotic oppositional barnacles mating optimizer-based deep learning (BCOBMO-DL) Model for Secure IoT environment. The BCOBMO-DL technique exploits BC technology with DL-based intrusion detection to protect the IoT environment. In the BCOBMO-DL technique, the linear scaling normalization (LSN) approach can be used to scale the input data into a uniform format. In addition, the BCOBMO-DL technique designs the COBMO algorithm for electing the optimal subset of features. For intrusion detection, the self-attention bidirectional gated recurrent unit (SA-BiGRU) model is applied with fractal theory. Finally, the reptile search algorithm (RSA)-based hyperparameter tuning process is utilized for tuning the parameters based on the SA-BiGRU model. Furthermore, the BC technology is useful for achieving security in the IoT network. The experimental evaluation of the BCOBMO-DL method takes place on the benchmark NSLKDD dataset. The widespread experimental outcomes highlighted the enhanced detection outcomes of the BCOBMO-DL algorithm over other models.

FRACTAL ANALYSIS IN ALTERNATIVE INVESTMENTS: UNVEILING DYNAMICS AND DIVERSIFICATION BENEFITS

The realm of alternative investments, comprising commodities, real estate, private equity, and other non-traditional asset classes, presents a rich landscape for investors seeking diversification and enhanced returns. Amidst the complexities of these markets, fractal analysis emerges as a powerful tool for unraveling the intricate dynamics and exploring diversification benefits. This paper delves into the application of fractal theory in alternative investments, assessing its efficacy in analyzing asset price movements, correlations, and portfolio diversification. Through a comprehensive review of the literature and empirical evidence, we elucidate how fractal analysis techniques can deepen our understanding of alternative investment markets and inform strategic decision-making. Key findings highlight the ability of fractal analysis to capture nonlinear dependencies, self-similarity, and multi-scale dynamics across various asset classes. The study demonstrates how these insights enhance portfolio resilience, optimize asset allocation, and improve risk-adjusted returns. Additionally, this research contributes novel perspectives on integrating fractal methods with portfolio management and risk diversification strategies.

Pore Structure and Heterogeneity Characteristics of Deep Coal Reservoirs: A Case Study of the Daning–Jixian Block on the Southeastern Margin of the Ordos Basin

To clarify the micropore structure and fractal characteristics of the Danning–Jixian block on the eastern margin of the Ordos Basin, this study focuses on the deep coal rock of the Benxi Formation in that area. On the basis of an analysis of coal quality and physical properties, qualitative and quantitative studies of pore structures with different pore diameters were conducted via techniques such as field emission scanning electron microscopy (FE-SEM), low-pressure CO2 adsorption (LP-CO2A), low-temperature N2 adsorption (LT-N2A), and high-pressure mercury intrusion (HPMI). By applying fractal theory and integrating the results from the LP-CO2A, LT-N2A, and HPMI experiments, the fractal dimensions of pores with different diameters were obtained to characterize the complexity and heterogeneity of the pore structures of the coal samples. The results indicate that the deep coal reservoirs in the Danning–Jixian block have abundant nanometer-scale organic matter gas pores, tissue pores, and a small number of intergranular pores, showing strong heterogeneity influenced by the microscopic components and forms of distribution of organic matter. The pore structure of the Benxi Formation exhibits significant cross-scale effects and strong heterogeneity and is predominantly composed of micropores that account for more than 90% of the total pore volume; the pore structure is affected mainly by the degree of coalification, the vitrinite group, and the ash yield. Fractal analysis reveals that the heterogeneity of macropores is greater than that of mesopores and micropores. This may be attributed to the smaller pore sizes and concentrated distributions of micropores, which are less influenced by diagenesis, resulting in simpler pore structures with lower fractal dimensions. In contrast, mesopores and macropores, with larger diameters and broader distributions, exhibit diverse origins and are more affected by diagenesis, reflecting strong heterogeneity. The abundant storage space and strong self-similarity of micropores in deep coal facilitate the occurrence, flow, and extraction of deep coalbed methane.

