Search Results (693)

This study develops and analyzes a four-state nonlinear policy–feedback dynamical system that couples a system stressor, an accumulated burden, a signed mitigation–response variable, and a signed policy-pressure variable. The proposed model represents governance response through a smooth threshold-centered feedback mechanism, in which the policy-pressure dynamics depend continuously on the deviation of the stressor from a prescribed reference threshold. Unlike reduced-order formulations with purely exogenous interventions, the present framework generates endogenous interactions among stress accumulation, burden evolution, mitigation response, and policy adjustment. The qualitative analysis establishes local well-posedness in the admissible phase domain, conditional nonnegativity of the accumulated burden, and boundedness of trajectories on admissible intervals. An autonomous effective system is then derived to characterize quasi-stationary mean behavior of the periodically forced dynamics. For this effective system, local stability is investigated using Gershgorin estimates and Routh–Hurwitz criteria, leading to explicit analytical conditions for local asymptotic stability and a critical policy-responsiveness threshold associated with possible Hopf-type oscillatory transitions. The analysis highlights the stabilizing role of mitigation damping and cubic saturation in regulating the feedback loop. To approximate the nonlinear system, a Physics-Informed Neural Network (PINN) surrogate is constructed by embedding the governing equations into a differentiable residual loss while enforcing the initial conditions analytically. The accumulated burden is represented through an admissible neural-network ansatz to preserve the well-definedness of the logarithmic coupling term, while the mitigation–response and policy-pressure variables remain signed in accordance with the model formulation. Numerical validation against reference ode45 solutions across two governance regimes shows maximum absolute errors of order ${10}^{-3}$, indicating that the PINN provides a reliable differentiable surrogate for the coupled policy–feedback dynamics. The resulting framework offers a foundation for future inverse modeling, parameter estimation, and data-assimilation studies involving policy responsiveness, intervention thresholds, and burden- suppression effects. Full article

A global, uniform two-variable analytic approximation for the modified Bessel function $I_{\nu }\left(x\right)$ is presented, valid for all real x and for orders $-1/2\le \nu \le 1/4$. The approximation is constructed using a two-variable multipoint quasi-rational approximation (MPQA) approach, in which the argument x and the order $\nu$ are treated simultaneously as independent variables. The method consistently incorporates the power-series expansion at small arguments and the asymptotic behavior at large arguments, leading to an explicit analytic representation that preserves the correct limiting behaviors. The resulting approximation remains suitable for analytical differentiation and integration, while all parameters are obtained from linear equations, avoiding numerical fitting procedures. A numerical analysis over the entire domain considered shows excellent agreement with the exact function. The largest relative error observed is ${\epsilon }_r=0.0213$, occurring at $\nu =-0.34$ and $x=2.56$. These results indicate that the proposed approximation provides an accurate and efficient analytic representation of $I_{\nu }\left(x\right)$ throughout the investigated domain. Full article

Dixit et al. proposed an asymptotic drag scaling method for zero-pressure-gradient flat-plate turbulent boundary layers based on the approximation $M\sim U_{\tau }^2\delta$, where M is the kinematic momentum rate through the boundary layer, $U_{\tau }$ is the friction velocity, and $\delta$ is the boundary-layer thickness. In the present paper, an explicit Reynolds-number-dependent correction to this approximation is derived from the logarithmic mean-velocity profile. Integration of the log law across the layer yields $M\sim U_{\tau }^2\delta f\left(R{e}_{\tau }\right)$, where $R{e}_{\tau }=\delta U_{\tau }/\nu$ is the friction Reynolds number and $f\left(R{e}_{\tau }\right)$ is given analytically. Application of the correction to the dataset compiled by Dixit et al. shows that the corrected scaling gives an exponent consistent with the asymptotic value $-1/2$ within bootstrap confidence intervals, whereas the uncorrected formulation does not. The correction should be viewed as a leading-order amendment, since the derivation uses the logarithmic law outside its strict range of validity. Full article

