Non-pharmaceutical interventions play a vital role in mitigating the outbreak of infectious diseases. During the COVID-19 pandemic, numerous countries and regions imposed multiple rounds of intervention measures to curb the spread of the disease. To get insights into optimal implementation of one-round interventions, two optimal control problems were examined under distinct time horizons, based on whether the herd immunity was reached. Here, one-round interventions were defined as interventions that operated under a budget constraint. Both of optimal control policies were proved to be purely bang-bang with a single switch, but their timings of onset were different. These results may provide some theoretical support for developing intervention strategies.

In this paper, we study the number of solutions to the inhomogeneous binary quadratic congruence ax² + bxy + cy² + dx + ey ≡ n (mod q), where q is a prime power. We treat the cases where q is an odd prime power and a power of 2 separately. By combining exponential sum methods, quadratic Gauss sums, and Hensel’s lemma, we obtain exact formulae for the number of unrestricted solutions over $\left(\mathbb{Z}_{q}\right)^{2}$ and the number of solutions with both variables restricted to units over $\left(\mathbb{Z}_{q}^{*}\right)^{2}$. This paper simultaneously handles a mixed quadratic term and two separate linear terms, extending several previously studied diagonal, homogeneous, and symmetric special cases.

For a class of strict-feedback nonlinear systems involving unknown time-varying parameters and external disturbances, an event-triggered adaptive prescribed-time control method is proposed. First, a prescribed-time adjustment function containing only a single reciprocal term is introduced to unify the multi-stage high-order scaling in conventional prescribed-time backstepping into a single first-order scaling, thereby simplifying the structure of the prescribed-time scaling term and reducing the complexity of the recursive controller design. Second, an adaptive law is developed to adaptively compensate for the effects of unknown time-varying parametric uncertainties, while a radial basis function neural network (RBFNN) is embedded into the backstepping framework to achieve real-time approximation and compensation of external disturbances. Moreover, an event-triggered mechanism is incorporated to operate synchronously with the controller, guaranteeing prescribed-time stability and substantially reducing the number of control updates compared with periodic sampling, thus saving communication and computational resources. Finally, numerical simulations demonstrate that the proposed control strategy achieves prescribed-time convergence and post-prescribed-time practical boundedness, while maintaining satisfactory robustness with significantly fewer control updates.

Full waveform inversion (FWI) is a powerful technique for reconstructing subsurface physical parameters. However, conventional FWI often suffers from cycle-skipping and local-minimum issues. With the rapid development of deep learning, data-driven methods have attracted increasing attention in FWI, where the inversion problem is reformulated as an end-to-end reconstruction task and neural networks are employed to directly estimate velocity models from seismic data. Most existing approaches rely on convolutional neural networks (CNNs) to extract local features from input data. However, compared with natural images, seismic data exhibit strong long-range dependencies that CNNs struggle to model effectively, since information associated with a single reflection interface is spatially distributed across the seismic data. Although CNNs perform well in computer vision, their local feature extraction mechanism limits reconstruction performance in FWI. To address this issue, we have proposed a novel FWI method based on the vision transformer (ViT). Unlike CNN-based approaches, the proposed method leverages the self-attention mechanism to model long-range dependencies in the input data. By stacking multiple transformer blocks, global physical information in seismic data is effectively encoded into the inversion process, resulting in improved reconstruction performance. Numerical results demonstrated that the proposed method outperforms CNN-based approaches in terms of reconstruction accuracy.

This paper investigated the identifiability of attack frequency and the parameter estimation problem in binary quantized finite impulse response (FIR) systems under Poisson-distributed replay attacks. First, the replay attack is modeled as a Poisson stochastic process, and the convergence properties of the attack frequency in the long-term statistical sense were analyzed, along with definitions and criteria for attack frequency identifiability. Under the assumption that the attack frequency is known, a compensated identification algorithm was established for parameter estimation, and the strong consistency and asymptotic normality of the estimators were established. Furthermore, for the case of unknown attack frequency, a joint estimation algorithm based on empirical statistics was proposed to achieve simultaneous identification of the attack frequency and system parameters. In addition, the optimal design criterion for the compensation ratio parameter was derived to reduce estimation errors. The proposed framework falls within the scope of resilient identification, where the estimator continues to recover system parameters from long-term statistics even under intermittent replay corruptions of the communication channel. Finally, numerical simulations under various attack frequencies, delay distributions, and noise levels were conducted to verify the effectiveness and robustness of the attack frequency identifiability analysis and the proposed algorithms.

