Search Results (1,324)

We study the incompressible fractional viscous–resistive magnetohydrodynamic system on ${\mathbb{R}}^n$ with fractional diffusion ${(-\Delta )}^{\alpha }$, where $\alpha \in (1/2,1]$, and with positive viscosity and resistivity coefficients $\mu ,\nu >0$. The problem is treated at the scale-invariant regularity $s_c={\displaystyle \frac{n}{p}}+1-2\alpha .$ For small divergence-free initial data in the critical Triebel–Lizorkin–Lorentz space ${\dot{F}}_{p,r}^{s_c,q}$, we construct a unique global mild solution. The main contribution is the use of the single-norm time–frequency space $m_m^{\prime }{\dot{F}}_{p,r}^{s_c,q}$, built on Meyer wavelets and the parabolic gauge $t{2}^{2\alpha j}$. This space keeps the critical spatial size, the short-time behavior, and the high-frequency decay in one norm. By using a Gevrey-weighted Duhamel formulation, we prove boundedness of the corresponding fractional heat propagators and establish the bilinear paraproduct estimate required for the fixed-point argument. Consequently, $e^{{\left(t{(-\Delta )}^{\alpha }\right)}^{\gamma }}(u,b)\in {\left(m_m^{\prime }{\dot{F}}_{p,r}^{s_c,q}\right)}^{2n}$ for some $\gamma >0$ depending on the parameters. This gives a Gevrey-type spatial smoothing effect, which is stronger than ordinary analyticity in the adopted scale. The restriction $\alpha >{\displaystyle \frac{1}{2}}$ enters through the factor $2^{j(1-2\alpha )}$, which supplies the high-frequency gain needed to close the critical bilinear estimates; in this sense it is sharp for the present method. The classical viscous–resistive case is recovered when $\alpha =1$. Full article

This work develops an analytical framework for nonlinear fractional partial differential equations that combine Kirchhoff-type terms, double-phase operators, and $\psi$-Hilfer fractional derivatives. This paper investigates two classes of problems involving variable-exponent growth conditions. The first problem analyzes general nonlinear sources and formulates the solution as a fixed point of a nonlinear operator. Precisely, by proving that the functional energy is coercive, hemicontinuous, and strictly monotone, we establish the existence and the uniqueness of weak solutions via monotone operator theory. The second problem incorporates a convection-type nonlinearity, which breaks variational structure and requires the more robust theory of pseudomonotone operators. Under suitable growth and mixed-order assumptions on the nonlinearity, we prove the existence of at least one weak solution. The main tools are grounded in variable-exponent Lebesgue and Musielak–Orlicz–Sobolev spaces, with compact embeddings, modular estimates, and fractional integral identities playing a key role in the proofs. We note that the results contribute to the mathematical modeling of phenomena involving nonlocal elasticity, viscoelastic materials, phase-transition media, and fractional dynamical systems where the stiffness of the medium depends on the total deformation (Kirchhoff effect) and the energy density alternates between distinct growth regimes (double-phase). The $\psi$-Hilfer derivative enhances the scope by enabling models with tunable memory and hereditary effects. Full article

In fixed-view interaction scenarios of unmanned restaurants, face detection models face two core bottlenecks: the mismatch between training data distribution and real deployment scenarios, and the misalignment between model feature capacity allocation and business priority. To address these problems, this paper takes YOLOv8n (You Only Look Once version 8n) as the baseline, proposes a unified Scale-Aligned Capacity Allocation (SACA) theoretical framework, and constructs an end-to-end Scale Distribution Reconstruction Network (SDRNet) for lightweight face detection. First, we define the SACA loss with KL (Kullback-Leibler) divergence as the core optimization objective, which mathematically characterizes the matching degree between model capacity allocation and real scene face scale distribution. Second, a two-stage scene-aware scale distribution reconstruction strategy is designed based on the SACA framework, which derives the core face scale interval of the unmanned restaurant scene through a monocular imaging model, and constructs a scene-adaptive training dataset based on the public WIDER FACE benchmark, which is highly consistent with the real scale distribution of unmanned restaurant scenarios. Third, three scale-aligned lightweight modules, including LFEM (Lightweight Feature Extraction Module), LDown (Feature Segmentation and Sparse Optimization Module), and MSCH (Multi-Feature Shared Convolution Module), are proposed to realize the priority allocation of model capacity to core interaction scales, achieving collaborative optimization of data distribution and model structure. Fourth, a 2 × 2 controlled experiment is designed to separate the independent contributions of the data strategy and architectural improvements, and the robustness of the proposed model is verified on the standard WIDER FACE benchmark. Finally, a scale-specific validation mechanism is established to conduct fine-grained evaluation of the model’s detection performance on faces of different scales, avoiding the overall indicator masking the accuracy fluctuation of core scenarios. Experimental results show that the parameters of the proposed model are reduced to 1.76 M (a decrease of 41%), and the computational complexity is reduced to 5.5 GFLOPs (Giga Floating-point Operations Per Second) (a decrease of 32%). The mAP@0.5 (mean Average Precision) of the core medium-scale face reaches 0.684, with the performance loss controlled within 2% compared with the baseline. On the standard WIDER FACE benchmark, the model maintains competitive detection accuracy under extreme lightweight compression, which fully verifies its robustness. On the NVIDIA Jetson Orin NX embedded platform, the inference frame rate of TensorRT-FP16 reaches 79.9 FPS (Frames Per Second), which fully meets the real-time deployment requirements of resource-constrained unmanned restaurant scenarios. Full article

