We investigate least-squares (LS) solutions of Sylvester-type matrix equations formulated via the general semi-tensor product (GSTP) of matrices. In particular, we consider generalized linear systems of the form A ⋉ x = B, where A and B are given rectangular matrices and x is an unknown column vector, with ⋉ denoting the GSTP that extends both the conventional matrix product and the semi-tensor product. By analyzing the derivative of the LS error associated with the equation, we show that LS solutions can be obtained by solving an equivalent linear system under the usual matrix product. Using matrix partitioning techniques, these results are further extended to several Sylvester-type equations, including A ⋉ X = B, X ⋉ A = B, and A ⋉ X ⋉ B = C, where X is an unknown matrix of compatible size. This framework unifies the classical and semi-tensor product cases under a generalized algebraic setting. Furthermore, we develop a gradient-descent iterative algorithm to compute approximate LS solutions efficiently. Numerical experiments confirm the convergence, capability, and effectiveness of the proposed method.

In this paper, we propose fractional variable transformation neural networks (fVTNNs, for short), a framework that embeds fractional variable transformation into neural networks (NNs), to systematically derive analytical solutions for nonlinear space-time fractional partial differential equations (fPDEs) via symbolic computation. This approach significantly enhances both the computational speed and the result precision by combining the robust approximation capacity of NNs with the exactness of symbolic computation. The output of fVTNNs, which consists of weights, biases, and activation functions, is taken as a trial function for the considered equation. In order to explain the feasibility of the proposed method, some examples are investigated. Hyperbolic function solutions and exponential function interactive solutions of these equations are obtained. The analytical solutions obtained using this method are accurate and have no calculation errors. To visualize the dynamic characteristics of the solutions, three-dimensional plots, contour plots, and density plots are employed. This research introduces a novel computational framework for obtaining exact solutions to fPDEs, with a broad applicability in science and engineering.

The authors considered a non-autonomous allelopathic phytoplankton competition model that incorporates the influence of fear effects that are often observed in natural biological phenomena. Based on Mawhin’s coincidence degree theory, sufficient conditions for the existence of periodic solutions were obtained. The authors validated their findings with an example and simulations showing that the constant coefficient case leads to dynamics that are neither steady state nor T-periodic, while the periodic coefficient case generates sustained T-periodic oscillations.

Monkeypox, a re-emerging zoonotic disease, has become a growing global public health concern due to its rapid transmission and high visual similarity to other dermatological conditions such as chickenpox and measles. This resemblance complicates early clinical diagnoses, particularly in resource-limited settings where laboratory testing capabilities are scarce. Deep learning methods capable of detecting monkeypox from skin images offer a promising alternative to manual inspection and PCR-based diagnoses. However, existing approaches are often limited by poor dataset quality, weak generalization ability, and insufficient model interpretability. To address these challenges, this paper proposes an attention-enhanced hybrid deep learning framework for the automated classification of monkeypox skin lesions. Specifically, the proposed model employs DenseNet-121 and EfficientNet-B4 as parallel convolutional feature extractors and integrates Convolutional Block Attention Modules (CBAM) to adaptively emphasize lesion-related features while suppressing background interference. Experimental results demonstrate that the proposed framework outperforms conventional deep learning baselines, thereby achieving an accuracy exceeding 91% and a Cohen’s Kappa score above 0.88 on the primary dataset. Furthermore, the model exhibits a strong generalization capability, thereby maintaining classification accuracies above 90% across multiple external public datasets. To enhance the transparency and clinical reliability, explainable artificial intelligence (XAI) methods are employed to visualize the model’s decision-making process. In addition, quantitative interpretability metrics are used to assess the reliability and consistency of the generated explanations, thus highlighting the model’s potential for practical clinical applications.

Bird activities like nesting and perching in transformer substations threaten power grid stability by causing short circuits and insulation failures. Existing bird repelling devices are inefficient due to the lack of accurate detection and positioning, leading to energy waste and safety hazards from continuous operation. To address this, this paper develops a tiny bird detection and location system guided by heterogeneous binocular images for precise, targeted repulsion. For well-lit scenarios, a two-stage contextual information enhancement network is proposed. It mines multiscale context to highlight tiny bird regions, fuses context with second-stage features via channel dimension enhancement, and uses spatial attention for accurate localization. For low-light or occluded scenes, a multiscale contextual feature enhancement network processes infrared images, adopting multibranch cross-level feature fusion and combining transformer with multisize convolution to suppress background and thermal radiation interference. Additionally, heterogeneous binocular cameras are calibrated to calculate bird spatial distance, integrating detection results with spatial information to drive a laser repelling device. Experimental results in real substation environments show the system meets engineering requirements for robustness and accuracy. The detection in visible images achieves an overall average precision of 59.8%, while the infrared detection outperforms advanced algorithms in key metrics. The spatial localization error is controlled within 4.9%, significantly improving bird expulsion success rate and reducing energy consumption. This work provides a reliable technical solution for safeguarding power grid operation and offers valuable references for tiny object detection in complex industrial scenarios.

