Let β > 0 and assume 0 < b ≤ β. Let n ≥ 2 denote an integer and let n − 1 < α ≤ n. The theory of u₀-positive operators with respect to a cone in a Banach space was applied to study eigenvalue problems for left focal boundary value problems for Caputo fractional linear differential equations. Under suitable conditions, it was first established that there exists 0 < δ < β such that if 0 < b < δ, the left focal boundary value problem had a unique solution, u ≡ 0. Then, the left focal point of the left focal boundary value problem was defined and criteria was established to characterize the left focal point with respect to the spectral radius of an associated compact linear fractional integral operator. In order to establish the criteria, properties of related families of Green’s functions were observed. The article closed with an application to a nonlinear boundary value problem.

Stochastic delay differential systems are widely applied in various fields, featuring stochasticity, time delay, and nonlinearity, which makes their analysis highly challenging. This paper investigates the existence, uniqueness, and averaging principle of solutions for a class of stochastic delay differential systems. First, by using delay matrix functions and rigorous theoretical derivation, we establish a theorem on the existence and uniqueness of solutions, which lays a foundation for further analysis. Second, under classical assumptions combined with inequality techniques and Itô’s formula, an averaging principle is derived, showing that the solution of the original system can be well approximated by that of the averaged system. Finally, numerical simulations are conducted to verify the correctness and practicality of the theoretical results.

In this paper, I present the properties and structures of Dedekind σ-complete directed partially ordered rings and algebras over R. Under certain conditions, such as property P₁, chain conditions on convex ideals, and strong Archimedean, structures are isomorphic to the real number field R with the usual total order or finite product or subdirect product of R.

Chinese herbal medicine (CHM) classification is an important and emerging topic in intelligent medicine. However, due to the limited local receptive field, existing Convolutional Neural Network-based methods are susceptible to background interference, which fails to capture discriminative visual cues and hampers the accurate recognition of visually similar herbal categories. To address these limitations, we propose a dual attention enhancement network for Chinese herbal medicine classification. Specifically, we introduce an object localization module to suppress background interference by accurately localizing the target region, thus guiding the network to focus on discriminative regions. Subsequently, we introduce a fused attention module to integrate horizontal and vertical directional information at different feature scales to capture long-range spatial dependencies and enhance the global contextual perception. Moreover, we propose a dual attention module composed of self-attention and cross-attention mechanisms to achieve explicit fusion and the semantic alignment of multi-scale features. Finally, we build a semantic feature enhancement module to further strengthen the inter-layer complementarity through adaptive semantic fusion, thereby improving the discriminative ability and robustness of feature representations. Extensive experimental results on two CHM datasets demonstrate that the proposed method outperforms existing state-of-the-art methods.

In this paper, the permanence and extinction of a class of non-autonomous competitive population systems existing in an impulsively polluted environment are investigated. By establishing a time-varying differential equation model that couples population competition, pollutant dynamics, and discrete impulsive disturbances, we employ the comparison theorem of impulsive differential equations and analytical methods to derive sufficient criteria guaranteeing population persistence and global extinction. Furthermore, by constructing a Lyapunov function, we obtain conditions for the global attractivity of the system solutions. The theoretical analysis indicates that the ultimate fate of the population is jointly determined by the time-varying growth rates, competition intensities, period of impulsive disturbances, and pollutant toxicity levels. This study provides a quantitative theoretical basis for assessing the ecological risks of sporadic pollution events in competitive communities.

This study investigates the spatial decay estimate for a coupled Moore-Gibson-Thompson (MGT)-Fourier system on a semi-infinite cylinder in R². The system is comprised of a third-order MGT displacement equation and a parabolic heat equation with symmetric coupling. Using a weighted energy functional and an integral-differential inequality, we establish the exponential decay along the axial direction. This result offers a generalized Saint-Venant principle for coupled dissipative thermoviscoelastic models, thus extending the classical analysis to higher-order systems and advancing our understanding of MGT equations in an unbounded domain.

