Search Results (2,472)

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

MNOs are increasingly adopting 5G Fixed Wireless Access (FWA) to meet household demands for high-performance services. This study evaluated the adoption and quality of experience (QoE) of 5G FWA through a multi-phase study. First, it utilized a systematic literature review to develop a structural equation modeling (SEM) framework. Second, questionnaire surveys from 42 industry experts and 52 end-users were administered to identify quality of service (QoS) and user experience (UX) factors. Finally, the SEM analysis showed that UX was not transferable between FTTx and 5G FWA, as the correlation (y = −0.052, t-value= −0.10) was statistically insignificant. The technical QoS FTTx does not influence how users perceive the technical QoS 5G FWA (y = −0.02, t-value = −0.12). Bandwidth and quality are the most critical drivers for 5G FWA success regarding UX, whereas latency, MoS, and throughput are vital for QoS. Exploratory Factor Analysis (EFA) for the UX and QoS parameters of 5G FWA showed strong internal consistency across all identified factors. The framework with fit indices reflected excellent model QoS (RMSEA = 0.08, CFI = 0.973, TLI = 0.965, CMIN/DF = 1.254 and GFI = 0.782) and UX (RMSEA = 0.08, CFI = 0.895, TLI = 0.881, CMIN/DF = 1.377 and GFI = 0.655). The mathematical SEM model provides empirical evidence of the role of the service factor as an observed parameter and introduces a validated theoretical framework QoE-SEM; this study assists decision-makers in the telecommunications industry in formulating strategic models for upcoming 5G FWA. Full article

Background/Objectives: Traditional protein contact analysis often fails to distinguish between local, sequence-driven motifs and global, tertiary scaffolding, which ensures structural determinism. While deep-learning models do not fully elucidate the ‘why’, they do reveal the underlying directional rules of the stability landscape. In this study, we analyzed 475 non-redundant Protein Data Bank (PDB) structures categorized into SCOP classes (all-α, all-β, α/β, α+β) of the hydrolase superfamily. Methods: To isolate the structural anchors of the global fold, we applied a sequence separation filter of ∣i − j∣ ≥ 6 and a precise spatial cutoff of 3–5 Å between Cα-only to construct asymmetric 20 × 20 frequency matrices, both raw and normalized, then present the former using a violin diagram. We developed a Pearson Correlation (PC) framework to analyze these matrices, providing high correlation when considered as vectors and giving the directionality (N-to-C vs. C-to-N) in protein folding when considered as matrices. Results: Our results reveal a hierarchical organization of tertiary determinism. Initial visualization of Residue–Residue Contact Frequency Matrices (RRCFMs), Z-score normalization (NRRCFM), and violin plots reveal the Universal Structural Grammar (USG) of interaction. Furthermore, a near-perfect PC (r = 0.99) as determined via inter-class Z-score correlation and inter-class PC demonstrates shared statistical interaction laws. In addition, PC Stage 1 (intra-class) analysis identified high symmetry, with around 80% of contacts exhibiting a very strong to strong positive correlations, while PC Stage 2 (inter-class) analysis demonstrated that around 50% of contacts exhibited very strong to strong positive correlations. Finally, we identified universal druggable pockets for drug discovery. Conclusions: This powerful mathematical framework provides a robust analytical tool for structure-based drug design. Full article

Longitudinal data, characterized by repeated measurements on the same subjects over time, are ubiquitous in biomedical sciences, economics, social sciences, and engineering. Analyzing such data presents unique statistical and computational challenges, including within-subject correlation, time-varying covariates, irregular observation times, informative dropout, and high dimensionality. While traditional statistical methods, such as linear mixed-effects models and generalized estimating equations, remain foundational, they often struggle with complex nonlinear dynamics, ultra-high-dimensional feature spaces, and very large sample sizes. Over the past two decades, machine learning (ML) and artificial intelligence (AI) methods have emerged as powerful complementary approaches to address these limitations. This review provides a comprehensive survey of mathematical and computational methods for longitudinal data analysis. We cover classical statistical models, penalized regression techniques, tree-based ensemble methods, kernel machines, Bayesian hierarchical models, and modern deep learning architectures, including recurrent neural networks, temporal convolutional networks, attention-based Transformers, neural ordinary differential equations, and generative models. We propose a unified taxonomy that organizes existing methods along two primary axes: the underlying mathematical framework and the analytical objective. For each category, we present detailed mathematical formulations, discuss key theoretical properties, examine computational considerations, and summarize representative reported applications drawn from the published literature. To increase the practical value of this review, we provide a cross-cutting comparison of method families against five key challenges (within-subject correlation, irregular sampling, missing data, high dimensionality, and scalability) and offer concrete guidance on method selection according to sample size, dimensionality, and analytical objective. Finally, we critically evaluate the strengths and limitations of these approaches, with particular emphasis on interpretability, scalability, handling of missing data, robustness to covariance misspecification, and uncertainty quantification. Full article