In this short note, we establish a connection between the algebraic multiplicity of eigenvalues of compact integral operators and the simplicity of zeros of a class of the Mittag–Leffler functions E_α,α(z) and E_α,2(z) for 1 < α < 2. Although these are two seemingly unrelated concepts in mathematics, our approach provides a functional-analytic criterion for determining the simplicity of zeros in this class of Mittag–Leffler functions.

We define the adjacency matrix A(G) and the degree diagonal matrix D(G) for a graph G. The Q-spectrum is the multiset of eigenvalues of the Q-matrix Q(G) = A(G) + D(G), counted with their algebraic multiplicities. A class of graphs is determined by their generalized Q-spectrum (DGQS for short) if any two graphs in the class have the same Q-spectrum and so do their complements, implying that they are isomorphic. Tian et al. introduced a new method to construct DGQS graphs by considering the rooted product graphs G ◦ Pk. They proved that when k = 2, 3, G ◦ Pk is DGQS for a special class of graphs G. In this paper, we extend this result and prove that, under the same conditions for G, the conclusion holds true for any positive integer k. While Wang et al. reported comparable results, our study arrives at these findings through an entirely different methodology. Our proof adopts the elementary properties of the resultants, providing an innovative approach to establishing the DGQS property for G ◦ Pk.

Let ($\mathcal{X}$, $\mathcal{Y}$) be a balanced pair in an abelian category $\mathcal{A}$. Denote by $\mathbf{K}_{\mathcal{E}-\mathrm{ac}}(\mathcal{X})$ the chain homotopy category of right X-acyclic complexes with all items in $\mathcal{X}$, and dually by $\mathbf{K}_{\mathcal{E}-\mathrm{ac}}(\mathcal{Y})$ the chain homotopy category of left $\mathcal{Y}$-acyclic complexes with all items in $\mathcal{Y}$. We establish realizations of $\mathbf{K}_{\mathcal{E}-\mathrm{ac}}(\mathcal{X})$ and $\mathbf{K}_{\mathcal{E}-\mathrm{ac}}(\mathcal{Y})$ as homotopy categories of model categories under mild conditions. Consequently, we obtain relative versions of recollements of Krause and Neeman-Murfet. We further present applications to Gorenstein projective and Gorenstein injective modules.

This paper investigated the structure and stability of solutions for simultaneously recovering density and bulk modulus in acoustic inverse scattering. The nonlinear scattering model was linearized via the Born approximation, and the inverse problem was casted as a 2 × 2 block operator equation. The off-diagonal blocks of this operator matrix captured the cross-coupling interactions between the two physical parameters. By employing the maximal Tseng generalized inverse, we developed an analytical formulation that remained valid even when diagonal blocks were non-invertible, overcoming limitations of traditional Schur complement methods. To address the inherent ill-posedness, we incorporated Tikhonov regularization into this linearized system, deriving explicit solution representations and establishing a rigorous stability estimate. Our analysis revealed that the strength of the off-diagonal interaction imposed a fundamental lower bound on the regularization parameter. As the parameter approached this bound, the reconstruction error exhibited singular behavior.

Graph neural networks (GNNs) have been widely studied for handling graph-structured data. Neighborhood awareness, a mechanism for integrating context into graphs, plays a crucial role in learning node embeddings in GNNs. Existing methods typically perform neighborhood-aware steps only at the node or hop level, which limits their ability to learn edge-level semantic information. There are two main challenges in extending neighborhood awareness to the edge level: (1) designing a learning framework with message propagation of edge embeddings, which supports the construction of edge embeddings to capture pairwise node relationships and propagate messages to perceive local context, and (2) developing a fusion approach that bridges node and edge embedding spaces, thereby compressing edge-level structural information into node space to enhance the original neighborhood awareness. In this study, we propose an edge-level neighborhood awareness network (ELNA-Net) that integrates both node- and edge-level context for more comprehensive neighborhood awareness. Specifically, we use the Taylor interaction effect to construct explicit interactions between pairwise nodes, which approximates edge embeddings and captures intricate non-linear correlations. Furthermore, the adaptive edge-aware propagation module adopts a line graph to switch the roles of nodes and edges, revealing potential dependencies among neighboring edges and promoting neighborhood awareness. Finally, we propose a cross-mapping fusion approach to integrate node- and edge-level contexts within the graph convolution operator. Experimental results demonstrate that ELNA-Net achieves superior performance in semi-supervised node classification and link prediction, outperforming state-of-the-art models.