This paper investigates the adaptive tracking control problem for a class of uncertain nonlinear systems subject to time-varying state constraints and input saturation. To avoid the repeated differentiation encountered in conventional recursive design, a command-filter-based predefined-time adaptive control scheme is developed. Fuzzy logic systems (FLSs) are employed to approximate the unknown nonlinear functions, while a fuzzy observer is constructed to estimate the unmeasurable states. In addition, an auxiliary compensation mechanism is introduced to reduce the influence of filtering errors, and a smooth saturation decomposition together with an auxiliary signal is incorporated to handle the mismatch between the designed control input and the actual actuator output caused by input saturation. By combining recursive design with a time-varying barrier Lyapunov function (BLF), it is shown that all closed-loop signals remain bounded, the prescribed state constraints are not violated, and the tracking error enters a small neighborhood of the origin within a predefined time. Finally, simulation results based on an RLC circuit example are provided to demonstrate the effectiveness of the proposed method.

In the field of machine learning, the solution of large-scale data optimization problems faces numerous challenges. Traditional conjugate gradient (CG) algorithms, though possessing excellent convergence properties, incur high computational costs when dealing with large-scale problems, thereby restricting their scope of application. On the other hand, stochastic gradient descent (SGD) algorithm, while being computationally inexpensive, has its convergence rates limited by the variance of gradient estimates, making it difficult to achieve satisfactory optimization results. To address these limitations, this paper proposes a biased stochastic three-term conjugate gradient algorithm. The proposed algorithm integrates the stochastic recursive gradient algorithm (SARAH) and a bandwidth-based step size strategy. Without incurring additional computational costs, it automatically incorporates upper and lower bounds on the step size, effectively balancing the flexibility and stability of the step size. Through the theoretical analysis presented in this paper, we demonstrate that the algorithm converges to a global optimum and analyze the linear convergence rate of the non-convex (λ-gradient dominated) objective functions. The numerical results of two machine learning models demonstrate the strong competitiveness from the biased stochastic three-term conjugate gradient algorithm with bandwidth-based step size (SCGBW) algorithm.

In smart grids, the interactions among diverse participants significantly affect system efficiency and social welfare. Considering that the real-time electricity pricing (RTP) mechanism under traditional social welfare maximization fails to account for the independent interests of various entities in the power system, a two-level load-utility balancing model for real-time pricing in smart grids is proposed in this paper, where the welfare of the demand side and the multi-energy supply side is collaboratively optimized and balanced. Furthermore, a multi-agent reinforcement learning (MARL) algorithm based on the centralized training and decentralized execution (CTDE) framework is designed for this model, and a multi-agent electricity market environment is constructed accordingly, comprising users, an aggregated power supplier, and a power market scheduling center (PMSC). The user agent is modeled with a heterogeneous utility function, the supplier agent is modeled with a profit function coordinating both traditional and renewable energy, while the PMSC agent is responsible for real-time pricing and cross-agent welfare balance coordination. Finally, simulation results show the effectiveness of the proposed model and algorithm in achieving welfare balance between the supplier and users. Compared with the pricing scheme without a welfare-balancing mechanism, the proposed model reduces the welfare gap between the supplier and users by approximately 46.9%. Compared with the non-dominated sorting genetic algorithm II (NSGA-II), the proposed method can achieve a comparable level of total social welfare.

This paper focuses on studying the existence and robust finite-time stability (FTS) for a more generalized class of nonlinear Caputo-Katugampola fractional differential equations with perturbation. The existence of mild solutions is proved by using the Sadovskii fixed-point theorem (SFPT). Then, sufficient conditions for robust FTS are obtained by applying the Jensen inequality and Hölder inequality. Finally, our theoretical results are supported by the research on the Lasota- Wazewska red blood cell (LWRBC) model.