Cosmic large-scale structure formation is commonly described in terms of the evolution of density fluctuations and correlation statistics. However, such approaches primarily characterize amplitude variations and do not directly capture the spatial rearrangement of mass distributions. Recent developments based on optimal transport theory have introduced a complementary perspective, in which structure formation is understood as a transport process in the space of probability measures equipped with Wasserstein geometry. In this work, we extend this framework by introducing transport–information geometry, which unifies transport geometry with information geometry. Within this formulation, cosmological states are represented as elements of the product space of probability measures and statistical manifolds, allowing gravitational mass transport and generative deformations associated with galaxy formation to be treated in a unified manner. Using entropic optimal transport, we demonstrate that Wasserstein geometry and Kullback–Leibler-based information geometry are connected within a single mathematical structure, leading to a geometric interpretation of cosmological evolution as a coupled transport–information process endowed with a dual-affine structure. In this picture, gravitational evolution corresponds to generative deformation associated with e-geometry, while observational processes, including finite sampling and survey selection, are described as mixing and projection in m-geometry. This dual-affine cosmology provides a unified framework in which gravitational transport, galaxy bias, observational effects, and nonlinear multi-stream structures are consistently incorporated. The resulting formulation offers a systematic basis for cosmological inference, data analysis, and stochastic descriptions of structure formation. Full article

We present a review on the application of the mathematical models for research on social processes, social structures, and actors in social systems. The scope of the review is not restricted to the classical applications of mathematics such as theory of probability, statistics, stochastic processes, differential equations, and game theory. We also discuss applications of the theory of networks for social network analysis and the numerical research on dynamics of social systems. The number of these applications has increased very fast in recent years. Special attention is given to the results from the area of sociophysics, where mathematical methodology is used to analyze social systems in cooperation with the models and concepts of physics. Another special topic in his review is connected to the results from econophysics, where the mathematical methodology and theories and methods of physics are used in the studies on the dynamics of economic systems. In addition, we give several examples for the application of mathematical methods to social systems: (a) application of difference equations to model the flow of substances in channels of networks; (b) analytical solution of nonlinear equations connected to the model of waves of popularity; (c) numerical results of the waves of popularity in a model that accounts for the change in the opinion of the supporters of the ideas for positive or negative popularity of a person, material item, or a piece of information (idea, theory, ideology, etc.) In the last case, we illustrate the effectiveness of the numerical analysis to discover new effects on the studied social system. The review ends with a large list of references. These references can be used as a guide of the way of new researchers to the large field of mathematical social dynamics. Full article

Fractional differential operators provide an effective approach for modeling complex technological processes, particularly physical phenomena in continuum mechanics characterized by memory and non-local effects. Different types of fractional derivatives require different numerical approximation schemes; in this study, the Caputo and Grünwald–Letnikov derivatives are considered. The aim of this work was to develop and validate a fractional differential model of longitudinal oscillations in a suspended monorail system that accounts for nonlinear and memory-dependent effects. In contrast to classical integer-order approaches, the proposed framework incorporates multiscale surface irregularity effects, including rail roughness, friction, and other disturbances influencing system dynamics, through a fractional-order formulation. A fractional differential mathematical model describing the motion of longitudinal oscillations of a large-sized cargo transported along a suspended monorail is proposed. A numerical algorithm based on finite-difference approximation of fractional operators was developed for its implementation. The scientific contribution lies in integrating multiscale surface irregularity effects into a fractional-order modeling framework to improve the accuracy of dynamic response prediction. Numerical experiments demonstrated the effectiveness of the approach, and the results were validated through comparison with existing models of monorail dynamics. Additionally, statistical validation based on correlation analysis confirmed good agreement with the experimental data. The proposed model can be applied to the design and optimization of suspended transport systems, improving vibration control, reliability, and operational safety under real dynamic loading conditions. Full article