In order to solve the engineering ships’ problem of difficult operation and low precision of cooperative control under complex sea states, this paper proposes a control strategy utilizing USVs to collaborate with an engineering mother ship for operational purposes. By using the nonlinear feedback method to improve the closed-loop gain-shaping algorithm, the tangent function is introduced to effectively solve the unstable problem of the control system for small USVs under complex sea states. Meanwhile, the motion of the mother ship is decoupled to establish the mathematical model of the headwind stationary state, and the improved closed-loop gain-shaping algorithm is applied to design the course-keeping controller and speed controller to effectively meet the dynamic positioning fixed-point control tasks of mother ships. The results show that the designed USV-engineering mother ship cooperative controller is effective under complex sea states. The control strategy is convenient and energy-saving, providing technical support for the improvement of informatization and the intelligence level of USV–mother ship cooperative control. Full article

The methodology described in Eurocode 8, Part 4, for calculating seismic effects on vertical cylindrical rigid and fixed steel storage tanks is programmed in MATLAB^®. The walls of the tanks are constructed of shell courses with varying thicknesses of sheet material. The strength conditions for the ultimate limit states of plasticity, elastic buckling, and elastoplastic buckling (“elephant foot”) are checked at many calculation points along the height of the storage tank. The thicknesses of the courses are determined to satisfy all strength conditions for different slenderness ratios of the tanks and for different volume capacities. Tanks with supported roofs and those with self-supporting roofs are considered, as well as open-top tanks. A mass per unit volume capacity is the criterion for optimization for different seismic loadings and steel grades. The criterion is not a smooth function because of the discrete thicknesses of the shell courses and their number. A smooth objective function is created for better parametric optimization analysis. The dependence of the optimal slenderness ratio on the volume capacity is determined, as well as the inverse dependence. The problem of the optimal number of storage tanks in a set of tanks with a given total volume capacity is also considered. Full article

Underwater image enhancement has been widely studied to improve visual quality; however, its impact on downstream geometric tasks such as sparse 3D reconstruction remains insufficiently understood. In particular, visually enhanced images do not necessarily lead to improved feature matching or reconstruction performance. This work addresses the problem of selecting appropriate preprocessing strategies for underwater Structure-from-Motion (SfM) pipelines from a task-oriented perspective. We propose a lightweight machine-learning-based preprocessing selector that predicts reconstruction performance from image statistics and recommends suitable enhancement strategies for each input sequence. To ensure reliable evaluation, we introduce a grouped leave-one-parent-sequence-out protocol that avoids overlap-induced bias common in clip-wise splitting. Experiments are conducted on challenging underwater datasets derived from the Real-world Underwater Image Enhancement (RUIE) benchmark, with the primary comparison variable defined as the number of reconstructed sparse 3D points. Supporting geometric variables, including the number of registered images, mean track length, and mean reprojection error, are recorded for interpretation. Results show that preprocessing choices significantly affect reconstruction outcomes and that the optimal strategy is scene-dependent. The proposed selector consistently improved over raw input on the evaluated grouped subset and remained competitive with a strong fixed preprocessing baseline. The grouped leave-one-parent-sequence-out protocol is intended to reduce overlap-induced bias common in clip-wise splitting and to provide a more conservative estimate of generalization. This work highlights the importance of task-aware preprocessing and reliable evaluation in underwater vision systems, offering practical insights for deploying enhancement strategies in real-world 3D reconstruction pipelines. Full article