This paper studies a nonlinear Schrödinger-type model for Bose-Einstein condensates with a fixed s-wave scattering length. Under a specific rescaling, the dimensionless Gross-Pitaevskii equation takes the form iΨ_t = −∇²Ψ − 2|Ψ|²Ψ. The model is rewritten in hydrodynamic form via a density-phase decomposition, and in the long-wave limit, it is asymptotically reduced to the classical Korteweg-de Vries (KdV) equation. For this reduced KdV equation, we present its Lax pair, Darboux transformation, and Bäcklund transformation, and obtain multi-soliton and positon solutions by means of the Hirota bilinear method. Furthermore, by coupling the tau functions of a single soliton and a first-order positon via the Wronskian determinant, we construct a mixed interaction solution, whose explicit expression contains nonlinear cross-coupling terms which characterize the collision dynamics between the two types of waves. An asymptotic expansion of the spectral problem yields infinitely many conservation laws, and we prove that all orders of global conserved quantities for the soliton-positon composite wave satisfy an exact superposition relation. These results demonstrate that the reduced KdV equation derived from the Bose-Einstein condensate model possesses a complete integrable structure, thus providing a theoretical reference for understanding nonlinear wave propagation in quantum fluids under the KdV approximation.

We study a class of nonlinear boundary value problems driven by mixed ψ-Riemann-Liouville tempered fractional derivatives of order $\alpha \in\left(\frac{1}{2}, 1\right)$ under homogeneous Dirichlet boundary conditions. The analysis is performed in a suitable fractional Sobolev-type space associated with ψ-tempered operators. We first establish sharp mapping properties for the ψ-Riemann-Liouville tempered fractional integral, proving its boundedness from L^p((a,b), ψ^′) into Hölder spaces $H^{\alpha-\frac{1}{2}}[a, b]$,b] with explicit constants depending on the incomplete gamma function. These estimates provide the functional framework required to handle the corresponding nonlocal problem. Using variational methods, we associate the problem with a C¹ energy functional on $\mathbb{H}_{\psi, 0}^{\alpha, \sigma}(a, b)$ and prove compactness via the Palais–Smale condition under optimal growth assumptions. We then establish a precise variational structure yielding: (i) existence of nontrivial solutions via minimization arguments, (ii) multiplicity results based on a mountain pass geometry, and (iii) existence of infinitely many solutions by means of symmetric critical point theory. Our results extend the current theory of fractional equations with respect to another function by incorporating tempered effects and explicit operator estimates, thus providing a unified and flexible framework for a broad class of nonlocal problems.

In large-scale, nonconvex optimization for machine learning, such as the training of spine large models, the balance between efficient high-order geometric features of nonconvex landscapes identification and computational stability remains challenging. This paper proposes a tensor stochastic three-term conjugate gradient (TSTCG) algorithm to address two issues under a stochastic CG framework, the insufficient curvature awareness of the stochastic gradient descent and the heavy reliance on line searches. By introducing an implicit correction mechanism based on third-order tensor information, the algorithm fuses function values and gradient differences to reconstruct evolution vectors, effectively enhancing search precision in ill-conditioned curvature regions. Simultaneously, leveraging a specially designed three-term denominator structure allows the algorithm to process sufficient descent property, line search reliance reduction, and efficient fixed step-size updates. Finally, the nonconvexity proofs and numerical experiments demonstrate that TSTCG outperforms other mainstream stochastic optimization algorithms in terms of convergence efficiency and stability.

This paper proposes a distributed relative-position-based cooperative pointing control scheme for stationary planar multi-agent systems toward a stationary target under switching topologies. First, the proposed scheme only requires the leaders’ orientation angles and relative position measurements, without relying on the agents’ states in a common global coordinate frame. Then, the cooperative control problem is decoupled into cooperative target localization and local pointing control. In the proposed scheme, the cooperative target localization problem is modeled as a distributed relative position estimation system, in which the system dynamics are determined by the switching topologies and the projection matrices induced by the leaders’ orientation angles. Under minimal information conditions, sufficient conditions are established to ensure the convergence of the proposed scheme by developing a novel error estimator and multiple Lyapunov functions. As a result, the generated reference orientations converge to the desired ones, and the local pointing controllers track them asymptotically. Finally, numerical simulations validate the effectiveness of the proposed method.