Many methods have been proposed for visualizing and interpreting the results of artificial intelligence (AI) algorithms. AI explainability (XAI) methods vary in mathematical basis, effectiveness, and scope of application. Knowing this, an important question arises: how do their results differ from a statistical point of view, and are some of them more useful than the others in certain scenarios? Our article aims to assess the robustness of the most popular AI models’ explainability visualization methods and to identify differences in the results obtained. We did this by analyzing fundamental convolutional neural network models that classified 598 cat images from the Oxford III-T Pet database and 580 filtered pictures of Boeing planes from the Aircraft Images Dataset. We performed a comparative analysis of the similarities between methods based on Class Activation Mapping (CAM), gradients, and Local Interpretable Model-agnostic Explanations (LIME). To evaluate them, we used Pearson Correlation Coefficient (CC), Matthews Correlation Coefficient (MCC), Spearman’s Rank, Structural Similarity Index Measure (SSIM), Kullback–Leibler divergence, Intersection over Union (IoU), and Soft IoU. To check the fidelity and robustness of the XAI methods, we used RandomCAM and ran an ablation test, checking for a decrease in prediction confidence as we gradually removed the least significant regions. Our results provide an up-to-date and broad comparative analysis of this field. They can serve as a reference point for machine learning scientists and engineers. Full article

This study proposes a condition-based maintenance monitoring method based on Geometry-based Optical Focus Metrology (GOFM) to detect wafer table edge deterioration early and enable proactive interventions before actual Critical Dimension (CD) bridge defects occur. In advanced Deep Ultraviolet (DUV) immersion photolithography, prolonged equipment operation mechanically wears the wafer table, inducing Edge-Roll-Off (ERO). Because conventional optical metrology struggles to separate this localized defocus from process noise, this work utilizes the existing GOFM technique to isolate the pure focus residual within the 140–147 mm radius region. To quantify this hardware-specific degradation, a mathematical dual-indicator system was constructed. This framework integrates a statistical threshold, the Range Percentile 97%, to reject baseline measurement noise, and a geometric variable, Slope × 3, to capture the topographical drop in the outermost 3 mm. Analysis of long-term time-series data from multiple High-Volume Manufacturing (HVM) scanners confirmed a strong correlation ($R^2=0.93$) between these indicators. Furthermore, we proved that the drift trajectory of Slope × 3 deterministically predicts mechanical failure prior to defect occurrence on production wafers. Based on these findings, an automated condition-based maintenance architecture was designed using an OR-logic decision gate. By triggering a preemptive table replacement at a quality-based critical warning threshold, this system converts routine time-based scheduling into a data-driven paradigm, maximizing both edge yield and equipment uptime. Furthermore, this proposed framework establishes a solid foundation for future extensions toward machine learning-based predictive maintenance. Full article

The proliferation of AI-generated content and personalized AI systems has sharpened two fundamental and related computational problems: the progressive erosion of authentic identity in AI-mediated representations, and the growing difficulty of distinguishing human-originated from AI-generated behavioral and textual streams. This paper proposes a rigorous computational framework in which digital identity is formalized as a holomorphic function of complex time T = (a + ib) ∈ ℂ, where the real component Re$\left(T\right)$ encodes chronological progression and the imaginary component Im$\left(T\right)$ spans a continuum from episodic memory ($Im\left(T\right) < 0$) through the present moment ($Im\left(T\right) = 0$) to prospective imagination ($Im\left(T\right) > 0$). We argue that holomorphicity—enforced via Cauchy–Riemann regularization during CTNN learning (Proposition 1)—provides a theoretically grounded encoding of identity coherence, and discuss its advantages over alternative mathematical choices, including Lipschitz continuity, ${\mathit{C}}^{\mathbf{\infty }}$ smoothness, piecewise analytic functions, and stochastic models. Under four explicit Assumptions 1–4 covering the Markovian structure and fixed context window of current LLM architectures, we establish via Lemmas 1 and 2 and Theorem 1 that AI-generated behavioral trajectories exhibit structural limitations in satisfying the Cauchy–Riemann conditions at temporal depths characteristic of human biographical memory—limitations that do not arise for human trajectories learned under CTNN regularization. Building on this result, we introduce the Human–AI Authenticity Discriminant (HAAD), a theoretically grounded classifier with a fully specified calibration algorithm and sensitivity analysis ($\mathit{\kappa }$ ΔAUROC ≤ 0.04 over ±30% perturbation). Five metrics—TCS, ISI, PAS, GAS, and HAAD—are derived analytically from the holomorphic structure. The algorithmic framework is instantiated on four real-world datasets: MovieLens 25M, the Pushshift Reddit corpus, the Stack Overflow Data Dump, and the LIAR dataset. On the LIAR benchmark, TDT-HAAD achieves AUROC = 0.82 (95% CI: [0.79, 0.85]), exceeding a RoBERTa-based LLM detector baseline (AUROC = 0.75, DeLong p < 0.01); an ablation study supports the structural contribution of each component. A credibility harvesting signature is detectable 45.3 ± 12.1 days before standard temporal models reach statistical significance. Full article