In 2003 W. Takahashi and M. Toyodaestablished the weak convergence of an iteration process to solve a variational inequality problem induced by an inverse strongly-monotone mapping. Recently we proved that for the same iterative process, most of its exact iterates are approximate solutions of the variational inequality. It was also shown that the iteration process for solving a variational inequality problem for an inverse strongly-monotone mapping generates approximate solutions in the presence of computational errors. In this work we employ the Cimmino algorithm in order to generalize these results for common approximate solutions of a finite family of variational inequalities with inverse strongly-monotone mappings and of a finite family of fixed point problems in the presence of computational errors. Full article

In this paper, we propose a secure protocol to compute the quadratic optimisation problem under a three-party outsourcing architecture in the scenario of cyber-physical systems. To enable real-world implementation, we propose an encoding framework that uses a fixed-point expression and a truncated-mapping scheme to map real numbers into multiple data blocks, improving the protocol’s efficiency. Based on this, we define the recovery operations for decryption, addition, and multiplication. Considering computations involving three parties to solve the quadratic optimisation problem, we thoroughly analyse privacy issues during the interaction process. Then, a secure protocol is developed by designing privacy-preserving addition, multiplication, and comparison protocols based on the additive homomorphic encryption scheme. The data blowup and “0”-privacy leakage problems are addressed specifically for the gradient descent process by designing a secure addition protocol for block data and a secure comparison protocol. The efficiency and security of the proposed protocol are formally analysed in depth. Finally, through intensive experiments, we demonstrate the efficiency and security of our protocol. Full article

In this paper, we introduce and systematically study the class of interpolative Geraghty-type contractive mappings within the framework of complete bicomplex-valued metric spaces (bi-CVMS). We prove seven new results: (i) a fixed point theorem for a single interpolative Geraghty contraction; (ii) a common fixed point theorem for a pair of such mappings; (iii) a fixed point theorem for interpolative Reich–Rus–Ćirić type contractions in bi-CVMS; (iv) a coincidence point and common fixed point theorem for weakly compatible maps; (v) a fixed point theorem for Jaggi-type hybrid contractions in bi-CVMS; (vi) a stability result for the Picard iteration associated with the main contraction; and (vii) an application theorem establishing the existence and uniqueness of solutions to a boundary value problem governed by a Caputo fractional differential equation. All results are furnished with complete proofs and non-trivial illustrative examples. Several well-known theorems—including those of Banach, Kannan, Reich, Geraghty, and their complex-valued analogues—follow as special cases. The paper significantly advances the fixed point theory in bicomplex-valued metric spaces. Full article

Airborne LiDAR (Light Detection and Ranging) data is widely used in urban modelling and three-dimensional spatial analysis studies. However, the irregular structure of LiDAR point clouds, varying point densities, and class imbalances observed in the datasets make semantic segmentation problematic. This study addresses the four-class semantic segmentation problem (unclassified, vegetation, ground, and building) on aerial LiDAR point clouds, with a particular focus on multi-class segmentation. The Oregon LiDAR Program dataset was obtained through the OpenTopography platform for use in this study. The point cloud data were resampled to 4096 points to ensure a fixed input size; for each point, the X, Y, and Z coordinates, along with the RGB and intensity features, were utilized. Experimental studies compared the proposed method with both baseline models (PointNet, PointNet++ MSG, and VoxelNet Lite) and recent state-of-the-art architectures, including Point Transformer, KPConv, and RandLA-Net. Additionally, the PointNet2 MSG Transformer model was developed based on the PointNet++ MSG architecture and includes a transformer-based feature fusion module. Different loss functions and training configurations were evaluated, and the effects of ensemble learning and test-time augmentation strategies on model performance were analyzed. The experimental results show that the proposed approach achieved a mean Intersection over Union (IoU) of 51.74% and an accuracy of 61.50% on the test dataset. These results demonstrate that combining multi-scale feature extraction with transformer-based feature fusion is an effective approach for semantic segmentation of LiDAR point clouds and multi-class segmentation tasks. Full article

