Copulas provide a flexible framework for building bivariate probability models that reflect specific dependency patterns. This work introduces a discrete form of the Type-I extreme value (Gumbel) distribution within a copula-based structure. Key mathematical and statistical characteristics are examined, including the joint probability mass function, survival function, hazard rate, conditional expectation, joint probability generating function, and dependency properties such as positive quadrant dependence and total positivity of order two. The bivariate discrete Gumbel model demonstrates strong performance in handling asymmetric data and proves particularly useful for capturing extreme and outlier observations. Its joint hazard rate function adds further flexibility, making it suitable for modeling a range of failure rate behaviors. Parameter estimation is carried out using the maximum likelihood method, and a thorough simulation study evaluates the bias and mean squared errors across various sample sizes. To illustrate its practical relevance, the model is applied to three different real-world datasets: football match outcomes, nasal drainage severity scores, and lens defects involving surface and interior faults.

The static convolutional network is designed with a restricted sense field, but it limits the global feature extraction. While dynamic convolution addresses the issue of limited static receptive fields, it struggles to perform well in discontinuous liver regions, liver tumor border regions, and microtumor segmentation. To alleviate the above issues, we proposed a network, GS_NeXt, based on graph theory and the Segment Anything Model (SAM) for liver and tumor segmentation from CT scans. First, we employed the feature extraction module of ConvNeXt-v2 to learn features across channels, enabling the network to focus on critical liver and tumor regions. Second, we utilized the SAM with frozen weights to extract more comprehensive global information, thereby enhancing feature representation. Third, we applied graph reasoning to globally model unstructured local features, improving the network's understanding of CT images in discontinuous liver regions, liver-tumor boundaries, and microtumor areas. Finally, we incorporated a deep supervision mechanism to facilitate the learning of multi-scale features throughout the network. We evaluated the proposed segmentation method for two publicly available abdominal liver tumor CT datasets. On the LiTS17 dataset, GS_NeXt achieved 97.74% and 87.25% on the Dice scores, 1.01 and 2.23 mm on the average symmetric surface distance (ASD), and 3.68% and 22.60% on the volume overlap errors (VOE) for liver and tumor segmentation, respectively. On the 3DIRCADb dataset, it achieved 97.31% and 87.36% on the Dice scores, ASD values of 1.01 and 2.12 mm, and VOE scores of 3.56% and 21.56% for liver and tumor segmentation, respectively.

In this paper, a fractional order differential algebraic predator-prey system with Holling type functional response was presented. The dynamic behavior of populations under the influence of economic benefits was thoroughly investigated. Specifically, with economic benefit as the bifurcation parameter, the existence of singularity-induced bifurcation was analyzed based on the theory of differential algebraic system. After that, a state feedback controller was designed to eliminate the singularity induced bifurcation and stabilize the system near the corresponding interior equilibrium. Furthermore, an optimal control problem was formulated, and the necessary conditions were derived. In addition, the validity of the theoretical results was confirmed through extensive numerical simulations.

Entity alignment, as a fundamental technique in knowledge graph integration, aims to identify the equivalent entity pairs that refer to the same real-world entity across different knowledge graphs. In domain-specific knowledge graphs, based on the atomic divisibility characteristics of entity categories, these categories can be classified into atomically divisible and atomically indivisible types. However, traditional methods for entity alignment in atomically indivisible knowledge graphs often fall short when dealing with tasks that involve more than one atomically divisible entity category. To address such a challenge, we propose a novel multi-embedding entity alignment approach based on matching information interaction over atomically divisible entities (MiAD). MiAD effectively exploits the matching information interaction between low-level entities and high-level entities to enrich the semantic representation of high-level entities and to enhance the reliability of the alignment results. The high-level entities are atomically divisible, and the matching result among the low-level entities can provide a significant hint for the entity alignment of high-level entities by analyzing their inherent interrelationships. To tackle the insufficient utilization of these matching information interactions, we propose a multiple hierarchical embedding strategy, which consists of a structural semantic aggregator, an attribute semantic aggregator, a corroborative aggregator, and a joint aggregator. This method leverages both the intra-level interactions and cross-level interactions to generate the hierarchical embeddings of high-level and low-level entities, respectively, from multiple perspectives (i.e., relations, attributes, and corroborative textural descriptions). As a result, MiAD significantly improves the accuracy of alignment outcomes and strengthens the overall structural and semantic understanding of higher-level entities. Comprehensive experimental results demonstrate that MiAD achieves substantial performance improvements compared with seven baseline methods on the Chinese recipe dataset Ta- da.