Trustworthy analytics for healthcare require models that are not only accurate but also interpretable and robust under distributional perturbations. In this paper, we propose an interpretable stacked ensemble framework that repurposes TabNet from an end-to-end classifier into an attention-guided feature generator for downstream learners. We constructed a dual-channel stacking architecture in which TabNet-derived embeddings and original tabular features were fed into heterogeneous gradient-boosted base learners (XGBoost and LightGBM) to enhance representation diversity, and were integrated by an interpretable logistic-regression meta-learner. For transparent and unbiased evaluation, we employed nested stratified cross-validation with fixed-budget hyperparameter tuning with systematic ablation studies. Experiments on the public Wisconsin Diagnostic Breast Cancer dataset showed that the proposed model achieves strong and stable performance (average accuracy 97.8% ± 1.0% under nested cross-validation) compared to a single TabNet baseline and conventional ensemble variants. Moreover, we assessed robustness under out-of-distribution-style covariate perturbations by injecting Gaussian noise at varying intensities, demonstrating that the stacking design mitigates the noise sensitivity of TabNet-derived representations and maintains a balanced sensitivity–specificity trade-off via adaptive thresholding. Overall, the proposed framework provides a reproducible template for combining deep tabular representation learning with explainable ensemble decision-making toward reliable data science applications in high-stakes domains.

We establish Lyapunov-type inequalities for systems of second order quasilinear differential equations with Dirichlet boundary conditions. By incorporating the positive parts of the coefficient functions, we correct and extend several known inequalities for quasilinear systems. Our results cover constant and variable coefficient cases and lead to explicit lower bounds for generalized eigenvalues. These inequalities generalize classical scalar results and improve the existing estimates within the literature. As an application, we obtain explicit and computable lower bounds for generalized eigenvalues of coupled quasilinear operators.

In this paper, we propose a vegetation-water coupled system that explicitly incorporates saturated water absorption and an intraspecific competition term. We first establish the global stability of the boundary equilibrium, then analyze the stability and Turing instability of the positive equilibria, demonstrating that the equilibrium with low vegetation density is consistently unstable, while the other positive equilibrium exhibits Turing instability when the vegetation diffusion coefficient is sufficiently small or the water diffusion coefficient is sufficiently large. We derive a priori estimates for nonnegative steady-state solutions via the maximum principle, conduct a detailed qualitative analysis of steady-state bifurcations at simple and double eigenvalues, and establish criteria for the bifurcation direction. Finally, through numerical simulations, we present the dynamical behaviors near the bifurcation points and simulate the evolution of vegetation patterns under different parameter settings, and find that as the water diffusion coefficientd₁ and the intraspecific competition coefficient σ increase, Turing patterns transition from spot-like to stripe-like structures; conversely, as the precipitation parameter a increases, the patterns shift from stripe-like to spot-like. These findings advance our understanding of the mechanisms governing vegetation pattern formation in water-limited ecosystems and provide a theoretical framework for predicting ecosystem responses to environmental changes.

We study a one-parameter family of polynomial J-fractions whose coefficients depend polynomially on the index and on an integer parameter u ≥ 0. After a factorial normalization of the denominator sequence, the difference of two consecutive convergents factors into the fixed central-binomial Apéry-type term $1 /\left(n^{2}\binom{2 n}{n}\right)$ and a rational factor determined by a normalized polynomial family P_u. We construct P_u by an explicit parameter-raising operator. We then prove a parameter-shift telescoping identity, which allows induction on u and gives

In this paper, we study a generalized hybrid fractional thermostat model involving the Caputo fractional derivative and the Riemann–Liouville fractional integral under nonlocal hybrid boundary conditions with feedback control. The problem describes a class of heat regulation systems in which the controller’s response depends on both the process’s thermal memory and sensor measurements taken at interior points of the domain. Working in the Banach space of integral-type Hölder functions J_α,β, we establish sufficient conditions for the existence and uniqueness of solutions to the associated boundary value problem. This functional framework is more suitable than the classical spaces C(E₀) and C^α(E₀), since it provides stronger regularity, better stability under fractional integral operators, and improved continuity properties for the nonlinear superposition terms involved in the hybrid structure. Our analysis is based on fractional calculus techniques, an appropriate measure of noncompactness, and Darbo-type fixed-point theorem. The obtained results extend several existing thermostat and hybrid fractional models in the literature. Finally, a numerical example is presented to illustrate and verify the theoretical findings for different values of α and β.

This paper presents a high-order numerical method for nonlinear Volterra integral equations of the third kind (VIE3) with weakly singular kernels. To overcome the accuracy loss caused by the unbounded derivatives of the solution near the origin, we propose a fractional Adams-Simpson- type method on a graded mesh. The stability and convergence of the proposed scheme, along with detailed error estimates, are rigorously established. It is shown that the method achieves an optimal convergence order of 4 − α − λβ, provided the mesh grading exponent λ is chosen appropriately. Numerical experiments are presented to validate the theoretical convergence results and to demonstrate the effectiveness of the method in resolving initial singularities.