This paper investigates explicit coefficient-based estimates for a class of fractional Hermite functions defined through finite power series with Gamma-function coefficients. These functions may be viewed as a fractional Hermite-type family associated with the Caputo fractional derivative of order $\alpha \in (0,1]$. An explicit representation of the fractional derivative is obtained as a finite sum of monomials with computable Gamma coefficients. This representation is used to derive a preliminary uniform estimate on bounded intervals $[0,R]$ with an explicit constant depending on $\alpha$, n, and R. Consistency with the integer-order setting is established by showing that, when $\alpha =1$, the construction reduces to a Hermite-type polynomial family and the Caputo derivative coincides with the ordinary derivative. Explicit asymptotic formulas are obtained for the associated coefficient envelope as $R\to 0^{+}$ and $R\to \infty$. Numerical experiments up to degree $n=7$ show that the ratio between the coefficient envelope and the computed supremum norm remains below approximately $1.45$ for the tested parameter range. In addition, a weighted $L^2$ estimate is derived with respect to a fractional Gaussian-type weight, yielding an explicit coefficient-based bound. The estimates obtained in this work are preliminary in nature, being based on coefficient-wise majorization, and are not claimed to be optimal. Determining sharp constants and establishing genuine norm-comparison inequalities remain open problems. The results presented here provide a rigorous starting point for the study of explicit coefficient-based estimates for fractional Hermite functions and suggest several directions for future research in fractional approximation theory. Full article

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

This paper develops a boundary-sensitive asymptotic theory for nonparametric conditional U-statistics smoothed by support-adapted asymmetric kernels when the response variable is subject to Missing-at-Random observation. The problem lies at the intersection of three well-established but traditionally separate lines of research: conditional U-statistics, asymmetric smoothing on constrained supports, and incomplete-data inference under MAR sampling. The contribution of the paper is not a novelty claim concerning any of these components in isolation. Rather, it consists in deriving a kernel-specific and MAR-aware limit theory for their simultaneous occurrence, where the estimators are nonlinear complete-case ratios of localized U-statistics and the localization devices are point-dependent approximate identities adapted to the geometry of the covariate support. The analysis covers three principal classes of support-respecting smoothers: Dirichlet kernels on the simplex, Bernstein polynomial smoothers, and multivariate beta kernels on hypercubes, with an additional extension to mixed continuous–categorical regressors. These smoothing schemes are not translation-invariant, and their local moments, effective support, normalizing constants and $L^2$-masses vary with the evaluation point, especially near the boundary. Consequently, their incorporation into conditional U-statistics requires more than a direct transfer of ordinary asymmetric-kernel regression theory. The numerator and denominator of the estimators are localized U-statistics whose stochastic expansions are governed by Hoeffding projections, including canonical components that must be controlled uniformly over the conditioning domain. Under regularity, smoothness and positivity assumptions adapted to the MAR setting, we establish uniform consistency, weak and strong uniform convergence rates, stochastic expansions and asymptotic normality. The results are obtained both on fixed compact subsets and on interior regions approaching the boundary, thereby identifying how support geometry enters the bias and stochastic normalizations. A central feature of the theory is the separation between the deterministic effect of complete-case sampling and its stochastic effect. For the complete-case estimator, the natural deterministic equivalent is obtained by replacing the design density f with the effective complete-case density $pf$, where p is the propensity score. Thus, the MAR mechanism may enter higher-order deterministic bias constants through the local design tilt, whereas the leading stochastic dispersion reflects the loss of effective information through propensity score factors. The precise variance constants and normalizing rates remain kernel-specific, depending on the local $L^2$-structure of the Dirichlet, Bernstein or beta smoothing device. The paper should therefore be viewed as a MAR extension and refinement of the complete-data asymmetric-kernel conditional U-statistic theory. It provides a common probabilistic architecture for several boundary-adapted smoothing schemes while retaining the kernel-dependent bias operators, variance constants, boundary regimes and Hoeffding-projection structures required for sharp asymptotic interpretation. Numerical experiments illustrate the finite-sample behavior predicted by the theory and highlight the interaction between support-adapted smoothing, boundary effects and incomplete response observation. Full article