Fractal Characteristics of the Relationship between Industrial Water Use Efficiency and Economic Development at Multiple Temporal and Spatial Scales

The fractal theory is introduced to explore the stable fractal characteristics of the correlation between industrial water use efficiency (IWUE) and economic development level (EDL) at different spatial and temporal scales. The research was carried out based on statistical data including the water consumption per 104 yuan of industrial added value (as IWUE) and per capita GDP (as EDL) of 336 cities in China from 1998 to 2017 and 142 countries across the world in 2017. The results show that during 1998-2017, the IWUE and the EDL of Chinese cities have improved gradually. Although the difference of IWUE and EDL among cities has gradually narrowed, the fractal dimension of the correlation between them has remained at around 1.45 in the past 20 years. The overall fractal characteristics is stable while the value of individual city is dynamic with time. Spatially, relationship between IWUE and EDL also maintains a stable fractal pattern for cities in different sub-regions and international countries. The stable fractal characteristics of EDL and IWUE at different spatial-temporal scales lays a theoretical foundation for further exploration of the structure of WUE regulation model and management target.

Transforming frontiers: The next decade of differential equations and control processes

Mathematics serves as the fundamental basis for innovation, propelling technological advancement. In the forthcoming decade, the convergence of differential equations and control processes is poised to redefine the frontiers of scientific exploration. The integration of artificial intelligence and machine learning with differential equations is set to inaugurate a new era of problem-solving, enabling the extraction of latent physical insights and accelerating solution discovery. Multi-scale modeling, with its capacity to span disparate physical domains, has the potential to resolve long-standing puzzles in fields such as fluid mechanics and nanoscience. Furthermore, the integration of fractal geometry with differential equations holds the promise of novel perspectives for understanding and optimizing complex systems, ranging from urban landscapes to turbulent flows. The integration of artificial intelligence (AI) with control innovations is poised to play a pivotal role in the development of next-generation technologies, with the potential to transform diverse sectors such as medicine, communication, and autonomous systems. This paper explores these developments, highlighting their potential impacts and emphasizing the necessity for interdisciplinary collaboration to leverage their full potential.

Mechanical Properties and Microscopic Fractal Characteristics of Lime-Treated Sandy Soil

In order to reveal the intrinsic mechanism of the mechanical properties of lime-treated sandy soil from a microscopic perspective, triaxial tests were conducted to analyze the macroscopic mechanical characteristics of sandy soil with different lime contents (0%, 5%, 8%, and 12%). The changes in the microstructure of the lime-treated sandy soil were studied through scanning electron microscopy, energy-dispersive spectroscopy, and mercury intrusion tests, combined with fractal theory for quantitative characterization. The results indicate that the stress–strain curve of lime-treated sandy soil can be divided into four stages: linear elastic, non-linear, failure, and residual strength. With the increase in lime content, the peak stress and cohesion first increase and then decrease, while the internal friction angle first decreases and then increases, suggesting the presence of an optimal threshold for lime content between 5% and 12%. The failure mode transitions from diagonal shear failure to bulging failure, significantly enhancing stability; both the fitted Mohr–Coulomb and Drucker–Prager failure criteria effectively reflect the failure patterns of the specimens in principal stress space. The results based on the three fractal dimensions demonstrate that lime-treated sandy soil exhibits clear fractal characteristics, with the highest fractal dimension value at a lime content of 8%, corresponding to the highest overall strength. In addition, the fractal dimension shows a binomial relationship with pore characteristic parameters and shear strength parameters; it can effectively characterize the complexity of the microstructure and accurately predict changes in shear strength parameters.