This paper investigated the exponential synchronization (ES) problem of reaction-diffusion neural networks over directed networks under an asynchronous sampled-data event-triggered intermittent control scheme. By integrating event-triggered communication and intermittent control mechanisms, hybrid control framework was developed to reduce communication transmissions and control activation time. A Lyapunov–Krasovskii functional (LKF) involving historical-state integral terms was constructed, and sufficient synchronization conditions were derived using graph-theoretic techniques and inequality methods. The proposed criterion guaranteed exponential convergence of synchronization errors and avoided Zeno behavior. Numerical simulations demonstrated that the proposed strategy achieved satisfactory synchronization performance with reduced communication requirements.

In this paper, we consider a complete strongly stable hypersurface Mⁿ with constant mean curvature (CMC) H in Nⁿ⁺¹(κ), where 2 ≤ n ≤ 5, and prove that Mⁿ is totally umbilical if H² + κ > 0 and the L² norm curvature satisfies some growth conditions. In addition, we obtain that a complete stable hypersurface Mⁿ in Nⁿ⁺¹(κ) (n = 3, 4, 5) with CMC is a geodesic sphere if − H² < κ ≤ 0 and Mⁿ has some finite L^p norm curvature.

In-the-wild facial expression recognition (FER) is inherently affected by ambiguous expressions and subjective annotations, making the observed labels only partially reliable. Conventional one-hot supervision nevertheless treats these labels as ground truth and may drive deep networks to memorize corrupted annotations. This paper proposes RDAS, a reliability-driven adaptive supervision framework for noisy-label FER. The central idea was to use label reliability not merely as a sample weight, but as a variable that determines the supervision policy of each training sample. RDAS enhances expression-sensitive representations by combining RGB appearance with structural high-frequency cues, estimates label reliability with an exponential moving average (EMA) teacher through the margin between the annotated class and its strongest competing class, and assigns reliability-specific objectives to high-, medium-, and low-reliability samples. Reliable samples retained hard-label learning, ambiguous samples received teacher-guided soft correction, and unreliable samples were prevented from dominating training with misleading gradients. Experiments on RAF-DB, AffectNet, and FERPlus datasets showed that RDAS improved both standard recognition performance and robustness under synthetic label noise. On RAF-DB, RDAS achieved an average accuracy of 89.70% on the original dataset and 86.15% under 30% label noise. Ablation studies provided quantitative support for reliability modeling and adaptive supervision, while the visual analyses provided complementary qualitative evidence consistent with the proposed mechanism.

Jump-diffusion models, such as Merton’s and Kou’s models, are essential in option pricing for capturing the discontinuous jump characteristics of asset prices. However, the nonlocal integral term in the resulting partial integro-differential equation (PIDE) presents substantial numerical challenges. This paper proposes an efficient and high-order numerical framework for pricing European and American options under these models. We employ the barycentric rational interpolation (BRI) collocation method for spatial discretization to achieve high-order accuracy and ensure numerical stability. For temporal discretization, the second-order implicit–explicit backward differentiation formula (IMEX- BDF2) is utilized, and the nonlocal integral term is evaluated via the Gauss–Legendre quadrature. To handle the early exercise constraint of American options, the operator splitting method is integrated into the framework. A rigorous error estimation and convergence analysis are provided to establish the mathematical reliability of the scheme. Numerical experiments demonstrate that the proposed method is highly efficient, providing accurate estimates for option values and stable numerical approximations for their Greeks (delta and gamma), validating its effectiveness for solving PIDEs in financial engineering.