In this paper, we developed a numerical scheme based on barycentric rational interpolation to solve the generalized modified Camassa–Holm–Degasperis–Procesi (CH–DP) equation. Using the proposed approach, we applied a direct linearization technique to transform the nonlinear generalized modified CH–DP equation into an equivalent linear formulation. Approximate solutions were constructed using barycentric rational interpolation basis functions. The spatiotemporal domain was discretized via barycentric rational interpolation, and a corresponding differentiation matrix was derived. Furthermore, theoretical analysis of error distribution and convergence properties was presented. Analytical results and numerical experiments validate the scheme’s computational efficiency and high accuracy in solving the generalized modified CH–DP equation.

In this study, we propose a mirage search optimization algorithm that combines dual-chaos mapping and an opposition-based learning tactic, which we call the dual-chaos mapping opposition-based learning mirage search optimization (DMOMSO) algorithm. First, when initializing the population, a chaotic initialization method was used. This method takes advantage of the chaotic characteristics of iterative mapping to generate an initial population with high dispersion. Thus, the algorithm can more reliably reach the global optimum in a shorter time. Before the formal iteration process starts, a chaotic sequence is generated via chaotic mapping, which is used to update the iterative parameters, enabling them to vary dynamically throughout the iterative process. This helps the algorithm avoid becoming stuck in local optima, as well as enhancing its optimization performance and accelerating its convergence. In each iteration, the parameters are adjusted using the opposition-based learning method, which helps to cover the solution space more effectively, find good solutions faster, and explore a wider area. We conducted simulation experiments to assess the DMOMSO algorithm’s effectiveness, in terms of its performance on the CEC2017 and CEC2022 benchmark function sets. In this way, we carried out a comparative analysis between the DMOMSO algorithm and seven other algorithms, namely, MSO, SSA, AO, FOX, WOA, HHO, and COA. The experimental results indicated that the DMOMSO algorithm outperforms the original mirage search optimization algorithm, being able to more effectively find optimal solutions and converging faster. The eight considered algorithms were also tested in the context of practical optimization problems, including the design of hydrostatic thrust bearings, the optimization of Himmel Blau’s function, and industrial refrigeration system design. Based on the results obtained, DMOMSO is more effective for solving complex engineering problems.

The purpose of this article is to investigate a class of nonlinear p-Laplacian Hadamard fractional differential equations. By employing two fixed–point theorems of the sum operator, we get the uniqueness of a positive solution for such p-Laplacian equations. In addition, an iterative sequence can be constructed to approximate the unique positive solution. The validity of the results is demonstrated through two illustrative numerical examples in the final of this paper.

In this paper, we present a method for constructing Darboux-type curves by using spherical curves and smooth functions. We show that all Darboux-type curves can be generated using this method. We discuss the relationship between Darboux-type curves and rectifying curves. Moreover, we establish a connection between Darboux-type curves and geodesics on a rectifying developable surface from the viewpoint of curves on ruled surfaces.

This work addresses active vibration control for fractional-order systems based on a multiple region eigenvalue assignment. First, we propose the selection criteria of fractional-order multiple stability regions and define multiple stability regions by generalized linear matrix inequality. Based on the inverse eigenvalue problem theory, we give the sufficient condition for solving the feedback control matrix under multiple region eigenvalue assignment and present the expression of the feedback control matrix. Then, we propose a numerical algorithm for solving this problem, which improves the control performance. Finally, we verify the feasibility and effectiveness of the proposed method through numerical examples of fractional-order simulation and actual physical systems.

