Search Results (99)

This paper develops a unified asymptotic theory for conditional single-index U-statistics and the associated conditional U-processes in the setting of locally stationary functional time series subject to missing-at-random response mechanisms. The proposed framework addresses, within a single nonparametric inferential architecture, three major sources of complexity in modern functional data analysis: infinite-dimensional covariates, smoothly time-varying stochastic dynamics, and incomplete response observations. The methodology is based on a class of kernel-type estimators combining temporal localization, functional single-index smoothing, and inverse-propensity correction. Temporal localization captures the gradual evolution of the underlying regression structure, the single-index projection provides an effective dimension-reduction mechanism for functional covariates, and the propensity adjustment restores the target conditional functional under the MAR sampling scheme. The principal contribution of the paper is the establishment of weak convergence, in a suitable space of bounded functions, for the resulting propensity-adjusted conditional U-process indexed by a general class of measurable kernels. Under absolute regularity conditions, local stationarity assumptions, small-ball probability requirements, entropy restrictions of VC type, and uniform consistency of the propensity-score estimator, the normalized process is shown to converge weakly to a tight centered Gaussian process. The limiting covariance structure explicitly reflects the interaction between temporal smoothing, functional concentration, dependence, and the random loss of responses. In parallel, uniform convergence rates are derived for the associated conditional single-index U-statistic estimators, thereby quantifying the respective contributions of smoothing bias, stochastic fluctuation, local-stationarity approximation error, and missingness-induced variance inflation. A substantial part of the analysis is devoted to the technical difficulties created by the simultaneous presence of dependence, nonstationarity, functional covariates, and incomplete observations. The proofs combine Hoeffding-type decompositions adapted to weighted incomplete data, blocking and coupling arguments for absolutely regular triangular arrays, refined entropy bounds for kernel-indexed function classes, and small-ball probability techniques for functional covariates. The MAR mechanism is incorporated via inverse-propensity weighting, and its effects on the effective sample size, asymptotic variance, and bias structure are made explicit. The theory also provides a rigorous foundation for bandwidth selection through blocked, propensity-adjusted cross-validation and clarifies its relation to the corresponding oracle risk. The proposed framework encompasses a broad class of statistical learning and inference problems involving pairwise or higher-order functionals of functional time series. In particular, it applies to conditional Kendall-type functionals, discrimination problems, metric learning with incomplete labels, and conditional independence testing under local stationarity. A simulation study illustrates the finite-sample behavior of the proposed estimators and supports the theoretical findings across varying regimes of temporal nonstationarity, serial dependence, functional concentration, and response missingness. Overall, the results provide a mathematically rigorous and methodologically flexible foundation for inference from evolving functional data when dependence, infinite dimensionality, and incomplete observation are present simultaneously. Full article

We analyze the standard multi-particle particle swarm optimization (PSO) algorithm with global-best (all-to-all) topology and constant hyperparameters on smooth strongly convex objectives. By rewriting the PSO velocity recursion as a stochastic heavy-ball method acting on a time-varying quadratic surrogate defined by the personal and global bests, and by applying a Lyapunov drift argument in the style of stochastic momentum analyses, we obtain mean-square convergence of particle positions to the unique minimizer and convergence of the best-so-far objective gaps. The deterministic PSO obtained by fixing the random coefficients at their mean values appears as a noise-free special case of the same Lyapunov framework. Full article

