We develop a unified mathematical and computational framework linking fractional calculus, fractal geometry, scaling laws, and artificial intelligence. We establish a universal scaling-shift theorem demonstrating that fractional operators systematically transform fractal and multifractal spectra of self-affine stochastic processes. For any self-affine process with Hurst exponent [Formula: see text], we prove that a fractional derivative of order [Formula: see text] produces a new process with scaling exponent [Formula: see text], implying a fractal dimension shift [Formula: see text] We further derive a multifractal spectral deformation law [Formula: see text] which demonstrate how fractional dynamics rearranges local singularities. To connect theory with observations, we present an AI-facilitated framework of neural operators that can compute fractal dimensions and detect fractional orders from observations. Numerical experiments confirm the theoretical predictions and demonstrate strong agreement between analytical scaling laws and AI-inferred parameters.

This paper investigates the informational efficiency and cross-dependence structure of sustainability-oriented equity indices under major geopolitical and climate-related shock events from a complex-systems perspective. We analyze four sustainability-linked S&P indices — Environmental, Clean Energy, Smart Transportation, and Agribusiness — alongside the S&P Global BMI as a broad market benchmark, using daily data from 2016 to 2025. The empirical framework combines entropy–information quantifiers based on ordinal patterns with multifractal detrended cross-correlation analysis (MFDCCA), allowing us to capture nonlinear, scale-dependent, and regime-sensitive dynamics beyond standard correlation measures. Informational efficiency is assessed through permutation entropy and Fisher information jointly embedded in the Shannon–Fisher causality plane, while cross-market dependence with geopolitical risk and climate physical risk is examined through multifractal scaling exponents and spectra across distinct crisis periods, including COVID-19, the Russia–Ukraine war, the Israel–Hamas conflict, and a recent trade-policy shock. The results reveal substantial heterogeneity across sustainability sectors. Transportation-related indices appear closer to the random-efficiency benchmark, whereas Environmental and Agribusiness indices exhibit stronger local temporal structure. Shock episodes generate pronounced, but sector-specific, departures from baseline efficiency. Multifractal analysis uncovers persistent and highly heterogeneous cross-correlations between sustainability indices and external risk factors, which intensify during periods of elevated uncertainty, particularly for Clean Energy and Transportation under geopolitical stress and for Environmental and Agribusiness sectors under climate physical risk. Overall, the findings demonstrate that sustainability-oriented equity markets are not informationally or structurally homogeneous and that external geopolitical and climate risks transmit into these markets in a nonlinear, multifractal, and scale-dependent manner. These results have implications for portfolio diversification, risk management, and the modeling of sustainability-linked financial systems under extreme events.

We develop a PDE framework on anisotropic fractal structures, which are fractal materials that could have different dimensions in each spatial direction. In particular, we develop the notion of distributional fractal gradient and the appropriate weighted Sobolev spaces for an equivalent continuum problem. We prove a number of related results, such as a new Poincaré Lemma and Poincaré inequality. One of the main results is a new scalar trace theorem, which allows for the solvability of the fractal Laplace and Poisson equations with non-homogeneous boundary data. Along the way, we track the dependence in the various estimates on the fractal dimension in each direction and the length of the original fractal solid.

Enhancing the heat and mass transfer performance of working fluids remains a critical challenge to pursue sustainable and energy-efficient technologies. Although regular working fluids have superior thermo-physical properties to pure base fluids, they often face limitations that hinder their adoption in multifunctional applications. To overcome these challenges, this study develops a novel, comprehensive and physically consistent mathematical model for an unsteady, electrically conducting ternary hybrid nanofluid composed of graphene oxide (GO), cerium oxide (CeO[Formula: see text], and hexagonal boron nitride ([Formula: see text]-BN) suspended in an environmentally friendly ionic liquid (Ethyl-3-methylimidazolium tetrafluoroborate) [EMIM][BF 4 ]. The model integrates magnetic effects, radiation heat transfer, viscous dissipation, Joule heating, and coupled thermo-diffusion effects. A fractal–fractional derivative operator is employed to generalize the governing equations, while the local radial basis functions (RBF) scheme is used to solve them numerically. Computational and graphical results reveal that CeO 2 suppresses the fluid velocity due to increased inertial resistance, while the dispersion of [Formula: see text]-BN significantly enhanced the thermal profile, resulting in a higher Nusselt number. Furthermore, higher values of Dufour and Soret numbers enhance the coupled heat and mass transfer rates, indicating the model’s potential to design advanced heat exchangers and smart cooling devices. These findings provide valuable guidelines for designing compact heat exchangers and thermal energy storage systems for applications in renewable energy and microelectronics cooling.