The periodic Sylvester tensor equation (PSTE) arises in multidimensional periodic systems. This paper proposes a tensor finite-time convergent gradient neural network (Tensor-FTCGNN) with a global adaptive gain for solving a class of PSTEs. Using the Einstein product, the periodically coupled tensor equations are formulated as a unified linear operator equation on a product tensor space, and the corresponding adjoint operator and global gradient are derived. Tensor linear operator theory and Lyapunov analysis are employed to establish the global finite-time convergence of the ideal continuous-time model and to derive an explicit upper bound for its settling time. For practical numerical implementation, a regularized Tensor-FTCGNN is introduced, which reaches an ϵ-dependent high-accuracy region in finite time and subsequently converges exponentially to the exact solution. Numerical experiments compare the proposed regularized model with representative gradient neural network (GNN) models, verify the theoretical finite-time upper bound through the observed Phase-I entrance times, and evaluate its convergence behavior over the tested tensor sizes. Finally, the proposed framework is applied to cooperative periodic trajectory reconstruction for a multiarm robotic system, illustrating its potential for multidimensional robotic applications.

This paper establishes a Filippov integrated pest management model with the fear effect and prey refuge, aiming to investigate economic-threshold-based intermittent control strategies and their dynamical behavior. First, the existence and local stability of the equilibria of the two subsystems are discussed, and the nonsmooth dynamics of the Filippov system are analyzed. Furthermore, sliding bifurcations, boundary equilibrium bifurcations, and global sliding bifurcations are investigated through numerical simulations. The results indicate that the control effect is highly sensitive to the threshold value, and as the threshold varies, the system exhibits a series of complex bifurcation phenomena. Furthermore, pest refuge behavior may weaken the effectiveness of control measures. Although the fear effect can suppress pest growth, an excessively strong fear effect may reduce prey availability for natural enemies, thereby limiting their population growth.

Neighborhood rough sets (NRS) are commonly applied to attribute reduction because they can process numerical data directly without discretization. Granular ball neighborhood rough sets extend NRS by adaptively determining neighborhood radii according to the data distribution. However, existing granular ball generation and reduction algorithms can be further enhanced in terms of granule representation, radius computation, and attribute evaluation. To address these issues, an optimized granular ball neighborhood rough set model (OGNRS) and a corresponding attribute reduction algorithm are proposed. In this model, a scoring function integrating density and centroid-based distance is employed to select centers, and a median absolute deviation-based strategy is introduced to determine the radii of granular balls. In addition, the minimal-redundancy-maximal-relevance criterion is incorporated into the reduction process to provide search guidance and reduce redundancy among selected attributes. Experiments on 14 UCI datasets demonstrate that OGNRS achieves competitive overall performance in terms of classification accuracy, reduct size, and computational efficiency.

Cooperative benefit allocation becomes difficult when coalition worths are imprecise and decision rights are asymmetric. This study develops a power-weighted interval-valued core framework that integrates asymmetric voting power with satisfaction-based interval coalition rationality. Individual voting power is derived from coalition criticality in a weighted voting game and aggregated at the coalition level through a least-squares rule, while the resulting max-min allocation model is solved using a bisection procedure based on monotonic feasibility. In a stylized, normalized three-player example motivated by marine fishery insurance, one optimal allocation generated by the proposed model satisfies the adopted efficiency and coalitional rationality conditions, whereas a selected power-adjusted interval Shapley benchmark does not under the same parameterization. Parameter sensitivity analysis further illustrates how voting weights, decision thresholds, and the grand coalition worth interval are associated with changes in power indices and the distribution of allocation uncertainty. The numerical results provide methodological illustrations under a controlled parameterization and demonstrate the feasibility and computational implementation of the proposed solution procedure.

Motivated by the stochastic optimal control problem, in this paper we systematically study a class of mean-field backward stochastic Volterra integral equations with jumps (MF-BSVIEs with jumps, for short). More precisely, by a special kind of BSVIEs with jumps, we present the well-posedness of MF-BSVIEs with jumps in the sense of adapted M-solutions. For completeness, both the well-posedness of mean-field forward stochastic Volterra integral equations with jumps (MF-FSVIEs with jumps, for short) and the comparison theorem for MF-BSVIEs with jumps are directly proved. Based on this, by virtue of the duality principle, we are able to establish Pontryagin’s type maximum principle for optimal control problems of MF-FSVIEs with jumps. We illustrate our results with an application to the linear-quadratic optimal control problem of MF-FSVIEs with jumps.