In this work, we presented an extended persistence for filtration of graded subgroups by defining a relative homology in this setting. Our work provided a more comprehensive and flexible approach to obtaining an algebraic invariant and overcoming the limitations of the standard approach. As the application of mathematical tools in data analysis requires stability-meaning small perturbations in the input data should induce only small changes in the output-our main contribution was the development of a stability theorem for extended persistence modules of filtration of graded subgroups. This theorem was established using an extension of the definition of interleaving, along with the rectangle measure and functor extension. We demonstrated the theorem’s utility by applying it to extended persistence modules obtained from the path homology of directed graphs and the homology of hypergraphs, two important examples in topological data analysis.

This paper studies left (right) Drazin invertible elements in a semisimple Banach algebra and provides the characterizations of the left (right) Drazin invertible elements. In addition, the paper proves that the left Drazin spectrum is compact. Finally, it is shown that being Drazin invertible is equivalent to being semi-B-Browder.

In this work, we consider the stochastic Newell-Whitehead-Segel (SNWS) equation, which is forced in the Itô sense by multiplicative advection noise. We demonstrate that the solution of the SNWS equation can be discovered by solving certain deterministic counterparts of the Newell- Whitehead-Segel equation with an extra diffusion term (DNWS), and then the results are combined with a solution for stochastic ordinary differential equations. We use two techniques to solve the DNWS equation: the (^G′/G) -expansion method and the exp(−ψ(η))-expansion method. The purpose of studying the Newell-Whitehead-Segel equation with multiplicative noise is to explore how stochastic fluctuations influence the stability and evolution of patterns near a bifurcation. Incorporating noise provides a more realistic description of physical systems and reveals impacts that are omitted from the deterministic model, such as changing thresholds, different pattern structures, and increased variability. Therefore, we use a MATLAB tool to exhibit numerous 3D and 2D graphs that show how multiplicative advection noise impacts the solutions of the SNWS equation.

We establish global well-posedness for a two-species competitive reaction-diffusion system in bounded two-dimensional domains under Neumann boundary conditions. The system incorporates spatially heterogeneous growth continuous over the closed domain and time-dependent bounded asymmetric intervention. A sharp threshold condition relating the target species’ promotion strength to its natural decay rate is necessary and sufficient to maintain the carrying-capacity constraint globally in time, with the two species’ total density not exceeding 1. Meanwhile, the unit square with both species’ densities ranging between 0 and 1 remains positively invariant, regardless of the target species’ promotion strength. Solutions are classical and unique, uniformly bounded, and Lipschitz-continuous with respect to initial data. Numerical simulations show that asymptotically applied interventions that satisfy the threshold condition result in a near-complete eradication of the competitor species at full efficiency. The promoted species establishes a heterogeneous spatial distribution shaped by growth heterogeneity. This framework extends classical diffusive Lotka-Volterra models and provides rigorous theoretical support for ecological management and misinformation suppression strategies.

In natural ecosystems, population dynamics are influenced by white and colored noise, which drive the population system to transition between multiple states. Therefore, it is essential to investigate predator-prey models that incorporate the concurrent impacts of white and colored noise. In this paper, we examined a three-dimensional stochastic predator-prey model involving dual prey species and a single predator by incorporating the refuge effect and habitat selection behavior under Markovian switching. By leveraging the theoretical framework of stochastic differential equations, its stochastic persistence was analyzed. Additionally, by constructing suitable Lyapunov functions, the conditions for an ergodic stationary distribution were deduced. Additionally, the extinction of populations and the asymptotic properties of solutions were explored. The findings revealed that habitat selection behavior has a significant and detrimental impact on the corresponding prey, while the refuge effect positively influences the prey and the predator. Ultimately, numerical simulations were performed to validate the theoretical outcomes. The results of these simulations strongly confirm the accuracy of the theoretical deductions.

This article study a coupled one-dimensional hyperbolic system with Lord-Shulman thermal heat law on the first equation, and a general weak inherent damping, along with time-dependent coefficient and time-varying delay term on the second equation. Using the multiplier method, through a suitable Lyapunov functional, we prove a general decay stability result that encompasses both exponential and polynomial decay estimates as special cases. This result extends and generalizes existing findings in the literature related to the system under study.