This study investigates the oscillatory properties of solutions for a general class of neutral differential equations with multiple delays. Using Riccati and comparison techniques, we establish five distinct oscillation theorems that address the limitations of previous results in this topic. Our criteria not only extend and generalize earlier findings but also reduce the required constraints. Notably, they provide sharper results when applied to special cases like Euler’s equation. The novelty and effectiveness of the proposed oscillation criteria are illustrated through a detailed analysis of a given example, supported by tables and figures.

This work is concerned with a Kirchhoff-Schrödinger-Poisson (KSP) system posed in a bounded domain of $\mathbb{R}^{3}$. The model features a singular nonlinearity $\alpha v^{-\tau}$ with 0 < τ < 1, together with a coupling term of the form $\varphi|v|^{q-2} v$, where 2 < q < 3. The singular term destroys differentiability of the energy functional while the nonlocal potential φ_v causes compactness issues. Using nonsmooth critical point theory, we establish a key estimate linking the weak slope with the derivative of the regular part, prove the Palais-Smale (PS) condition, and characterize critical points as weak solutions. By means of Ekeland’s variational principle and the mountain pass theorem, we establish the existence of a constant Γ > 0 with the property that the system admits two distinct positive solutions whenever $\alpha \in(0, \Gamma)$.

Octonion-valued neural networks (OVNNs) provide a powerful framework for modeling and processing high-dimensional data due to their eight-dimensional normed division algebraic structure. However, their inherent non-commutativity and non-associativity, coupled with ubiquitous time delays and stochastic disturbances in real-world systems, make the analysis of their dynamical behavior a significant challenge. This paper focuses on the complex oscillatory dynamics, specifically pseudo almost automorphy, within a class of stochastic higher-order Hopfield neural networks (NNs) based on octonions. To adequately characterize the stochastic processes involved, a novel concept of pseudo almost automorphic stochastic processes in finite-dimensional distributions is first proposed. Subsequently, by fixed point theorems and employing inequality techniques, sufficient criteria are established for the existence and global exponential stability of pseudo almost automorphic solutions in finite-dimensional distributions for the considered octonion-valued stochastic higher-order Hopfield NNs with time-varying delays. The obtained results are not only new for the octonionic system, but also remain novel even when the system degenerates to its real-valued counterpart. Furthermore, the analytical framework developed herein offers a general methodology applicable to studying pseudo almost automorphic dynamics in other types of complex-valued NNs.

We investigate the metric geometry of gyrogroups, a class of group-like structures whose binary operation is generally nonassociative. In particular, we extend the notion of the word metric from finitely generated groups to gyrogroups. This extension enables any gyrogroup to be viewed as a metric space, providing a suitable framework for proving a Mazur–Ulam-type theorem and for analyzing its algebraic and combinatorial structure via the associated right Cayley graph in a manner analogous to the classical setting of groups.

This work investigated the fault-resilient output-feedback control problem for delayed memristor-based neural networks (MBNNs) subject to actuator faults and external disturbances. To overcome the practical limitation that full-state measurements are often unavailable, an output-feedback controller was designed to rely solely on measurable outputs. By constructing a Lyapunov-Krasovskii functional and employing the Bessel-Legendre inequality together with the generalized reciprocally convex combination inequality, a sufficient condition was established to simultaneously guarantee $\mathcal{H}_{\infty}$ and $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ disturbance attenuation for the considered MBNNs. Decoupling techniques were then introduced to eliminate nonlinear coupling terms, and the controller synthesis was formulated in terms of linear matrix inequalities that are solvable with existing convex optimization tools. Finally, a numerical example was provided to corroborate the reduced conservatism of the proposed analysis condition and the effectiveness of the synthesis method in achieving the desired dual-performance guarantees.