In this research, we work on the generalization of the classical Hermite–Hadamard–Mercer (HHM)-type inequalities by establishing a more generalized form with the addition of the beta function. The inequalities are very important in mathematical analysis, especially when estimating the bounds of the convex function within an interval. We also build new trapezoidal-type inequalities for differentiable convex functions in terms of the beta function with more precise estimations. This research contributes important results of novel variants of fractal Hölder’s, power mean and Young’s inequalities. One of the most important roles of these estimates is their flexibility since they are readily translatable into familiar integral estimates, like traditional integral inequalities, Riemann–Liouville (RL) fractional integral inequalities and [Formula: see text]-Riemann–Liouville ([Formula: see text]-RL) fractional integral inequalities. This identification places them at center stage in classical and fractional calculus. Lastly, to illustrate the applicability of our findings, we provide several applications of these newly derived inequalities, indicating their potential contribution in mathematical analysis and other related areas.

The edge-Wiener index, as an edge-based variant of the Wiener index, serves as an important topological descriptor for characterizing network structures. In this paper, we study the skeleton network derived from the classic fractal — the level-3 Sierpinski triangle — and investigate the exact analytical expression of its edge-Wiener index. [Formula: see text]By introducing the finite patterns method (Wang et al.), the geometric relations between edge pairs are classified into four fundamental patterns, and corresponding distance recursion relations are established. Based on pattern classification and self-similar measures, we derive an exact formula for the edge-Wiener index with respect to the iteration order [Formula: see text].

This work implemented the faster_rcnn_resnet_101_coco neural architecture in Python with TensorFlow using Scientific Machine Learning to monitor face mask usage for contagion prevention. Convolutional neural networks were used in a supervised model trained on images captured from students aged 18–35. Training was conducted in environments with and without GPUs, evaluating execution time and prediction accuracy. The predictive model was assessed using the confusion matrix, error, precision, recall, and [Formula: see text]1-score. It was trained on 7200 proprietary images. Based on functional tests, the system was validated.

In this paper, we introduce a new concept of dimension on fractal networks, the so-called Hausdorff Steiner [Formula: see text]-dimension. It is defined by a natural generalization of the Hausdorff dimension in Zeng’s work. In particular, this new dimension is a non-trivial higher estimate for the Hausdorff dimension. We explore its basic properties and determine the exact values of some networks. Meanwhile, we discuss the changes under different graph operations, including single node sum and the Cartesian product of graphs. Within a specific class of networks, the Hausdorff Steiner [Formula: see text]-dimension and the Hausdorff dimension exhibit an exact quantitative relationship.

The effective representation of complex systems relies heavily on optimizing network models that capture their intricate and dynamic interdependencies. This study introduces a novel category of polygonal, fractal networks characterized by distinct alteration patterns and structural properties. Through a comprehensive examination of the statistical and topological characteristics of these networks, including clustering coefficient, average path length, degree distribution, and network density, we investigate their properties. A rigorous analysis combining mathematical modeling and empirical evaluation reveals that these fractal networks possess emergent scale-free and small-world properties, which are essential for conferring robustness and efficient information flow. Our findings provide valuable insights into the dynamics of complex systems and lay the groundwork for understanding and designing more efficient network-based systems in socioeconomic and biological contexts. This analysis not only theoretically confirms predictions related to fractal and complex networks but also enhances understanding of the adaptability and scalability of these structures.

In this study, a fractal-based model for apparent permeability is developed by embedding the tree-like branching network and capillaries within a porous matrix. The model incorporates multiple gas flow mechanisms, including slip flow and Knudsen diffusion. The results indicate that apparent permeability decreases with increasing pore pressure or decreasing maximum parent branch width. At low pressure and small maximum parent branch width, the flow is predominantly governed by Knudsen diffusion, while at high pressure and large maximum parent branch width, slip flow dominates. As pressure decreases or the maximum parent branch width decreases, the flow transitions from slip flow to Knudsen diffusion. In addition, higher porosity and larger fractal dimensions both contribute to higher apparent permeability. Moreover, the geometric parameters of the branching structure, such as the length ratio, width ratio, branch aspect ratio, and branch level, significantly influence apparent permeability. This model reveals the coupled effects of multiple transport mechanisms within the branching structure. It enhances our understanding of gas flow in the complex porous media of shale reservoirs.