This paper investigates a discrete diffusion carbon emission-absorption model with periodic boundary conditions derived via the piecewise constant argument scheme. The existence of equilibrium points is established, and sufficient conditions for the local asymptotic stability of the positive equilibrium are derived through eigenvalue analysis. Then, uniform boundedness of positive solutions is proved, and the global asymptotic stability of the interior equilibrium is established by an iterative method and the comparison principle for difference equations. Furthermore, the model is shown to undergo a flip bifurcation when a critical parameter threshold is reached, leading to period-doubling dynamics and chaotic behavior. The influence of spatial diffusion is examined through a Turing instability analysis, yielding conditions for diffusion-driven instability and spatial pattern formation. Finally, Decision Tree and Random Forest classifiers are used as proof-of-concept tools to efficiently approximate the analytically derived stability regions using Monte Carlo-generated data. Both classifiers successfully reproduce the analytical stability structure, while the Random Forest classifier provides higher accuracy and smoother stability boundaries. Numerical simulations support the theoretical results and illustrate the stability and bifurcation phenomena exhibited by the model. These findings indicate that the proposed framework is useful for analyzing carbon emission-absorption dynamics and that machine learning can serve as an efficient computational surrogate for identifying stability regions in nonlinear dynamical systems. Full article

This study develops an analytical and computational framework for coupled transient gas redistribution induced by simultaneous localized injection and withdrawal in transmission pipelines. The aim is to describe source–sink interactions within a single transmission system, unlike conventional approaches that treat inflow and outflow processes independently. The governing equations of one-dimensional non-stationary isothermal compressible gas flow are transformed into a diffusion-type formulation using Charny regularization. The pipeline is divided into three interacting regions connected through pressure-continuity and mass-flux coupling conditions. Closed-form Laplace-domain solutions are derived for the dimensionless pressure field, and a practical Laplace-domain approximation is used for computational evaluation of transient pressure profiles. The results reveal a characteristic balancing point separating injection-dominated and withdrawal-dominated regions and show rapid convergence toward a quasi-steady redistribution regime. A pressure-deviation-based objective function is introduced to evaluate hydraulic disturbance, and the optimization analysis shows that the minimum disturbance occurs under a near-balanced source–sink operating condition. The obtained pressure profiles, asymptotic behavior, and regional redistribution patterns confirm the physical consistency of the proposed model. The framework provides a mathematically interpretable basis for analyzing coupled redistribution dynamics, hydraulic stabilization, and asymptotic equilibrium in gas transmission systems. Full article

East Coast Fever (ECF) causes approximately one million livestock deaths annually in sub-Saharan Africa, posing a significant threat to livestock. The wildlife–livestock interface complicates disease management, as wildlife serve as reservoirs. This study developed a Continuous Time Markov Chain (CTMC) model incorporating the wildlife–livestock interface to analyze ECF dynamics. Using the Galton–Watson approximation, we assessed the probability of disease extinction following the introduction of infected hosts or vectors. The probability of disease extinction calculated from the branching process is shown to be in good agreement with the probability approximated from numerical simulations. The disease dynamics of the deterministic model and the CTMC model are compared to ascertain the effect of demographic stochasticity on ECF dynamics. Differences in model predictions and asymptotic dynamics between stochastic and deterministic models were evident. The deterministic and stochastic formulations should therefore be viewed as complementary modeling frameworks, with the deterministic model characterizing average epidemic dynamics and the CTMC model capturing the probabilistic variability and extinction behavior inherent in real transmission processes. These differences are crucial for intervention strategies earmarked to prevent outbreaks. Our analysis revealed a high probability of ECF extinction if the disease emerges from recovered carrier cattle. Finite time to ECF disease extinction is estimated using 10,000 sample paths, and it is shown that the epidemic duration is shortest if the disease is introduced by infectious cattle. The epidemic duration is longest when the disease is introduced by infectious ticks. Additionally, we observed that host interactions at the wildlife–livestock interface play a critical role in shaping ECF transmission and informing control strategies. Full article