Differential-algebraic equation (DAE) is widely used in engineering domains, such as fluid dynamics, multi-body dynamics, mechanical systems, and control theory, owing to their ability to effectively characterize dynamic variations and inherent constraints. In recent years, physics-informed neural networks (PINNs) have manifested remarkable advantages in solving both the forward and inverse problems of DAE by integrating physical prior knowledge into neural network models. Presently, PINNs-based approaches still encounter challenges of inadequate solution accuracy and limited generalization performance when dealing with DAE involving discrete events. This paper presented a physics-informed neural network that integrates the Legendre-Gauss-Lobatto (LGL) collocation method from spectral methods to solve the aforementioned DAE with discrete event. To further augment the accuracy and continuity of the solution, the model employed a time- domain decomposition strategy to construct the network architecture, thereby enabling high-precision continuous-time prediction of DAE. Numerical examples illustrated that the LGL-PINN can attain high-precision solutions of DAE. In comparison with the PINNs, the error between the predicted solution and the exact solution of the LGL-PINN was substantially reduced, with the accuracy improved by one to two orders of magnitude. Therefore, the proposed solution model demonstrated excellent computational accuracy for solving DAE problems involving discrete event.

In response to the problems of sparse temperature measurement point distribution and poor cross-interval continuity in the hot continuous rolling process, a full-process temperature field prediction and reconstruction method based on the combination of a physics-informed neural network (PINN) and a Bayesian-XGBoost surrogate model is proposed in this paper. First, a PINN model across five process areas (from the reheating furnace to the coiler) is established, taking time nodes as input and directly outputting discrete points of the temperature-time curve along the strip in each process area. Subsequently, a Bayesian-XGBoost surrogate model is used; at this time, hierarchical surrogate decision-making determines the number of prediction points automatically by the maximum error principle and thereby improves the prediction accuracy. Finally, a piecewise cubic spline interpolation algorithm is designed, based on curvature detection to achieve a precise high-precision temperature curve reconstruction of discrete prediction results. The experimental results show that this method has high accuracy in temperature curve reconstruction and good reliability in mechanical property prediction; it verifies the effect and practicability of the proposed framework in real hot continuous rolling processes.

In this paper, we investigate the existence of normalized ground state solutions for a class of quasilinear elliptic equations. More precisely, we consider the p-Laplace equation with a prescribed L^p-norm constraint. The problem involves a potential function and a nonlinear term with mass supercritical growth. By employing variational methods, we aim to find solutions with a given mass, where the parameter λ appears as a Lagrange multiplier. Under appropriate assumptions of potential and nonlinearity, we establish the existence of such solutions for any given positive mass.

A stochastic model for theme park ride waiting times is developed by modeling the waiting time as a continuous-time, discrete-state Markov process with state-dependent, time-varying transition rates. These transition rates are interpreted as a control term acting on the waiting-time process, allowing the model calibration task to be formulated as a data-driven optimal control problem. To solve this problem efficiently, we construct a physics-informed neural network (PINN) that embeds the Kolmogorov forward equation solver into its architecture. Under mild assumptions, we prove the existence of an optimal control, providing theoretical support for the learning procedure. Numerical simulations demonstrate the effectiveness of the PINN-based solution. The proposed framework provides an interpretable, physically consistent, and data-driven approach for modeling and forecasting ride waiting times.

This paper is devoted to investigating the existence of traveling wavefront solutions with a sufficiently large wave speed for a nonlocal diffusion model with nonlocal delay effect using a perturbation method. Our proof is based on an abstract formulation of the wave profile as a solution to an operator equation in a specific Banach space, combined with the Fredholm theory and the Banach contraction mapping principle. Several numerical simulations are presented to illustrate our main results.