From a theoretical perspective, it is worthwhile to use the idea of control to synchronize multiple individual assets. We regard personal assets as multi-agent systems, i.e., second-order multi-agent systems (SOMAS). The delayed feedback control for consensus and average quasi-consensus of delayed SOMAS is studied. First, two weight matrices are studied, where the interactions are not entirely cooperative into the SOMAS with mixed time-varying delays (the delay varies with time). Second, a delayed feedback control is designed based on two weight matrices where the interactions are cooperative to get the consensus about followers and the average quasi-consensus about leader and followers. The consensus and average quasi-consensus of the delayed SOMAS are established by the graph-theoretic technique and Lyapunov functions. Finally, some numerical simulations is given to verify the theory.

This paper is devoted to studying dynamic event-triggered quantized H_∞ control for networked control systems (NCSs) with stochastic deception attacks. To save limited system resources, a dynamic event-triggered scheme is offered, in which a new triggering error is introduced. A lower trigger frequency can be obtained by appropriately adjusting the triggering error. Then, considering the conventional deception attack, accumulated dynamic cyber-attack, and dynamic event-triggered scheme, a new quantized control model is constructed, and the stochastic deception attack is described by two independent Bernoulli distributed variables. Moreover, a new H_∞ performance criterion is given by using a customized Lyapunov-Krasovskii functional (LKF), and a novel controller design approach is derived based on the criterion. Finally, some simulations are listed to verify the validity of the derived methods.

This paper analyzes a class of variational inequalities involving a double-phase degenerate parabolic operator with variable exponents, which arises in the valuation of American put options. By establishing energy estimates for the associated penalty problem and applying a limiting argument, the existence of a weak solution is obtained. The uniqueness of the weak solution is also discussed.

Pathological oscillations within the basal ganglia (BG) are a hallmark of Parkinson’s disease (PD), a prevalent neurodegenerative disorder. This study aims to evaluate the effects of delayed feedback on oscillatory activity in a PD model and to analyze the roles of synaptic weights and transmission delays within the feedback circuit. Results indicate that the model induced by delayed feedback increase leads to the transition in firing activity, progressing from regular gamma activity to pathological beta oscillation states, and may even transition into complex states. This transition is further modulated by synaptic weights, which could not only change the model from a steady state to an oscillating state, but also cause a high-saturation steady state when sufficiently large. Furthermore, sensitivity analysis based on polynomial chaos expansions enables the quantification of the contributions of transmission delays and synaptic weights to the oscillation frequency variance of the model. The analysis shows that with increasing delayed feedback, the transmission delay parameter that contributes the most to model oscillation shifts from transmission delay within the BG circuit to that of the cerebral cortex circuit. Synaptic weights between the external globus pallidus and the subthalamic nucleus have a significant impact on oscillatory activity when delayed feedback is large. Our findings provide new directions for further clinical treatment of PD.

This paper investigated the stability of a class of neutral-type stochastic delayed neural networks with Markov switching. Under a general decay rate and weaker conditions on the neutral term, sufficient conditions for stability in the p-th moment, almost sure stability, and actual stability were established by constructing appropriate Lyapunov functions and applying the nonnegative semimartingale convergence theorem. The theoretical analysis was validated via MATLAB simulations using the Euler-Maruyama method.

We present radial basis function (RBF) collocation methods for time-dependent space-fractional problems on general bounded domains. Building on a recently developed approach for accurately computing the integral fractional Laplacian of any RBF, we design collocation schemes for fractional heat and Stokes equations using extended-domain techniques. In particular, we propose a numerical Leray projection method for fractional Stokes problems, where both the discrete projection operator and the collocation scheme are formulated on extended domains to handle complex domains. Numerical results demonstrate the effectiveness of the proposed methods in solving time-dependent nonlocal problems on complex domains.

In this paper, we introduce two methods for constructing matrix-valued cosine and sine functions defined on a discrete domain. The first method develops an algorithm to compute the n × n matrix-valued cosine and sine of a given square matrix A with some restrictions on its eigenvalues. The second method derives a formula based on the Jordan canonical form to compute each term of a Jordan block in the discrete matrix-valued cosine and sine of a given square matrix A. To illustrate the utility of our approach, we present some examples.