Geopolymer gel concrete (GPC) is a kind of environmentally friendly concrete, which has become a potential alternative material to replace ordinary concrete. Traditional mix design of GPC is carried out under experimental conditions, which is time-consuming and labor-intensive. Geopolymer concrete (GPC) is intended for use in hydraulic structures, which are often exposed to water environments. Water flow exerts significant abrasion and erosion on these structures. If the abrasion resistance (AR) of the material is poor, the service life and service quality of hydraulic structures will be substantially reduced under the action of water flow. Therefore, AR is a key performance indicator for GPC in hydraulic engineering applications. This abrasion resistance can be enhanced by using fibers (for example, steel fibers, polyvinyl alcohol (PVA) fibers, and basalt fibers) and nanomaterials. Furthermore, there is a complex nonlinear relationship between the proportions of fibers and nanoparticles added and the properties of GPC. In this study, the circular ring test method and the underwater steel ball test method were conducted to investigate the AR of nano-SiO₂ (NS) and hybrid fiber (NHF) reinforced geopolymer gel concrete (NHF-GPC). A backpropagation (BP) neural network (BPNN) model optimized by the Grey Wolf Optimizer (GWO) (GWO-BPNN) is established to predict the abrasion resistance strength (ARS) and the abrasion rate of NHF-GPC based on the circular ring test method. In addition, the ARS, abrasion rate, and average abrasion depth (AAD) based on the underwater steel ball test method were also predicted. The results indicate that the GWO-BPNN model demonstrates superior performance over the standard BPNN, exhibiting higher prediction accuracy, better fitting performance, and faster convergence speed. Specifically, for the circular ring test method abrasion rate prediction, GWO-BPNN reduced the root mean square error (RMSE) by 30.3% and lowered the mean absolute percentage error (MAPE) to 8.4%. The GWO-BPNN model established in this study can provide efficient and reliable theoretical support for the optimization of the NHF-GPC mix design. Full article

We develop an operational, measurement-first framework for the geometry of locally finite cell complexes, in which length is defined as a count of face crossings, and curvature is read off from the discrepancy between a measured radius and a radius reconstructed from boundary, area, or volume counts using the same yardstick. We prove that the count metric is geodesic on every locally finite complex, and we introduce a unified small-ball/small-sphere curvature estimator that is valid in dimensions two through four with a single closed-form expression. By comparison with the standard small-ball volume expansion of a smooth conformal metric $g=e^{2u}{g}_0$, we establish a quantitative identification theorem with explicit rate $O(a/r+r)$, which optimizes to $O\left(\sqrt{a}\right)$ at $r\asymp \sqrt{a}$. We extend the construction to directional (sectional) estimators via Fermi tubes around geodesic two-slices, assemble the curvature operator, Ricci tensor, and scalar curvature in three dimensions, and prove a measured Gromov–Hausdorff convergence theorem for the rescaled count metric. All hypotheses are verified explicitly on Voronoi complexes of conformal metrics. Throughout, we are explicit that the discrete construction is interpreted via, and its asymptotic validity is established by comparison with, the smooth Riemannian theory; the contribution is the unified counts-only protocol with rigorous convergence rates, not a reformulation of curvature itself. Full article

Quantiles are among the most fundamental constructs in probability theory and statistics, intrinsically linked to order structures, stochastic dominance, and the principles of robust statistical inference. Although the univariate theory of quantiles is by now classical and well developed, their generalization to multivariate settings remains mathematically subtle and methodologically demanding. In particular, extending the notion of “location within a distribution” beyond one dimension raises delicate questions of geometry, ordering, and equivariance. Within this landscape, the spatial—or geometric—formulation of multivariate quantiles has emerged as a rigorous and conceptually unifying framework capable of reconciling these issues. In this work we advance this paradigm by introducing a kernel-based estimation procedure for nonparametric conditional geometric quantiles of a multivariate response $Y\in {\mathbb{R}}^q$ ($q\ge 2$) given a functional covariate X that takes values in an infinite-dimensional space. The data are assumed to form a strictly stationary and ergodic process, while the responses may be subject to a missing-at-random mechanism, a feature of substantial practical relevance. Our analysis establishes strong consistency of the proposed estimator, characterizes its optimal convergence rate, and derives its asymptotic distribution. These limit theorems, in turn, provide the theoretical foundation for constructing asymptotically valid confidence regions and for performing inference in multivariate quantile regression with functional covariates. The theoretical developments rest on natural complexity conditions for the involved functional classes together with mild smoothness and regularity assumptions. This balance between generality and mathematical precision ensures that the resulting methodology is not only robust in a rigorous probabilistic sense but also widely applicable to contemporary problems in high-dimensional and functional data analysis. The proposed methodology is numerically investigated through simulations and is implemented in a real data application. Full article