Following publication of this article [1], we identified an error in the presentation and interpretation of the out-of-bag permutation-and-predict variable importance results shown in Figures 4 and 6. The quantity labeled and discussed as OOB PaP was, in fact, a scaled version of the measure. As a result, the figures and portions of the accompanying discussion did not represent the unscaled OOB PaP metric. Figures 4 and 6 have been updated to include both the raw unscaled OOB PaP measure and the scaled version, OOB scale. The corrected versions of Figures 4 and 6 are reproduced below. Corresponding revisions have been made to the discussion and conclusions. Specifically, several statements previously attributed to OOB PaP should instead be attributed to OOB scale. The revised manuscript also includes discussion of the behavior of the unscaled OOB PaP metric. These corrections do not affect the underlying analyses, methodology, or overall conclusions of the study regarding the relationships among variable importance measures. Rather, they provide a more accurate characterization of the behavior of the out-of-bag permutation-based importance metrics. Additionally, Figures 7–10 have been updated to improve clarity and reduce redundant plotting information.

This paper studies a diffusive predator–prey model with Smith growth, a ratio-dependent Holling type III functional response, and Leslie–Gower predator dynamics. By analyzing the characteristic equations, we derive conditions for Turing instability and a linear stability threshold that signals the onset of Hopf-type oscillations. For the Turing branch, we prove the local existence of non-constant steady states bifurcating from the positive equilibrium, treating simple cases using the Crandall–Rabinowitz theorem. Numerical simulations are then carried out to illustrate these theoretical results and to explore the system’s behavior near the intersection of the Turing and Hopf curves, where long-lived transient spatial patterns, spatially inhomogeneous periodic solutions, and complex spatio-temporal oscillatory patterns are observed. Overall, this work offers a mathematical account of how nonlinear growth, ratio-dependent predation, and spatial diffusion together shape the dynamics of this predator–prey system.

Effectively controlling industrial pollution is a critical global challenge, yet the strategic interaction between local and central governments often leads to unstable or unpredictable policy enforcement, creating significant difficulties for long-term environmental governance. In this study, we developed an evolutionary game model within a system dynamics (SD) framework to analyze the strategic behaviors of local and central governments. We proposed and compared three mechanisms, the assessment mechanism for ecological progress (AMFEP), the ecological governance supervision and punishment mechanism (EGSPM), and a hybrid model combining both, to stabilize the evolutionary path. Simulation results demonstrated that while the basic model and individual mechanisms often failed to converge, the hybrid model successfully drove the system to a stable equilibrium, increasing the probability of local governments implementing environmental regulation strategies from 0.2 to 0.167. This finding indicated that a combined appraisal mechanism is significantly more effective in guiding local governments toward proactive environmental enforcement while central governments maintain a lighter supervisory role, offering a robust and practical policy tool for sustainable industrial pollution management.

Global energy demand keeps rising amid economic expansion and population growth. Although renewables like wind and solar have grown fast, fossil fuels (especially petroleum) remain central to the global energy mix, underpinning industry, households, policy, and stability. This article presents an improved hybrid forecasting framework, denoted as the VMD-WTD-GRU model with mode-specific denoising, for daily average petroleum consumption, which adopts a two-stage decomposition-targeted denoising strategy. The framework combines the models of variational mode decomposition, wavelet threshold denoising, and the gated recurrent unit networks for petroleum energy forecasting, where different models are designed for time-series data decomposition, noise elimination, sub-series forecasting, and integration, respectively. In general, the proposed hybrid model has achieved excellent predictive performance on multiple evaluation indicators when compared with several candidate models, especially demonstrating that the mode-specific denoising strategy can effectively reduce noise interference while preserving useful trend information in low-frequency components. Such a framework can be easily implemented to forecast petroleum consumption in real application scenarios.

This paper was devoted to solving the large-scale trust-region subproblem (TRS), a core subproblem in nonlinear optimization with extensive applications in scientific computing fields such as the Lorentz eigenvalue problem, Tikhonov regularization, nonlinear least squares, and so on. The generalized Lanczos trust-region (GLTR) method is one of the most popular iterative algorithms for large-scale TRS. In this method, the approximate solution converged more slowly than the approximate Lagrange multiplier. Inspired by this characteristic and the generalized minimal residual method (GM- RES) and minimal residual method (MINRES), we proposed a minimal residual method based on the GLTR framework (MRTR). The MRTR method fixed the approximate Lagrange multiplier from GLTR and sought the approximate solution that minimized the residual norm within the Krylov subspace. We established some theoretical results to characterize the relationship between the approximate solutions of MRTR and GLTR. The experimental results showed that the MRTR method outperformed the GLTR method in terms of iteration numbers and CPU running time, and its residual norm convergence curve was smoother with better numerical stability, while achieving approximate optimal values with the same high precision as GLTR.

