This study numerically investigates the effects of an aligned magnetic field on the behavior of Williamson nanofluid traversing a stretched sheet that is submerged in a porous medium while considering radiation and heat source/sink effects, along with convective limits on the border. The ordinary differential equations are derived from the corresponding partial differential equations using the similarity alteration technique. These equations are then solved using the fourth-order Runge-Kutta method combined with shooting practice. The impact of various values for the restrictions on the flow field profiles is analyzed and visualized graphically. To show the changes, graphs and tables are used for skin friction, Nusselt numeral, and Sherwood numeral across diverse flow conditions. An elevation in the magnetic restriction and permeability restriction contributes to the velocity field decreasing while improving the temperature and concentration distribution. The skin friction coefficient, Sherwood numeral, and Nusselt numeral are found to exhibit a decreasing trend in relation to the magnetic restriction, aligned angle restriction, Forchheimer numeral, and permeability restriction.

The Hermite–Hadamard inequality obtained a prominent place in the field of mathematical inequalities since its discovery. This inequality has been inspiring mathematicians to continuously generalize, improve and refine it. This paper explores a novel approach of establishing conticrete forms of the Hermite–Hadamard–Mercer-type inequalities within the framework of Caputo fractional derivative operators through the application of separable sequences. The main results of this paper are attributed to the derivation of conticrete Hermite–Hadamard–Mercer-type inequalities by considering three [Formula: see text]-tuples and using the property of convexity within the framework of fractional calculus. By employing various vectors, bases, and their dual bases, we derive a series of subsequent corollaries from the main inequalities. These derivations yield both new and previously established inequalities as special cases of the primary results. The remarks at the end of these corollaries extend these results to diverse sequence types, including nondecreasing sequences in P-mean, star-shaped, monotonic, synchronous and convex sequences, illustrating the broad utility of the proposed methodology.

Modern complex network theory reveals that real-world systems possess not only small-world and scale-free topological features, but may also exhibit statistical self-similarity and fractal scaling. Within the framework of fractal geometry and iterated function systems, this paper investigates the geometric dimension of a class of networks generated from binary structures. Inspired by the Frostman measure theory, we adopt the definition of the Frostman dimension for a family of networks and systematically compute the dimension for networks derived from the full binary tree and from a restricted subtree where consecutive left-child nodes are forbidden. Through precise combinatorial analysis and asymptotic estimation, we prove that the Frostman dimension of the binary tree is log2/log3, while the dimension of Fibonacci network-whose node growth follows the Fibonacci sequence-is log [Formula: see text], where [Formula: see text] is the golden ratio. The results show that the geometric dimension of such binary networks coincides with the fractal dimension of classical sets such as the Cantor set, and in the Fibonacci case, is closely related to the Fibonacci entropy. This establishes a deeper connection between network science and fractal geometry. The study provides new theoretical insights and computational methods for understanding the fractal properties of networks under regular and constrained growth mechanisms.

This study explores the fractional (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation, a higher-order nonlinear model known for capturing complex wave behaviors influenced by spatial and dispersive effects. To obtain a variety of solitary wave solutions including dark, bright, and mixed solitons-three advanced analytical approaches are employed: the Riccati modified extended simple equation method, the modified F-expansion method, and the newly improved generalized exponential rational function method. The model is first simplified into an ordinary differential equation using a fractional wave transformation. Mathematical simulations illustrate how key parameters affect wave propagation, highlighting the strength and flexibility of the proposed techniques in handling complex nonlinear systems. By demonstrating the effectiveness of these modern analytical methods and revealing distinctive features of nonlinear dynamics, this research offers valuable insights into higher-dimensional nonlinear equations and wave phenomena.