The applicability of a highly efficient sixth-order convergent method, originally proposed by Kansal et al., is extended in this study to a Banach space setting. The initial development of this method relied upon Taylor series expansions in ${\mathbb{R}}^n$ and the assumption that the nonlinear operator is sufficiently differentiable. This vague condition implies the existence of high-order derivatives that are not actually utilized by the algorithm. This study transcends these limitations by establishing convergence based solely on generalized continuity conditions of the first Fréchet derivative. By dispensing with these strong smoothness requirements, the domain of applicability is significantly widened. We derive computable radii for the ball of convergence and establish error bounds under local analysis. Furthermore, a rigorous semi-local convergence analysis is presented, a feature previously absent in the literature for this specific scheme, utilizing a majorizing sequence technique to guarantee the existence and uniqueness of the solution. The theoretical results are validated through numerical experiments, which demonstrate that the method converges even when the standard sufficiently differentiable conditions are violated. Full article

This study presents a novel softsign-function-based fractional-order proportional–integral–derivative (softsign-FOPID) controller optimized using the fungal growth optimizer (FGO) for the stabilization and precise position control of an unstable magnetic ball suspension system. The proposed controller introduces a smooth nonlinear softsign function into the conventional FOPID structure to limit abrupt control actions and improve transient smoothness while preserving the flexibility of fractional dynamics. The FGO, a recently developed bio-inspired metaheuristic, is employed to tune the seven controller parameters by minimizing a composite objective function that simultaneously penalizes overshoot and tracking error. This optimization ensures balanced transient and steady-state performance with enhanced convergence reliability. The performance of the proposed approach was extensively benchmarked against four modern metaheuristic algorithms (greater cane rat algorithm, catch fish optimization algorithm, RIME algorithm and artificial hummingbird algorithm) under identical conditions. Statistical analyses, including boxplot comparisons and the nonparametric Wilcoxon rank-sum test, demonstrated that the FGO consistently achieved the lowest objective function value with superior convergence stability and significantly better (p < 0.05) performance across multiple independent runs. In time-domain evaluations, the FGO-tuned softsign-FOPID exhibited the fastest rise time (0.0089 s), shortest settling time (0.0163 s), lowest overshoot (4.13%), and negligible steady-state error (0.0015%), surpassing the best-reported controllers in the literature, including the sine cosine algorithm-tuned PID, logarithmic spiral opposition-based learning augmented hunger games search algorithm-tuned FOPID, and manta ray foraging optimization-tuned real PIDD². Robustness assessments under fluctuating reference trajectories, actuator saturation, sensor noise, external disturbances, and parametric uncertainties (±10% variation in resistance and inductance) further confirmed the controller’s adaptability and stability under practical non-idealities. The smooth nonlinearity of the softsign function effectively prevented control signal saturation, while the fractional-order dynamics enhanced disturbance rejection and memory-based adaptability. Overall, the proposed FGO-optimized softsign-FOPID controller establishes a new benchmark in nonlinear magnetic levitation control by integrating smooth nonlinear mapping, fractional calculus, and adaptive metaheuristic optimization. Full article

Reciprocal recommendation requires satisfying preferences on both sides of a match, which differs from standard one-sided settings and often involves hierarchical structure (e.g., skills, seniority, education). We present biLorentzFM, which is a multi-objective framework that integrates hyperbolic geometry into factorization machine architectures using Lorentz embeddings with learnable curvature and manifold-aware optimization. The approach addresses whether a geometric structure aligned with hierarchical relationships can improve reciprocal matching without requiring major architectural changes. On a large-scale recruitment dataset from Kariyer.Net (1,150,302 interactions, 229,805 candidates), the model achieves candidate and company AUCs of 0.9964 and 0.9913 respectively, representing 6.6% and 6.0% improvements over the strongest Euclidean baseline while maintaining practical inference latency (2.1 ms per batch). Cross-validation analysis confirms robustness (5-fold: 0.9813 ± 0.0002; 3-seed: 0.9964 ± 0.0012) with very large effect sizes (Cohen’s d = 2.89–3.08). Although the per-epoch training time increases by 23.5% due to manifold operations, faster convergence (12 vs. 18 epochs) reduces the total training time by 17.8%. Cross-domain evaluation on Speed Dating data demonstrates generalization beyond explicit hierarchies with a 2.8% AUC improvement despite lacking structured taxonomies. Learned curvature parameters differ by entity type, providing interpretable indicators of hierarchical structure strength. Ablation studies isolate contributions from geometric structure (6.6%), learnable curvature (4.7%), multi-objective learning (2.1%), and explicit feature interactions (0.6%). A systematic comparison reveals that Lorentz embeddings outperform Poincaré ball implementations by 4.4% AUC under identical conditions, which is attributed to numerical stability advantages. The results indicate that pairing standard recommendation architectures with geometry reflecting hierarchical relationships can provide consistent improvements for reciprocal matching, while limitations including cold-start performance, computational overhead at an extreme scale, and static hierarchy assumptions suggest directions for future work on adaptive curvature, fairness constraints, and dynamic taxonomies. Full article