This study introduces a novel cortico-thalamo-basal ganglia-pedunculopontine nucleus (PPN) (CTBGP) computational framework to investigate how PPN-related pathways control beta oscillations. We observe that PPN-thalamic pathways exert bidirectional Hopf bifurcation control over thalamic beta oscillations, with coupling strength adjustments shifting stable/oscillatory state boundaries; PPN-cortical projection stabilizes cortical beta oscillations through supercritical/subcritical Hopf transitions dependent on coupling strength; PPN-GPi projection modulates cortico-thalamic beta oscillations via interactions with GABAergic GPi-thalamic pathways, enabling coexistence of supercritical/subcritical bifurcations; PPN-STN projection strongly suppress basal ganglia beta oscillations by elevating STN-GPe network activity to saturated states. Notably, three direct PPN inputs (EPN-PPN, STN-PPN, GPi-PPN) collectively regulate beta oscillations through the PPN-STN-GPe axis. This work provides the computational evidence that PPN pathways dynamically control beta oscillations across CTBG subcircuits via bifurcation mechanisms. The identified PPN-STN-GPe axis and thalamic/cortical projections offer novel targets for DBS and pharmacological interventions.

Let S be a commutative Noetherian ring and a a proper ideal of S. We give several bounds of a-depth of S-complexes and S-modules, investigate the behavior of a-depth and a-Cohen-Macaulay modules under tensor product with a faithfully flat S-module. Furthermore, we establish the Foxby equivalence of a-Cohen-Macaulay S-modules.

This work investigates the nonlocal diffusion-wave equation governed by the time-fractional Caputo derivative. We first establish that the solution is t-analytic through the application of Fourier expansion and the properties of the Mittag-Leffler functions. Building on this, we apply the Laplace transform to demonstrate the subordination principle for solutions of parabolic and hyperbolic equations in the context of the nonlocal diffusion-wave equation. Second, we prove the uniqueness and conditional stability of a solution to an inverse problem involving the determination of the spatially varying source term based on interior information from a subdomain. We alslo introduce a novel framework termed Laplace-based physics-informed neural networks (L-PINNs), which is tailored for determining source terms in nonlocal diffusion-wave systems. We substantiate the proposed approach through a series of numerical experiments, demonstrating its superior accuracy and computational efficiency.

This paper investigated the Hopf bifurcation in fractional-order ring neural networks incorporating multiple time delays (leakage, transmission, and distributed delays) and reaction-diffusion effects. By introducing virtual neurons into the original system, a new model capable of equivalently describing the influence of distributed time delays was constructed. The critical conditions for the emergence of pure imaginary roots in the characteristic equation were analytically determined using the Coates flow graph and the holistic element method. Furthermore, by selecting the leakage and transmission delays as bifurcation parameters, explicit criteria for local stability and the existence of Hopf bifurcation were established. The theoretical findings were substantiated through numerical simulations.

This study investigates a numerical method for incompressible flows governed by unsteady Navier–Stokes equations, proposing a characteristic finite element method based on two Gauss integrations. The proposed method is compared and analyzed with the classical two-grid method and variational multiscale methods, respectively, with error estimates conducted. The superiority of the characteristic finite element method based on two Gauss integrations is verified through numerical examples with exact solutions. For a high Reynolds number of 10⁶, both algorithms remain stable. The computational results show strong agreement with those reported in well-established literature. Furthermore, simulations of the lid-driven cavity flow are performed, and the computational results are in excellent agreement with those of Ghia Ghia and Shin.

Mars exploration has increased the interest in bio-inspired flapping-wing micro aerial vehicles due to their high maneuverability and efficient performance in low Reynolds number environments. This work presents an optimization of the compound performance of two tandem pitching and heaving aerofoils in the Martian environment by using the hybrid method of the reinforcement learning (RL) and the immersed boundary-finite difference method (IB-FDM). The searching parameters include the horizontal and vertical spacing, phase shift, and mean angle of attack (AoA) of the hindwing at a Reynolds number of 1100. It is found that the optimal phase shifts lie close to in-phase and out-of-phase motions, while intermediate phase shifts tend to produce suboptimal results. For the in-phase cases, an increase in AoA and a decrease in horizontal spacing led to an increase in the compound performance. For the out-of-phase cases, a similar trend is observed until the optimal value is reached, after which the performance begins to decline. The effects of vertical spacing vary vastly on a case-to-case basis, depending on other motion and positional parameters.

In this paper, we investigated the existence and uniqueness of solutions for a class of interactive fuzzy Caputo–Katugampola F-correlated fractional differential equations with time delay. First, we gave the definition of the correlated integral and derivative for interactive fuzzy Caputo–Katugampola F-correlated systems. Second, we presented an equivalent integral formulation and, under suitable assumptions, established the existence of solutions to fuzzy interactive Caputo–Katugampola F-correlated fractional-order differential equations with time delay by applying Schauder’s fixed-point theorem. Finally, a numerical example was presented to validate the effectiveness of the proposed approach.