This paper introduced a block diagonal preconditioner that incorporated two distinct preconditioned matrices for solving saddle point problems. Then we established some convergence conditions for the resulting iterative method under appropriate conditions. Moreover, a spectral property of the preconditioned matrix was also provided. Numerical experiments demonstrated that the proposed preconditioner exhibited superior performance compared with several recently developed preconditioners.

This work investigates the selection principle of the critical speed in a three-species Lotka–Volterra competitive system with nonlocal diffusion. The system incorporates symmetric con- volution kernels for interspecific couplings and, after a suitable change of variables, is recast into a cooperative framework. By applying the method of super- and subsolutions, we construct a novel pair of upper and lower solutions for the traveling-wave ordinary differential equations. This construction furnishes new analytic criteria that distinguish between linear and nonlinear selection of the minimal propagation speed. In particular, our results show that under suitable parameter regimes, the minimal wave speed coincides the linear prediction from the system’s unstable equilibrium (linear determinacy), whereas sufficiently strong competition among the three species causes the minimal wave to travel faster than the linear theory predicts (nonlinear determinacy). These findings advance the understanding of wave speed selection in reaction–diffusion systems with nonlocal interactions by delineating the boundary between linear (pulled) and nonlinear (pushed) invasion fronts.

As a critical branch of optimization theory, adaptive optimization algorithms mitigate the limitations of classical gradient descent, such as slow convergence and vulnerability to local optima, particularly in high-dimensional, non-convex environments. This is achieved by dynamically adjusting learning rates and per-parameter update rules. By integrating metaheuristic strategies with adaptive gradients, a dynamic self-adaptive optimization framework (DSOF) is proposed in this paper. The DSOF introduces a history-informed second-order momentum mechanism, which leverages recent gradient statistics to improve robustness under non-stationary objectives. Additionally, it maintains population diversity through evolutionary crossover and mutation operations to prevent premature convergence, and adaptively contracts or expands parameter bounds to balance exploration and exploitation. Experimental results demonstrated that, under the evaluated settings, the DSOF improves optimization stability by reducing the gradient-variance fluctuation coefficient from 0.418 to 0.382 in the modified national institute of standards and technology classification task, achieves a 100% global optimum hit rate on the 50-dimensional Ackley function, and reduces NAS-Bench-201 search time to 25 hours with only 1.02 GPU-days. These results verify the empirical effectiveness of the DSOF in improving stability, global exploration capability, and computational efficiency.

This work studied the local well-posedness of the Cauchy problem for a one-dimensional variational wave system arising from the theory of nematic liquid crystals. The system is degenerate and singular at spatial infinity. We applied the method of characteristics and the contraction mapping principle to establish the local existence and uniqueness of classical solutions. A suitable weighted L^∞ norm was introduced to ensure that the solution has no loss of regularity and the existence time of the solution is independent of the spatial variable.

We developed a game-theoretic model of an electronics manufacturing supply chain with cross-competition, in which each supplier trades with two common buyers simultaneously, and analyzed firms’ innovation investment strategies with spillover effects. The model’s structure and parameter space were motivated by key characteristics of the electronics manufacturing industry, such as rapid technological iteration, high spillover intensity, and cross-product competition. The results indicate that when the competition is weak, both process innovation and product innovation yield greater benefits to suppliers. However, fierce competition or increased investment scale can drive equilibrium shifts from innovations to non-innovations. Compared with product innovation, process innovation demonstrates greater resilience to competitive pressures in achieving dominance. At high spillover intensities, the benefits of product innovation can be completely negated, even in the presence of weak competition. This study constructs an innovation decision-making system for electronics manufacturing with cross-competition, providing guidance for innovation decisions in cross-competitive enterprises.