In intuitionistic fuzzy 2-normed spaces, there are numerous symmetries in the topological structures and mapping theorems. In this work, we present the concept of an intuitionistic fuzzy 2-normed space(IF2NS) and demonstrate its structural properties using illustrative examples. This approach unifies and broadens the scope of both classical 2-normed spaces and intuitionistic fuzzy normed spaces when specific conditions are met. We introduce the idea of fuzzy open balls and explore the convergence of sequences with respect to the topology derived from the intuitionistic fuzzy 2-norm. In addition, we define left and right N-Cauchy sequences relative to the topologies ${\tau }_N$ and ${\tau }_{N^{-1}}$ and analyze their convergence characteristics. Special attention is given to the inherent symmetry of the 2-norm, where the magnitude of a pair of vectors remains invariant under exchange of arguments, and to the balanced interaction between membership and non-membership functions in the intuitionistic fuzzy setting. This intrinsic symmetry is further reflected in the proofs of the open mapping and closed graph theorems, which naturally preserve the symmetric structure of the underlying space The paper culminates with the formulation and proof of the open mapping theorem that can be considered for its symmetric properties and the closed graph theorem in the context of IF2NS, thereby generalizing essential theorems of functional analysis to this fuzzy setting. Full article

Sulfur dioxide not only affects the ecological environment and endangers health but also restricts economic development. The reasonable prediction of sulfur dioxide emissions is beneficial for formulating more comprehensive energy use strategies and guiding social policies. To this end, this article uses a multiparameter combination optimization gray prediction model (PGM(1, N)), which not only defines the difference between the sequences represented by variables but also optimizes the order of all variables. To this end, this article proposes an improved algorithm for the Dung Beetle Optimization (DBO) algorithm, namely, CSLDDBO, to optimize two important parameters in the model, namely, the smoothing generation coefficient and the order of the gray generation operators. In order to overcome the shortcomings of DBO, four improvement strategies have been introduced. Firstly, the use of a chain foraging strategy is introduced to guide the ball-rolling beetle to update its position. Secondly, the rolling foraging strategy is adopted to fully conduct adaptive searches in the search space. Then, learning strategies are adopted to improve the global search capabilities. Finally, based on the idea of differential evolution, the convergence speed of the algorithm was improved, and the ability to escape from local optima was enhanced. The superiority of CSLDDBO was verified on the CEC2022 test set. Finally, the optimized PGM(1, N) model was used to predict China’s sulfur dioxide emissions. From the results, it can be seen that the error of the PGM(1, N) model is the smallest at 0.1117%, and the prediction accuracy is significantly higher than that of other prediction models. Full article