Crude oil markets are characterized by noise and multifractal features, which undermine the reliability of traditional hedge ratio estimation models. Therefore, this study develops a denoising-multifractal dual intelligent integration framework for estimating optimal hedge ratios in the Shanghai crude oil futures (INE) and crude oil spot markets, with a primary focus on minimizing spot risk while enhancing hedging returns. The framework begins with an innovative use of the Complementary Ensemble Empirical Mode Decomposition (CEEMD) approach to remove high-frequency noise, improving data quality. It then employs the Multifractal Detrended Cross-Correlation Analysis (MF-DCCA) method to capture the multifractal structure of crude oil futures and spot markets. Building on these results, the Flower Pollination Algorithm (FPA) is employed to integrate hedge ratios in a two-stage manner. First, local hedge ratios are aggregated across time scales within each fluctuation level. Second, hedge ratios are further integrated across different fluctuation amplitudes. This unique design allows the model to fully exploit the multi-scale and multi-fluctuation information for deriving the optimal hedge ratios. Empirical analysis confirms the existence of significant noise and multifractal properties in crude oil markets. Moreover, the hedging results show that the proposed model outperforms all competing methods, achieving higher accumulated returns, hedging effectiveness (HE), and Sharpe ratios in most cases. The study offers an effective hedge ratio estimation method for investors in the INE crude oil futures and spot markets.

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. By introducing the finite patterns method (Wang, Yu and Xi, Fractals, 2017), 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 n.

In this paper, we introduce a new concept of dimension on fractal networks, the so-called Hausdorff Steiner k-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 k-dimension and the Hausdorff dimension exhibit an exact quantitative relationship.

The fractal Rosenau-Burgers equation is proposed and its fractal variational principles were established by Wang and Xu, et al in their work (Fractals, 2024, 32, 6: 2450121). In this study, the more general variational principles with free parameters are developed using the semi-inverse method (SIM) via introducing a new trial-functional taking some parameters. The results reveal that the fractal variational principles obtained by Wang and Xu are the special cases of the general fractal variational principles in this paper.

Gas-water relative permeability significantly affects the efficiency of fluid transport. However, due to the complex fracture-pore structures in coal, accurately predicting this property remains a challenge. This study presents a novel gas-water relative permeability model developed based on the fractal characteristics of coal fracture-pore structures. The model utilizes a tree-like bifurcation network and a curved capillary bundle approach to characterize the roughness of the fracture-pore system. By incorporating the distribution theory of gas-water phases in annular flow and integrating the cubic law with the momentum balance equation, two permeability solution methods were established, grounded in two-phase flow regimes and multi-scale structural features. Moreover, the structural clogging mechanism is incorporated into the model to further improve its applicability and accuracy. To validate the model, theoretical results were compared with experimental data obtained using a self-developed experimental system. Additionally, a sensitivity analysis was conducted to evaluate the influence of various parameters. Several key factors were found to affect coal permeability: permeability K g was positively correlated with maximum fracture opening (e smax ), maximum pore diameter (r max ), fracture opening fractal dimension (D e ), pore distribution fractal dimension (D r ), and length ratio (χ); and negatively correlated with fracture tortuosity fractal dimension (D T ), pore tortuosity fractal dimension (D t ), characteristic length (L), bifurcation angle (α), and residual water saturation (S w ). Notably, fracture structure parameters exert a much greater influence on permeability than pore structure parameters, highlighting the critical role of fracture networks in controlling permeability in fracture-pore such as coal rock. This model provides parameter support for developing CO 2 storage efficiency prediction models and offers scientific guidance for designing fracture network regulation strategies in CO 2 storage schemes.

Images acquired in real-world applications such as medical imaging, remote sensing, and hyperspectral imaging are frequently degraded by mixed noise, where Gaussian perturbations, impulsive outliers, and structured artifacts coexist. Conventional restoration methods, typically developed and evaluated on grayscale or RGB images with only one or three channels, often struggle to handle such heterogeneous degradations. In this work, we propose a denoising framework that integrates fractal priors and deep image prior (DIP) for mixed-noise image restoration, which can be naturally applied to multi-channel data such as hyperspectral images (HSI). Specifically, the noisy observation is decomposed into four components: the clean image, Gaussian noise, sparse impulsive noise, and structured stripe noise. Each component is penalized with tailored regularizations, including a Frobenius norm for Gaussian noise, an ℓ 1 norm for impulsive noise, and a mixed ℓ 2,1,1 norm for structured noise. By combining the complementary capabilities of DIP and fractal modeling, a fractal regularization term enforces nonlocal self-similarity, whereas DIP provides a strong implicit prior that captures global image structure for faithful reconstruction. The overall problem is formulated as a constrained variational optimization and efficiently solved using the Alternating Direction Method of Multipliers (ADMM). Extensive experiments on grayscale, RGB, and hyperspectral datasets demonstrate that the proposed framework effectively suppresses diverse noise types while preserving fine structural details, achieving superior robustness compared with existing approaches.