This paper investigated λ-biharmonic hypersurfaces in $\boldsymbol{L}^{\boldsymbol{m}} imes \mathbb{R}$, where L^m represents an Einstein space, and ℝ denotes the real line. We demonstrated that such hypersurfaces with some curvature conditions are either of two types: minimal or vertical cylinders over λ-biharmonic hypersurfaces in L^m. Particularly, when the Einstein space L^m has constant sectional curvature, we classify λ-biharmonic hypersurfaces as totally umbilical or semi-parallel.

The rapid development of image deraining technology has made it possible to effectively restore images captured even in severe rainy weather, thereby alleviating the decline in image quality. However, existing image deraining solutions typically use overly deep neural networks, which are prone to gradient vanishing and explosion. This often results in the restored image losing essential background details. To address this important issue, this paper proposed an enhanced dual-encoder vision transformer UNet for single-image rain removal. First, a dual-branch heterogeneous encoding structure was formed by integrating two encoders with different architectures. These encoders extract rain streaks by leveraging their distinct characteristics. Specifically, the first encoder was based on a convolutional neural network, termed the residual encoder network, which utilizes the local receptive field capabilities of convolutional kernels to capture both rain streaks and corresponding background details. Concurrently, the second encoder, the parallel vision transformer encoder network, employs self-attention mechanisms to model long-range dependencies and establish global contextual relationships between rain streaks and background details. Subsequently, the rain streak information extracted by the two encoders was fused. The decoder then performs the rain removal task. To preserve spatial consistency and overall image integrity, high-frequency detail refinement was facilitated through residual connections between the encoders and the decoder. Evaluations on synthetic and real-world datasets confirm the algorithm’s robustness in processing rain streaks of varied density. The restored images exhibit enhanced visual clarity, improving visibility for computer vision applications.

In this paper, we propose a high-order numerical method for the 2D time-fractional convection-diffusion equation with weakly singular solutions, where the time Caputo fractional derivative is approximated by using the L2-1_σ formula on a nonuniform graded mesh, and the space derivatives are discretized by a fourth-order compact finite difference scheme on a uniform mesh. The fully discrete compact ADI scheme is established by adding a high-order term. The stability and the convergence analyses of the scheme are analyzed in L²-norm by using the discrete energy method. It has been proved that the introduced numerical scheme has spatial fourth-order convergence, and temporal optimal (1 + α)-th order convergence. The numerical results show that the error estimates are sharp.

It is a proven fact that nowadays, thanks to convolutional neural networks that implement skip connection mechanisms, we can train increasingly deeper, more accurate, and efficient networks. These networks successfully address the degradation problem previously experienced in very deep networks. Among the most popular are ResNets and DenseNets. ResNets introduce skip connections using summation, while DenseNets employ concatenation. The summation mechanism in ResNets can limit the adaptation of prior information to the specific needs of each layer. In contrast, the DenseNet concatenation mechanism can become computationally expensive as convolutional blocks attempt to process all accumulated prior information. Therefore, in this paper, we proposed a new attention- based skip connection mechanism: Attention Shortcuts. This mechanism allows convolutional blocks to process the most relevant prior information, reducing computational burden. We conducted a preliminary experimental study adapting the proposed mechanism to the ResNet-50 backbone and compared its performance to the original ResNet.

Due to their simplicity, low memory requirements, strong convergence properties, and ability to solve problems of high dimensions, the conjugate gradient (CG) methods are widely used to solve linear and non-linear unconstrained optimization problems. The Polak-Ribiére-Polyak (PRP) is considered as one of the most efficient CG methods in practical computation. However, theoretically, its convergence properties are poor. Therefore, many variants of PRP with good numerical results and good convergence properties have been developed, such as Gilbert and Nocedal method (PRP⁺), Wei-Yau-Liu method (WYL), and Yousif et al. method (OPRP). In this paper, based on PRP⁺ and OPRP methods, we proposed another modified version of PRP that inherits all the convergence properties of PRP⁺ and OPRP and has improved numerical results. To show the efficiency and robustness of the new modified method in practice, it was compared with PRP⁺, WYL, and OPRP when they are all applied under the strong Wolfe line search. At the same time, the new method was applied in deep learning to obtain ideal parameters of some neural network (NN) models during the training process.