Within computer-aided geometric design (CAGD), Said–Ball curves are primarily adopted in domains such as 3D object skeleton modeling, vascular structure repair, and path planning, owing to their flexible geometric properties. Techniques for curve degree reduction seek to reduce computational and storage demands while striving to maintain the essential geometric attributes of the original curve. This study presents a novel degree reduction model leveraging Euclidean distance and curvature data, markedly improving the preservation of geometric features throughout the reduction process. To enhance performance further, we propose a multi-strategy enhanced coati optimization algorithm (MSECOA). This algorithm utilizes a good point set combined with opposition-based learning to refine the initial population distribution, employs a fitness–distance equilibrium approach alongside a dynamic spiral search strategy to harmonize global exploration with local exploitation, and integrates an adaptive differential evolution mechanism to boost convergence rates and robustness. Experimental results demonstrate that the MSECOA outperforms nine highly cited agorithms in terms of convergence performance, solution accuracy, and stability. The algorithm exhibits superior behavior on the IEEE CEC2017 and CEC2022 benchmark functions and demonstrates strong practical utility across four engineering optimization problems with constraints. When applied to multi-degree reduction approximation of Said–Ball curves, the algorithm’s effectiveness is substantiated through four reduction cases, highlighting its superior precision and computational efficiency, thus providing a highly effective and accurate solution for complex curve degree reduction tasks. Full article

Due to the limited difference in maneuverability between the pursuer and the evader in three-dimensional space, it is difficult for a single pursuer to capture the evader. To address this, this paper proposes a strategy where three pursuers intercept one evader and introduces a Q-learning-cover algorithm. Based on the motion models of the pursuers and the evader in three-dimensional space, this paper presents a region coverage scheme based on the Ahlswede ball and analyzes the convergence upper bound of the Q-learning-cover algorithm by designing an appropriate Lyapunov function. Through extensive model training, the successful capture of the evader by the pursuers in a three-on-one scenario was achieved. Finally, numerical simulation experiments and hardware-in-the-loop simulation experiments are presented, both of which demonstrate that the proposed Q-learning-cover algorithm can effectively realize the three-on-one encirclement and interception of the evading target. Full article

The Sharpness-Aware Minimization (SAM) optimizer connects flatness and generalization, suggesting that loss basins with lower sharpness are correlated with better generalization. However, SAM requires manually tuning the open ball radius, which complicates its practical application. To address this, we propose a method inspired by linear connectivity, using two models initialized differently as endpoints to automatically determine the optimal open ball radius. Specifically, we introduce distance regularization between the two models during training, which encourages them to approach each other, thus dynamically adjusting the open ball radius. We design an optimization algorithm called ’Twin Stars Entwined’ (TSE), where the stopping condition is defined by the models’ linear connectivity, i.e., when they converge to a region of sufficiently low distance. As the models iteratively reduce their distance, they converge to a flatter region of the loss landscape. Our approach complements SAM by dynamically identifying flatter regions and exploring the geometric properties of multiple connected loss basins. Instead of searching for a single large-radius basin, we identify a group of connected basins as potential optimization targets. Experiments conducted across multiple models and in varied noise environments showed that our method achieved a performance on par with state-of-the-art techniques. Full article

The main goal of this paper is to study Jarratt-like iterative methods to obtain their order of convergence under weaker conditions. Generally, obtaining the $p^{th}$-order convergence using the Taylor series expansion technique needed at least $p+1$ times differentiability of the involved operator. However, we obtain the fourth- and sixth-order for Jarratt-like methods using up to the third-order derivatives only. An upper bound for the asymptotic error constant (AEC) and a convergence ball are provided. The convergence analysis is developed in the more general setting of Banach spaces and relies on Lipschitz-type conditions, which are required to control the derivative. The results obtained are examined using numerical examples, and some dynamical system concepts are discussed for a better understanding of convergence ideas. Full article

Multistep methods typically use Taylor series to attain their convergence order, which necessitates the existence of derivatives not naturally present in the iterative functions. Other issues are the absence of a priori error estimates, information about the radius of convergence or the uniqueness of the solution. These restrictions impose constraints on the use of such methods, especially since these methods may converge. Consequently, local convergence analysis emerges as a more effective approach, as it relies on criteria involving only the operators of the methods. This expands the applicability of such methods, including in non-Euclidean space scenarios. Furthermore, this work uses majorizing sequences to address the more challenging semi-local convergence analysis, which was not explored in earlier research. We adopted generalized continuity constraints to control the derivatives and obtain sharper error estimates. The sufficient convergence criteria are demonstrated through examples. Full article