This paper addresses the estimation of the multi-component stress–strength reliability when both the strength variables and the stress variable follow the one-parameter Garhy distribution. Data are assumed to arise from a unified hybrid censoring scheme, which generalizes both Type-I and Type-II hybrid censoring. A closed-form expression for the reliability parameter $R_{m,k}=P(\mathrm{at}\mathrm{least}m\mathrm{of}(X_1,\dots ,X_k)>Y)$ is derived, enabling efficient computation. Three estimation procedures are developed: maximum likelihood estimation (MLE), Bayesian inference using Markov chain Monte Carlo (MCMC) with non-informative priors, and the Tierney–Kadane Laplace-type approximation for posterior moments. For each method, we provide complete mathematical derivations, including the likelihood function under unified hybrid censoring, the posterior conditionals, and the asymptotic distribution of the reliability via the Delta method. Furthermore, Bayesian estimation is extended to asymmetric loss functions, and posterior propriety is formally proven. To check the suitability of the proposed methods, a real data application on generator failure times in power systems is presented. Full article

Existing ground-slotted coplanar waveguide (CPW) spoof surface plasmon polariton (SSPP) low-pass filters (LPFs) remain constrained by the difficulty of achieving a wide stopband while maintaining a compact size, as well as by undesired radiation leakage arising from their open-aperture slot configuration. To address these issues, a grounded coplanar waveguide spoof surface plasmon polariton (GCPW-SSPP) low-pass filter based on a multi-arm star-shaped slot (MASS) loading topology is proposed. An equivalent-circuit interpretation and full-wave dispersion analysis show that the multi-arm slots introduce enhanced distributed reactive loading, thereby lowering the asymptotic frequency and enabling compact SSPP implementations. The near-field characteristics further demonstrate tighter electromagnetic confinement, as reflected by an approximately 48% reduction in the electric-field confinement width along the z-direction. To alleviate the trade-off between miniaturization and wide-stopband performance in cascaded SSPP LPFs, the single-cell S-parameters of the proposed topology are investigated. A single MASS unit exhibits a sharp cutoff and a deep transmission notch, allowing a wide stopband to be obtained with fewer cascaded cells. Radiation characteristics are subsequently quantified by a loss-decomposition method, and the MASS topology is found to suppress the radiation leakage of open-aperture ground-slotted structures, yielding a maximum radiation-loss reduction of approximately 75%. To validate the design methodology, a MASS-loaded GCPW-SSPP LPF is designed, fabricated, and measured. The measured results are in good agreement with the simulated ones, confirming the effectiveness of the proposed scheme. By simultaneously achieving a wide stopband, compact size, and suppressed radiation leakage, the proposed filter offers a promising low-interference filtering solution for highly integrated microwave and RF front-end systems. Full article

In this paper, we introduce and study a Szász-type family of positive linear operators generated by Euler polynomials of negative order on $[0,\infty )$. The construction is based on an explicit finite representation of these polynomials with non-negative terms, which ensures the positivity of the corresponding kernel. We prove the basic properties of the operators and show that they can be represented as finite convex combinations of shifted classical Szász operators. We also provide a probabilistic representation of the kernel as a finite mixture of Poisson distributions, which clarifies the role of the parameter k and the resulting moment structure. The corresponding algebraic and central moment identities are derived and used to establish convergence on compact intervals and to obtain quantitative estimates in terms of the modulus of continuity, Lipschitz-type classes, and Peetre’s K-functional. Furthermore, Voronovskaya-type asymptotic results are obtained, including a quantitative form and a second-order asymptotic formula. Numerical tables and a graphical illustration are presented for selected test functions and parameter values, and the results are consistent with the theoretical convergence behaviour. The paper shows that Euler polynomials of negative order provide a positive and structurally tractable framework for constructing Szász-type approximation operators on the positive real axis. Full article