Input Aware Lightweight Cryptosystem for IoT Infrastructure – Design, Implementation and Validation on Reconfigurable Hardware

The rise of IoT has also brought technological developments and threats. Edge computing has become a vital solution for most networks and demands resource-constrained operations. This work presents a lightweight data encryption scheme based on the Sierpinski Triangle Fractal, Chaotic Logistic map and their FPGA implementation on Intel Cyclone IV E. The size of the fractal geometry emerges from the number of bits in the input. Hence, this algorithm is an input-aware cryptosystem that simultaneously handles all input bits using rotation, random data fitting and diffusion. Further, round keys are supplied through the logistic map, and three incarnations of their FPGA implementations were investigated against randomness. Worst-case inputs have been taken to evaluate the design, and the results are validated through standard metrics. Round keys and cipher data have passed the NIST SP 800–22 test suites, thus evidence of the randomness. The proposed cryptosystem consumes only 944 logic elements (0.8%) and cipher 27/128/256 bits of data concurrently on Cyclone IV E FPGA. Encryption time has been calculated as 400 ns for 50 MHz of operating clock.

Improving Ethical Leadership in Sustainable Public Health Through Fractal AI

This study explores innovative, ethical leadership approaches using artificial intelligence (AI) and fractal geometry in public health while fostering sustainable business practices within public health systems. The research employs a qualitative methodology based on case studies, secondary data analysis, and fractal-based AI algorithm evaluations. It examines advanced algorithms' technical applications in public health settings, improving data privacy, copyright, and intellectual property protection. The study finds that fractal algorithms offer robust solutions for promoting ethical leadership in AI-driven public health systems. Fractal geometry's complexity and self-similarity improve predictive modeling, resource allocation, and system transparency while ensuring legal and ethical compliance. By applying fractal algorithms, public health organizations can improve privacy protection, intellectual property management, and ethical governance. The study highlights the need for further research on practical applications, optimization of fractal algorithms, and overcoming the computational demands associated with their deployment in public health. Ethical leadership approaches supported by fractal algorithms can drive more equitable and secure public health interventions, enhancing trust in AI-driven solutions and reducing healthcare access and outcomes disparities. This research presents a novel integration of fractal geometry and AI to address critical ethical issues in public health, providing innovative solutions for data privacy, intellectual property protection, and ethical leadership practices.

Nonlinear dynamic analysis of PEEK based polymer composite involute spline couplings with textured tooth

To improve the wear resistance of involute spline couplings and meet the requirements of lightweight design, this work focuses on PEEK (polyetheretherketone) based polymer composite involute spline couplings with tooth surface texture. A dynamic model and differential equation of PEEK based polymer composite involute spline couplings with tooth surface texture were established using the concentrated mass method. The arrangement of tooth surface texture was established based on fractal theory, and a calculation method for the dynamic meshing stiffness of the system was established using the potential energy method. The surface texture morphology parameters, texture arrangement, and material characteristics were considered in this method. The dynamic characteristics of different textured systems were analyzed, and the influence of geometric eccentricity and rotational speed on the dynamic characteristics of the system was explored. The results show that the dynamic meshing stiffness of the surface texturing system exhibits a nonlinear state, and texture 2-F exhibits good stiffness characteristics; Different texture systems exhibit different dynamic characteristics, among which texture 2-E exhibits poorer dynamic characteristics; Compared with complete alignment, the dynamic characteristics of large misalignment systems are more complex; Eccentricity of dynamic and static mixing at high speeds can alter the stability of the system and make it non periodic in motion. This lays a theoretical foundation for the precise design of PEEK based polymer composite involute spline couplings with surface texture and the analysis of service performance based on the design of similar components.