Transfer hydroformylation of alkenes with formaldehyde constitutes a green and sustainable route to aldehydes. In this work, the transfer hydroformylation of styrene with formaldehyde was efficiently catalyzed by [Rh(κ²-dppe)₂]⁺ (A), where dppe stands for 1,2-bis(diphenylphosphino)ethane. The reaction was found to be first order with respect to both Rh and substrate concentrations and fractional order with respect to formaldehyde concentration, in line with the behavior previously reported for 1-hexene. DFT was used to investigate the reaction mechanism by using ethene and [Rh(κ²-dpe)₂]⁺ (A), where dpe stands for 1,2-bis(phosphine)ethane, as simplified models of the substrate and catalyst, respectively, and by considering several functionals. The DFT calculations indicate that M06-L provides the most suitable description of the thermodynamic and activation parameters associated with the elementary steps. The combined analysis of kinetic results and the DFT calculations allowed us to propose a detailed catalytic cycle for this reaction, initiated by the reversible oxidative addition of formaldehyde to complex A to afford [Rh(H)(CHO)(κ²-dppe)₂]⁺ (B, ${\mathit{K}}_{\mathbf{1}}$). Coordination of ethene occurs through partial dissociation of one phosphorus atom of the diphosphine ligand, generating [Rh(H)(alkene)(CHO)(κ²-dppe)(κ¹-dppe)]⁺ (I_B, ${\mathit{K}}_{\mathbf{2}}$), followed by the transfer of the hydride to the alkene to give [Rh(alkyl)(CHO)(κ²-dppe)₂]⁺ (C, ${\mathit{k}}_{\mathbf{3}}$), which is considered the rate-determining step of the process. The cycle is completed by reductive elimination of propanal, thereby regenerating A. The overall activation energy calculated by DFT (Ea = 20.0 kcal mol⁻¹) is in good agreement with the experimental values determined for 1-hexene and styrene (20.1 and 22.9 kcal mol⁻¹, respectively). On the basis of these experimental and DFT results, a mathematical kinetic model with the canonical form $r_0={{\mathit{K}}_{\mathbf{1}}{\mathit{K}}_{\mathbf{2}}\mathit{k}}_{\mathbf{3}}{\left[Rh\right]}_o\left[alkene\right]\left[C{H}_{2}O\right]/\left({1+\mathit{K}}_{\mathbf{1}}\left[C{H}_{2}O\right]\right)$ was developed and fitted using a tandem LMFit/Bayesian approach, allowing the values of K₁ and K₂k₃ to be estimated, with comparatively low uncertainty. Overall, this integrated kinetic, computational, and statistical study provides a consistent mechanistic and quantitative framework for understanding the transfer hydroformylation of alkenes with formaldehyde. Full article

Cross-basin tropical cyclone variability may exhibit complex, non-linear dependence structures influenced by large-scale climate modes and potential regime shifts. Reliance on traditional linear correlation measures without accounting for structural changes can therefore lead to misleading interpretations of global storm relationships. This study investigates the regional dependence structures of tropical cyclone counts across six major ocean basins (NA, ENP, WNP, NI, SI, and SP) from 1980 to 2024. We adopt a two-stage analytical framework integrating changepoint detection and copula modeling to address non-stationarity in both marginal distributions and dependence structures. First, we identify a significant structural break in the year 2000 via a penalised likelihood applied jointly to the $d=6$-variate Poisson series, with inter-basin dependence captured by a latent Gaussian process (the construction used by Lund et al. (2025). This is mathematically equivalent to a Gaussian copula with Poisson margins (Genest and Nešlehová (2007)). Then, we apply bivariate copula models separately to the pre- and post-2000 regimes using the randomized probability integral transform with results averaged over 500 replications of the auxiliary uniforms to mitigate randomization noise. The results reveal substantial non-stationarity, most notably a 59% increase in North Atlantic storm frequency and a fundamental reorganization of global dependence structures, while dependence structures evolved from primarily symmetric and weak (dominated by Gaussian and Clayton copulas) to more complex and stronger dependencies (increased Frank and Gumbel copulas). Notably, a statistically significant ($p<0.001$) and strong negative dependence emerged between the Southern Pacific and Northern Indian basins ($\tau =-0.464$) in the recent regime. The inclusion of changepoint detection significantly improves model fit and reveals a fundamental reorganization of global tropical cyclone teleconnections, with enhanced coordination between basins in the contemporary climate regime. Modeling these regimes separately, as opposed to a single stationary period, uncovers a shift towards more complex, tail-dependent copula families (Gumbel, Clayton) in the recent era. These findings have important implications for climate risk assessment, seasonal forecasting, and understanding the impacts of climate change on global storm patterns. The proportion of Gumbel copulas (capturing upper-tail dependence) increased from 7% to 20%, while Gaussian copulas decreased from 53% to 33%, indicating more complex, extreme-value-focused dependencies in the contemporary climate. Due to small sample sizes ($n_1=20$, $n_2=25$), copula and dependence estimates are exploratory, not confirmatory. Interpretations reflect this power constraint, utilizing Benjamini–Hochberg adjustments for significance. Full article