We have proposed a robust and efficient variable selection method for ultrahigh-dimensional linear models with nonrandomly missing responses, leveraging modal regression. The propensity score function was specified by a semiparametric model and we introduced a two-step estimation procedure. In the first feature screening stage, the Pearson chi-square (PC) test statistic identifies significant predictors in the sparse propensity score model. The generalized method of moment (GMM) estimates parameters to obtain consistent estimation for the propensity score in the second stage. With the estimated propensity score, we suggested a feature screening and variable selection procedure based on the inverse probability weighting (IPW). A modified sure independence screening (SIS) method first reduces the model dimensionality, followed by a penalized modal regression approach to select significant covariates. The proposed procedure can deal with the ultrahigh-dimensional data with nonignorable nonresponse, and this modal-based procedure is robust against outliers and heavy-tailed errors. Additionally, we established the asymptotic properties of the estimators under mild regularity conditions. Simulation studies and real data applications confirm the method’s effectiveness in finite samples and practical settings.

In this study, we innovatively applied dynamic game theory to analyze blockchain adoption decisions in competitive environments. By constructing a multi-stage dynamic game model and solving for optimal strategies through backward induction, we overcame the limitations of the traditional static game analysis. Combining the mathematical modeling with numerical simulations, we identified a critical threshold effect of the competition intensity on the blockchain adoption equilibrium. Specifically, when the competition intensity fell below the threshold, blockchain adoptions emerged as the dominant strategy. Beyond this threshold, the equilibrium shifted to a mixed-strategy outcome. Furthermore, we developed a dynamic response mechanism that linked traceability preferences to profit functions. Our model provides a theoretical framework for analyzing blockchain adoptions in competitive settings, while the numerical results offer actionable insights for optimizing the blockchain system design.

Correlation filter algorithms are widely used in the field of object tracking due to their excellent tracking performance and real-time tracking efficiency. Traditional correlation filter tracking methods focus on three aspects of improving algorithm performance: feature representation, spatial regularization techniques, and temporal smoothness. However, these methods overlook the removal of redundant and interfering information in the filters, as well as the protection of filters in occluded scenes, which leads to poor performance in complex scenarios such as cluttered backgrounds and occlusions. It is noted that interfering information exhibits sparsity but often lacks the structural property of combinatorial sparsity. Inspired by this, this paper proposes a correlation filter object tracking algorithm based on spatial and channel attention mechanisms. Specifically, the algorithm introduces fiber group sparsity constraints in the spatial direction of the filter and slice group sparsity constraints in the row, column, and channel directions. In this way, the structural sparsity property of the filter is further exploited to allocate attention to the spatial and channel features, thereby removing redundant and interfering information. In addition, the reliability analysis of the best candidate samples is performed by a history template pool to decide whether the filter should be updated or not to avoid the tracking failure problem caused by filter degradation in occlusion scenarios. Experiments on several datasets show that the proposed method outperforms other state-of-the-art trackers and achieves good performance in terms of accuracy and robustness.

This study addressed the theoretical challenges faced by rank-dependence structures in Spearman’s footrule correlation coefficient. Two asymptotic representations were proposed to approximate its null distribution. The first approach simplifies dependencies by substituting empirical distribution functions with their population counterparts. The second employs Hájek projection technique to decompose the initial form into a sum of independent components, establishing rigorous asymptotic normality. Simulation experiments combined with real-world data analyses validated both representations, demonstrating their excellent approximation to the limiting normal distribution under the independence hypothesis.