Topological Analysis of Fractal Binary and Ternary Trees

Fractals are complex geometric objects that seem the same at different scales and have self-similarity. Because of this special characteristic, fractals can be used to simulate intricate biological processes. Fractal binary and ternary trees are novel data types that combine the productiveness of tree architectures with the ideas of fractal geometry. In addition to self-similarity and scalability, these trees have potential across a range of computer science and medical applications. The main goal of the study is to identify and analyze topological indices and graph entropy for fractal binary and fractal ternary trees. By examining indices such as the Randić index, Zagreb indices, and entropy measurements, the study aims to obtain a comprehensive knowledge of the structural complexity and information-theoretic properties of these fractal graphs. The study starts with the vertex and edge partitioning of fractal binary and ternary trees in order to distinguish different structural classes. Using these partitions, we obtained topological indices and graph entropy values for the fractal trees. Also, this study compares the topological indices for each fractal tree with the number of copies in the fractal dimension for a succession of graphs.

From Chaos to Fractals: A Review of Mathematical Modeling of Population Dynamics

The dynamic changes in population size are a core issue in ecological research and are crucial for understanding ecosystem stability and biodiversity. Chaos theory and fractal theory offer new perspectives for exploring population dynamics. Through mathematical modeling and computational methods, the complexity and intrinsic patterns of population changes can be revealed. This paper reviews the application of chaos theory and fractal theory in the study of population dynamics. The research covers the role of chaos theory in revealing the nonlinear characteristics of population dynamics and sensitivity to initial conditions, as well as the ability of fractal theory to describe population spatial distribution patterns and scale effects. Special attention is given to the application of the fractal model (_p_ model) proposed by Su et al. in analyzing species abundance distributions (SAD) within communities, and the potential value of chaos and fractal theories in practical ecological conservation and management. This paper finds that chaos theory and fractal theory provide powerful analytical tools for studying population dynamics, helping to understand the complexity and unpredictability of population size changes. The application of these theories not only enhances the understanding of population spatial structure and distribution patterns but also provides a scientific basis for biodiversity conservation and ecosystem management. Future research may further deepen the application of these theories, develop new mathematical models and computational methods to better predict and manage population dynamics. Through interdisciplinary collaboration, these theories are expected to play a greater role in ecological conservation and resource management

Assessing the global dynamics of Nipah infection under vaccination and treatment: A novel computational modeling approach.

In biology and life sciences, fractal theory and fractional calculus have significant applications in simulating and understanding complex problems. In this paper, a compartmental model employing Caputo-type fractional and fractal-fractional operators is presented to analyze Nipah virus (NiV) dynamics and transmission. Initially, the model includes nine nonlinear ordinary differential equations that consider viral concentration, flying fox, and human populations simultaneously. The model is reconstructed using fractional calculus and fractal theory to better understand NiV transmission dynamics. We analyze the model's existence and uniqueness in both contexts and instigate the equilibrium points. The clinical epidemiology of Bangladesh is used to estimate model parameters. The fractional model's stability is examined using Ulam-Hyers and Ulam-Hyers-Rassias stabilities. Moreover, interpolation methods are used to construct computational techniques to simulate the NiV model in fractional and fractal-fractional cases. Simulations are performed to validate the stable behavior of the model for different fractal and fractional orders. The present findings will be beneficial in employing advanced computational approaches in modeling and control of NiV outbreaks.

Understanding the role of movable oil in the complexity of shale pore structures based on fractal theory

Understanding the role of movable oil in the complexity of shale pore structures based on fractal theory

Local fractional Ostrowski type inequalities for generalized s-φ-convex functions on fractal sets

Local fractional Ostrowski type inequalities for generalized s-φ-convex functions on fractal sets

Revisiting holographic codes with fractallike boundary erasures

In this paper we investigate the code properties of holographic fractal geometries initiated in Pastawski []. We study reconstruction wedges in AdS3/CFT2 for black hole backgrounds, which are in qualitative agreement with the vacuum-AdS approximation using generalized entanglement entropy in Bao []. In higher dimensions, we study reconstruction wedges for the infinite, straight strip in AdSd+1/CFTd and clarify the roles of “straight” and “infinite” in their code properties. Lastly, we comment on uberholography from the perspective of complexity transfer and one-shot holography. Published by the American Physical Society 2025