This study presents a new version of the ZLindly (ZL) model that improves modeling flexibility while maintaining ease of analysis, allowing for the simultaneous accommodation of redundant zeros, thick-tailed behavior, and complex failure rate dynamics within a unified probabilistic framework. Marshall–Olkin (MO) theory facilitates this advancement. The MOZL hazard rate can exhibit several patterns, including increasing, decreasing, bathtub, or upside-down bathtub-shaped. These features enable the model to capture diverse reliability phenomena such as early-life failures, random shocks, and wear-out effects. Comprehensive theoretical investigations were conducted and shown to be governed by an interpretable dual-parameter mechanism, where the Marshall–Olkin parameter controls tail behavior and dispersion, while the scale parameter regulates skewness and hazard evolution. A likelihood-based approach was developed under Type-II censoring conditions, and rigorous evidence is provided for the existence and uniqueness. To address inferential uncertainty, both classical asymptotic confidence intervals and log-normal approximations were constructed. Within a Bayesian framework, independent gamma priors were assumed, and posterior inference was performed via an efficient Metropolis–Hastings algorithm. Bayesian point and credible estimators were obtained and compared with their classical counterparts. An extensive simulation study demonstrates that Bayesian estimators, particularly with informative priors, consistently outperform likelihood-based estimators in terms of bias, mean squared error, interval length, and coverage probability, especially for moderate sample sizes and higher censoring levels. Three engineering applications are provided to assess the practical utility of the MOZL model, where it provides superior goodness-of-fit relative to 15 competing models, including MO–Exponential, MO–Gompertz, MO–Nadarajah–Haghighi, MO–Exponentiated Weibull, and Birnbaum–Saunders, among others. Overall, the proposed MOZL distribution emerges as a flexible, interpretable, and computationally efficient lifetime model whose structurally meaningful parameter interactions enhance distributional balance and flexible hazard behavior, thereby contributing to modern symmetry-oriented distribution theory while offering valuable applications in reliability engineering, survival analysis, and applied statistical modeling. Full article

Imprecise probability models generated from data represent epistemic uncertainty by replacing the precise empirical distribution with a set of compatible probability distributions. When this set is described by reachable probability intervals, the induced bounds are tight, so the represented imprecision is not inflated by unattainable interval limits. This paper studies the informational effect of this replacement through the epistemic entropy gap, defined as the difference between the maximum entropy over the induced credal set and the Shannon entropy of the empirical distribution. The gap is a differential quantity: it measures the additional uncertainty introduced by the imprecise model beyond the observed frequencies. We analyze it for three reachable interval models generated from multinomial data: the Imprecise Dirichlet Model, the $ϵ$-contamination model and the approximated Non-Parametric Predictive Inference model. The analysis covers its main properties, its asymptotic behavior and its role in entropy equivalent calibration of model parameters. The results show that the entropy gap offers a common informational scale for comparing how different imprecise models represent the same empirical evidence, and helps interpret the degree of caution associated with limited data reliability and with empirical distributions that may otherwise lead to overconfident uncertainty assessments. Full article

This paper designs a novel aperiodic intermittent control (AIC) strategy using discrete-time observation information. It can stabilize unstable hybrid McKean–Vlasov stochastic differential equations and reduce control consumption effectively. Key contributions include the following: (1) Lévy noise is introduced into the hybrid McKean–Vlasov framework to describe discontinuous disturbances. We further derive the existence, uniqueness and generalized Itô formula for the above system. (2) A new distribution-dependent Lyapunov functional to prove moment finiteness, mean square, and asymptotic exponential stability is constructed. (3) We derive explicit ranges for the AIC time rate and observation intervals. By tightening the state error bound via an innovative technique, the control design constraints are effectively relaxed. (4) We prove the equivalence of exponential stability between the controlled system and its particle approximation. This approach avoids the computational intractability of the exact probability distribution. Finally, the efficacy of our method is demonstrated through a numerical example. Full article