This paper examines the rotational motion of a compound mechanical system comprising a rigid carrier body equipped with internal gyroscopic devices and a point mass that moves along a prescribed trajectory relative to the body. The system undergoes free motion in a uniform gravitational field. We derive the complete equations of motion accounting for the constant gyrostatic torque (GT) generated by internal rotors. Using asymptotic methods, we develop approximate dynamical equations valid under two distinct physical scenarios: (i) when the moving mass is small relative to the carrier mass and executes rapid oscillations and (ii) when the mass oscillates with small amplitude near a fixed location within the body, regardless of mass ratio. The accuracy and validity range of these approximations are rigorously established. For the first scenario, we have approached the idea that gyrostatic coupling fundamentally alters the system’s integrability properties while introducing beneficial stabilization mechanisms. We characterize families of permanent rotational states and analyze their stability using linear perturbation theory. The second scenario reveals that the approximate dynamics correspond to gyrostat motion rather than the classical Euler–Poinsot case. Comprehensive numerical simulations validate theoretical predictions and demonstrate applications to spacecraft attitude control problems. The results provide practical design guidelines for gyrostabilized systems with internal moving components. Full article

In this paper, we study a class of second-order fractional boundary value problems involving Θ-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the Θ-fractional framework. Existence results are obtained via Krasnoselskii’s fixed point theorem, while uniqueness is established using the Banach contraction principle under suitable Lipschitz-type conditions. The obtained results extend several earlier works on Caputo, Hadamard–Caputo, and Riemann–Liouville fractional derivatives. Two examples are presented to illustrate the applicability of the theoretical results. Full article

Investigating soil seepage considering free surface conditions under complex geological conditions is of great significance to ensure the safety of dams. In recent years, physics-informed deep learning (PINN) has become a cross-disciplinary hotspot for solving forward and inverse problems based on partial differential equations. However, the challenges in free surface simulation have confined the majority of current PINN research to seepage problems under fixed boundary conditions. To address the above issues, we propose a physics-informed deep learning-based approach for steady seepage in dams. In the proposed method, two different free surface approximation strategies are introduced to accommodate varying boundary conditions in the dam seepage problem. The first strategy approximates the free boundary by sampling points, while the second strategy approximates the free boundaries by an additional deep neural network. To validate the proposed methods, three benchmark cases with different boundary conditions have been conducted. The results indicate that the proposed approach effectively simulates steady seepage in dams. Both point-sampling and deep neural network-based free surface approximation strategies demonstrate high accuracy in predicting the location of the phreatic surface and the discharge of the seepage. Specifically, the prediction results are comparable in accuracy to analytical solutions and advanced numerical simulation methods. Full article

This study introduces a novel inertial-type iteration algorithm based on the Normal S iteration for the class of almost contraction mappings in Banach spaces. Traditional fixed point iterations often suffer from slow convergence and high computational cost; to address these limitations, the proposed framework incorporates an adaptive inertial-type parameter. We establish strong convergence of the algorithm and derive explicit a posteriori error estimates under weak contractive conditions. In addition, we demonstrate the asymptotic equivalence of the NS inertial-type trajectories with the classical Normal S iteration, provide a comprehensive weak $w^2$—stability analysis, and obtain sharp upper bounds for the data dependence problem. The practical performance of the algorithm is evaluated in two distinct computational domains: image deblurring via wavelet-based ${\mathrm{\ell }}_1$ regularization and the generation of complex fractal patterns, including Julia and Mandelbrot sets. Numerical results show that the proposed inertial-type iteration algorithm significantly outperforms existing methods—such as Picard, Mann, Ishikawa, and standard Normal S iterations—achieving faster convergence, higher PSNR values in image restoration, and more stable basins of attraction in fractal visualizations. These findings highlight the effectiveness and versatility of the NS inertial-type iteration algorithm approach for both theoretical analysis and real-world applications. Full article

This study proposes a new strongly convergent iterative framework obtained by combining a Krasnosel’skiǐ–Mann type subgradient extragradient process with a hybrid projection strategy and an inertial extrapolation mechanism. The method is applied to address hierarchical fixed-point problems (HFPPs) for nonexpansive and quasi-nonexpansive mappings as well as variational inequality problems (VIPs) involving a pseudomonotone operator in real Hilbert spaces. The proposed scheme employs step sizes that are restricted by the inverse of the Lipschitz constant of the underlying cost operator. Strong convergence of the iterates is achieved under mild hypotheses on the inertial parameter and control sequences. The method is further applied to problems arising in optimization and monotone operator theory. The results show that the proposed framework generalizes and integrates a number of existing approaches while offering improved computational performance. Full article