In this paper, a fault-tolerant output feedback maneuvering control scheme with prescribed performance for a class of random Euler–Lagrange systems with unmeasurable velocity and actuator faults is presented. First, an adjustable velocity observer is ingeniously constructed without additional dynamic compensation signals. Second, a projection operator is used to estimate the actuator fault factors. Based on the designed observer and projection operator, a static controller is designed to address geometric tasks with performance constraints, while a dynamic controller is developed to achieve speed allocation tasks. Finally, the effectiveness of the proposed control method is demonstrated through an illustrative example involving a two-link robotic system operating in a random environment.

We aimed to address the issues of low encryption complexity and insufficient resistance to statistical attacks in chaotic image encryption algorithms. This paper proposes a novel method that combines hash scrambling, and bit-level transformation, leveraging the synergy between chaotic mapping and spatial-frequency transformation. Firstly, a specific hash function is constructed using the cubic chaotic system to achieve efficient scrambling of the image. Subsequently, the frequency domain data is generated by the Fourier transform and represented in the form of an optimized complementary code. Finally, the encryption is completed by DNA primary diffusion, secondary diffusion based on adjacent blocks, and the neighbor bit deprivation operation, and the entire process is jointly regulated by the Chen chaotic system and the Sin-Tent-Cos chaotic system. The experiment demonstrates that this scheme exhibits excellent robustness and security, with all indicators surpassing those of traditional methods. In particular, its resistance to differential attacks is close to the theoretical optimal value, demonstrating the important application potential of spatial-frequency joint encryption.

Physics-informed neural networks (PINNs), built upon general neural networks, integrate physical information into the loss function, achieving remarkable results in solving partial differential equations (PDEs). Meanwhile, PINNs face challenges in solving complex interface problems, such as the Navier–Stokes/Darcy interface model considered in this paper. To address these challenges, we extend the general PINNs to propose subregion–identified physics-informed neural networks (SI-PINNs), which consist of three different neural networks for different subregions (Navier–Stokes, Darcy, and interface) with separated datasets. By identifying and classifying the regional features of the neural networks and datasets, the complexity of the physical information that SI-PINNs need to process is effectively split, while the networks also take all physical information into account during the training. Additionally, this splitting feature substantially reduces the training costs while enhancing or maintaining the accuracy of the general PINNs for the target model. Numerical experiments are performed to validate the effectiveness of the SI-PINN method.

In this paper, we considered a two-species chemotaxis-Stokes system with porous medium diffusion ∆n^m 1 in two-dimensional smooth bounded domains. We proved the existence of a time peri- odic solution for any m ≥⁶ 5 and any large periodic sources g¹(x,t) and g²(x,t).

This paper investigates observer-based networked control for stochastic nonlinear systems under false data injection (FDI) attacks and limited communication bandwidth. To mitigate the impact of FDI attacks while reducing communication load, we propose an FDI attack-resilient periodic encoding–decoding scheme based on uniform quantization. We first establish a detectability criterion for the stochastic nonlinear system under this periodic encoding–decoding scheme. Then, we derive a condition to further guarantee the input-to-state stability of the resulting closed-loop system. The condition, which enables the determination of the desired observer and controller gains, involves a series of linear matrix inequalities that are straightforward to verify using available MATLAB numerical tools. Finally, we validate the effectiveness and robustness of the proposed controller design through a case study.

This study begins with the nonlinear coupled Korteweg-de Vries equation and successfully solves its Lax pair using the prolongation structure method. Based on the obtained Lax pair, the Bäcklund transformation and superposition formula are further derived. These results provide a theoretical foundation for understanding the dynamic behavior of nonlinear systems and offer new approaches for constructing exact solutions. The study highlights the potential of the Bäcklund transformation and superposition formula in improving computational efficiency, enhancing model generalization capabilities, and increasing prediction accuracy. Furthermore, this paper explores the application prospects of these mathematical tools in the field of AI demonstrating significant advantages in simulating complex physical phenomena, such as waves and turbulence. Experimental results show that the AI model incorporating Bäcklund transformation improves computational efficiency by 40 percent and reduces prediction error by 25 percent in turbulence forecasting. Integrating traditional mathematical methods with modern AI technologies not only significantly enhances the performance of existing algorithms but also opens up new possibilities for addressing complex real-world problems. This research lays the groundwork for future interdisciplinary studies, encouraging more scholars to engage in this cutting-edge field. Through cross-disciplinary integration, it is expected to promote the joint advancement of nonlinear science and AI technology.