Cluster analysis of unlabeled categorical data is crucial in a wide range of practical applications, such as medical diagnosis, financial risk assessment, and recommendation systems. Unlike numerical data residing in explicit Euclidean spaces, categorical data consists of qualitative values without inherent ordering, making the definition of object similarity a critical yet challenging determinant of clustering success. Conventional approaches typically rely on single, predefined metrics (e.g., Hamming distance or context-based measures). However, these metrics are often constructed based on limited prior knowledge or specific statistical assumptions, failing to capture the complex, intrinsic structures of diverse datasets. Consequently, the mismatch between the defined metric space and the “true” data structure significantly hinders the performance of downstream clustering tasks. To address these limitations, this paper proposes a novel subspace and metric space co-learning framework named SBMS. Instead of relying on a static measure, SBMS introduces an adaptive learning paradigm that iteratively optimizes two coupled spaces: a metric space, where multiple complementary distance metrics are fused to provide a comprehensive similarity measure; and an attribute subspace, where attribute weights are dynamically adjusted based on cluster discrimination and compactness to identify the most relevant features for each cluster. Furthermore, we provide a theoretical analysis of the proposed method, discussing its computational complexity and demonstrating the convergence properties of the optimization algorithm. Extensive experiments on real-world public datasets from various domains illustrate that SBMS effectively bridges the gap between defined and true metric spaces, yielding superior clustering accuracy and stability compared to state-of-the-art baselines.

Utilizing evolutionary algorithms (EAs) to solve multitask optimization problems (MTOPs) simultaneously has gained significant attention because many real applications usually share similar properties or have a certain degree of correlations. In this new research area in the EA community, namely multitask optimization (MTO), a key challenge is transfering useful knowledge among different tasks efficiently. In this paper, a multitask differential evolution (DE) algorithm is proposed in which a simple variant of the canonical DE is adopted as the basic optimizer. Furthermore, an expert library module (ELM), which can extract useful knowledge from the historical optimization experiences of DE, is introduced as a complement for the DE to generate offspring. In the proposed algorithm, named MTDE-ELM, when the DE performs the mutation operator, if all individuals chosen to perform the mutation have the same skill factor, intratask knowledge transfer can be realized. Conversely, when individuals with different skill factors are adopted to execute the mutation, intertask knowledge transfer can be realized. In the ELM, evolutionary data of different tasks are used to train the feedforward neural networks (FNNs). After that, the trained FNN can realize potential knowledge transfer among different tasks. Moreover, based on the trained FNN, each individual can query its promising evolutionary direction, which can be regarded as an effective supplement for the DE to generate offspring. The comprehensive performance of the MTDE-ELM is demonstrated through comparisons with five other MTO algorithms. Furthermore, distinct properties of the newly introduced strategies are also confirmed by experimental analysis.

Environmental enrichment can destabilize predator–prey coexistence through a Hopf bifurcation, yet real ecosystems are finite and intrinsically stochastic. We investigate how mechanistically derived demographic noise shapes near-Hopf dynamics in the Rosenzweig–MacArthur model by systematically comparing two diffusion closures that share identical deterministic drift but differ solely in their predation-induced covariance structure. Starting from a continuous-time Markov chain description, we derive a full-covariance stochastic differential equation whose diffusion tensor inherits stoichiometric coupling, generating a negative prey–predator cross-covariance. Our exact nonlinear simulations demonstrate that the dominant near-Hopf phenomenon is the profound breakdown of the linear noise approximation itself. While the linear noise approximation predicts unbounded variance and spectral amplification, the true nonlinear quasi-cycles remain strictly bounded, rapidly driving the system into absorbing extinction boundaries. We conclude that accurate early warning inference in finite ecosystems depends not on resolving fine-scale stoichiometric covariance, but on properly accounting for nonlinear saturation and noise-induced boundary hitting.

This study investigates the threshold behavior of a population-varying stochastic susceptible- infectious-recovered-susceptible (SIRS) model driven by a logarithmic Ornstein-Uhlenbeck process. Introducing the logarithmic Ornstein-Uhlenbeck process to account for random environmental fluctuations enhances the biological significance of the model. By applying the Itô stochastic integral, we construct a suitable Lyapunov function, proving the existence and uniqueness of the global positive solution of the model, thereby ensuring biological feasibility. Then, a critical threshold parameterR^s₀ is derived: If $R_{0}^{s}>1$, the system admits a unique invariant probability measure; if R^s₀ < 1, the infection dies out almost surely around the disease-free equilibrium. Furthermore, near the quasi-equilibrium point, the invariant probability density admits a local Gaussian approximation and converges weakly to a normal distribution as the environmental noise tends to zero. Numerical simulations in MATLAB illustrate the theoretical results, further reveal the sensitivity of the stochastic threshold to the key parameters, and confirm that population variation influences the threshold structure and long-term infection level.