Transferring summary-level auxiliary information from source domains to enhance statistical analysis in target domains is increasingly important, yet settings involving right-censored survival data remain underexplored. To address this gap, we developed a general methodology for incorporating subgroup Kaplan–Meier survival probabilities at multiple time points, a widely reported form of auxiliary information. Our framework handles both estimation of an abstractly defined target parameter, such as a regression coefficient or a causal estimand, and hypothesis testing for it. We first constructed a target-only estimator that possesses a unified structure applicable to a broad class of target parameters. This estimator was then linked, via a leveraging factor, to a transitional element computed from the target dataset in a manner analogous to the available auxiliary information. We proposed and compared two versions of this element. The model-free version is universally applicable and robust, whereas the model-based version delivers improved efficiency under a partly independent censoring assumption that has often been overlooked in existing literature. Our method requires no individual-level source data and maintains low computational complexity. Theoretical derivations and numerical experiments together demonstrated gains in estimation efficiency and testing power. In addition, Python implementations of the proposed approaches were provided.

This study proposes an explainable computational intelligence framework for badminton stroke-quality assessment with large language model (LLM)-assisted coaching analytics. The proposed framework integrates stroke event segmentation, skeletal motion analysis, graph-based action recognition, stroke-quality assessment, and natural-language feedback generation. Racket–shuttlecock contact moments are first detected to extract individual stroke clips. MediaPipe Pose is employed to obtain skeletal joint trajectories, from which six stroke performance indices are formulated, including swing velocity index, wrist trajectory smoothness index, wrist posture index, elbow posture index, footwork efficiency index, and racket control index. These indices are integrated to generate interpretable stroke-quality ratings and radar-chart visualizations. For stroke classification, the proposed mST-GCN++ model achieved 94.34% accuracy across 11 badminton stroke categories, while the racket–shuttlecock contact detection module achieved an F1-score of 99.74%. The recognized stroke type, stroke performance indices, and assessment results are subsequently provided to an LLM-based module to generate personalized coaching feedback. Experimental results demonstrate that the proposed framework can effectively distinguish effective and ineffective stroke executions through interpretable quantitative assessments. Human-in-the-loop evaluation with badminton experts and players achieved an average score of 8.594/10, indicating that the generated feedback is technically accurate, consistent with observed movements, and useful for practical training. The proposed framework contributes by introducing six interpretable stroke performance indices that bridge low-level skeletal motion and high-level coaching knowledge, a hierarchical fuzzy inference system that performs transparent biomechanical reasoning for stroke-quality assessment, and a knowledge-guided LLM-assisted coaching framework that generates grounded natural-language coaching feedback from structured assessment results.

In this paper, we consider linear ill-posed inverse problems in Hilbert spaces under the classical power-type source condition. We propose a block stochastic gradient descent (Block SGD) algorithm, which generalizes the classical SGD method by allowing updates on data blocks rather than single indices. Under the power-type source condition, we derive a novel mean squared error bound between the true solution and the regularized solution, in terms of the step size, the batch size, and the block number. In particular, our estimate shows that the stochastic error decreases as the batch size increases, which is confirmed by numerical experiments.

Federated learning is a distributed learning paradigm in which clients collaboratively train a shared model through local computation and server-side aggregation of model updates, without sharing raw data. However, under non-independent and identically distributed data settings, the aggregation performance of conventional FL often degrades significantly. Although clustered FL is an effective approach to mitigating data heterogeneity, existing methods still exhibit several limitations, including high computational overhead from full-model parameter clustering, insufficient characterization of layer-wise difference under monolithic whole-model clustering and aggregation, and inadequate information utilization caused by hard clustering. To address these issues, we proposed a new layer-wise fuzzy clustered FL paradigm that uses the scaling factors of batch normalization layers as local data representations. Low-dimensional BN scaling factors were used to replace full-model parameters for lightweight clustering. Then, we introduced a layer-wise independent clustering and aggregation mechanism that performs client clustering and model aggregation separately for different network layers to better capture layer-specific heterogeneity. Finally, fuzzy C-means (FCM) was applied at each layer to enable fuzzy clustering, allowing each client to belong to multiple clusters and enabling membership-weighted aggregation for improved information utilization. Extensive experiments on the MNIST, EMNIST, CIFAR-10, and CIFAR-100 datasets with SIMPLE-CNN and VGG16 demonstrated the effectiveness of the proposed approach over several state-of-the-art clustered FL baselines, with up to 7.10% gains in test accuracy over the strongest baseline. Further ablation studies showed that both the proposed fuzzy clustering mechanism and layer-wise aggregation strategy provide significant benefits, and their combination achieved the best overall performance.