This paper studied a delayed diffusive predator-prey model with predator and prey harvesting terms. The existence of constant steady-state solutions and Hopf bifurcation were analyzed. Then by applying the central manifold theorem and the normal form method, the direction of the Hopf bifurcation and stability of the bifurcating period solution were studied. Numerical simulations were conducted to confirm the accuracy of the proposed theory. In addition, taking the predator-prey relationship between sharks and tuna as an example, this study investigated the impact of predator harvesting coefficients on the constant steady-state solutions of the system and the time required for the system to reach stability.

This article investigates a class of hyperbolic equations of the fractional Kirchhoff type with viscoelastic and nonlinear terms:

where [u]_s is the Gagliardo semi-norm of u, $\Omega \subset \mathbb{R}^{N}$ is a confined area featuring a smooth boundary, (−∆)^s is the fractional Laplacian with $s \in(0,1), 2<a<2 \gamma<b<2_{s}^{*}$,u⁰ and u¹ are the initial function. First, we obtain the existence of global solutions by combining the potential wells with the Galerkin method. Moreover, employing the perturbed energy approach, we systematically study the asymptotic behavior of solutions.

Brucellosis is currently recognized as one of the most serious zoonotic infectious diseases, caused by Brucella abortus, and is classified by the World Organization for Animal Health as a Category B zoonotic disease. In this study, we developed a new seven-compartment model of the transmission dynamics brucellosis in sheep, which accounts for the combined effects of vaccination, age structure, culling of infected sheep, and the impact of the contaminated environment. Our analysis demonstrates that the DFE point of the system is globally and asymptotically stable when the basic reproduction number R₀ < 1. Furthermore, we show that the disease persists when R₀ > 1, in accordance with the theory of uniform persistence. In the numerical simulation section, we fit data on the number of Brucella infections in sheep in Egypt from 1999 to 2010 and estimate the model’s parameters using the least squares method. Based on this, we propose four control strategies to establish the optimal control system. We applied Pontryagin’s maximum principle to derive the necessary conditions for optimal control and perform numerical simulations. A cost-benefit analysis was conducted from the perspective of sheep farmers. Our findings suggest that the most cost-effective strategy for reducing brucellosis infection rates in sheep is to uniformly house young sheep and vaccinate them simultaneously.

Graph contrastive learning (GCL) has emerged as a powerful self-supervised paradigm for graph representation. A critical challenge in GCL arises from the conflict between the contrastive loss function and the message-passing mechanism of graph convolutional network encoders, as the message- passing mechanism pulls neighboring nodes close, while the contrastive loss function pushes negative nodes apart, especially neighboring nodes. Prevalent methods typically tackle the conflict by identifying conflicting node pairs and subsequently ignoring them during training. However, the identification of conflicting pairs in these methods may be one-sided and even incorrect, likely deleting valuable pairs which are inherently non-conflicting. To overcome the limitation, we have proposed conflict refined graph contrastive learning (CR-GCL), a novel framework that leverages pseudo-labels to prevent ignoring non-conflicting negative pairs. Technically, CR-GCL incorporates three key components: 1) a perturbation augmentation module that creates augmentation graphs and generates representations; 2) a conflict quantification module that estimates the gradient impact of negative pairs by tracking representation similarity changes; 3) a conflict refinement module that utilizes pseudo-labels to enhance the identification of conflicting negative pairs, which are then ignored during training. Extensive experiments on six benchmark datasets revealed that CR-GCL significantly outperforms state-of-the-art methods, delivering superior node classification accuracy with various label rates.