Traditional credit scoring models treat lending decisions as static classification, ignoring the dynamic evolution of borrower risk and long-term profit optimisation. This paper reinterprets credit risk management as a discrete-time stochastic optimal control problem and integrates the Hamilton–Jacobi–Bellman (HJB) framework with deep reinforcement learning. Theoretically, we establish the equivalence between a discrete Markov decision process and the HJB equation, prove the existence and uniqueness of the optimal value function, derive the closed-form Riccati solution under linear-quadratic assumptions, and provide a convergence analysis of neural network value iteration. Empirically, using LendingClub loan data (2016–2018), we implement a PPO-based dynamic credit policy. The proposed model achieves an average reward of 1.6726 and a total reward of 867,613, significantly outperforming static baselines as well as a DQN baseline. Ablation experiments show that replacing the policy network with a linear mapping reduces the average reward by 40.8%, confirming the necessity of nonlinear function approximation. Sensitivity analysis and statistical tests (p < 0.001) confirm the robustness and significance of the gains. This work provides a rigorous mathematical foundation and empirical evidence for shifting credit scoring from static classification to dynamic optimisation. Full article

Modern machine learning systems can achieve remarkable predictive performance. Nevertheless, in several fields, this is not enough to produce acceptable solutions as we need formal guarantees of robustness, fairness, and interpretability. Most existing approaches treat these properties separately or introduce them through external constraints, which makes their interaction difficult to analyze. In this work, we develop a unified variational perspective that incorporates these requirements directly into the learning objective. Concretely, we model learning as the minimization of a composite functional that combines predictive risk, regularization, and additional terms that capture robustness, fairness, and interpretability. This viewpoint allows us to study these properties within a single mathematical framework. Under standard assumptions, we prove the existence of minimizers and show that the resulting solutions are Pareto-optimal for the associated multi-objective problem. We illustrate the framework using examples based on adversarial and distributional robustness, statistical fairness criteria, and a notion of interpretability. The analysis points out the trade-offs that inevitably arise. We also examine statistical aspects of the proposed objective and show that classical generalization guarantees can still be obtained under appropriate conditions. The resulting framework provides a flexible basis for designing reliable learning systems. Full article

This study examines failure in introductory programming courses, commonly known as CS1, in Chilean higher education by combining academic trajectory analysis with early-risk prediction models. We analyzed a cohort of 994 students from a Chilean technical university enrolled during the first academic semester of 2025, with a 46% failure rate, integrating pre-university academic and admission variables (e.g., mathematics and language indicators, as well as baseline diagnostic measures when available), sociodemographic information, and within-semester performance indicators. Group differences were assessed using non-parametric tests, and predictive performance was evaluated under two realistic information-availability scenarios: (i) pre-university variables only and (ii) variables available up to the first major written examination (C1). The results show statistically significant differences between students who passed and those who failed, with indicators of quantitative preparedness and, most notably, C1 performance emerging as the strongest signals of risk. In the pre-university scenario, models achieved acceptable discrimination (AUC ≈ 0.77), whereas incorporating C1 substantially improved discriminative performance (AUC ≈ 0.92) and increased precision in identifying at-risk students while reducing false positives. These findings support a staged institutional strategy: broad, low-cost preventive support before the semester begins, followed by more targeted and intensive interventions after C1, thereby enabling more efficient early-warning systems in high-stakes first-year courses. Full article