Without having to interview every person, we may draw conclusions about the whole population by estimating the population mean, a fundamental summary statistic that is obtained from a sample. For example, using a sample to estimate the population’s mean income, height, or academic achievement may provide crucial information that supports sociological study, strategic business choices, and government. In order to enhance the estimate of a population mean, this article proposes a ratio-cum-product exponential estimator. This estimator makes use of additional information gathered from two transformed auxiliary variables that have been collected within the setting of simple random sampling without replacement. The suggested estimator’s bias and mean squared error are articulated up to the first degree of approximation. This is done in order to validate accuracy. Additionally, we conduct empirical and thorough simulation studies, demonstrating that the proposed estimators consistently surpasses its competitors.

ABSTRACT This paper investigates a probabilistic extension of the multi-Stirling numbers of the second kind and a ‘poly' version of the probabilistic degenerate Lah-Bell polynomials. The probabilistic multi-Stirling numbers of the second kind are defined with respect to a random variable satisfying specific moment conditions, generalizing previous work on probabilistic Stirling numbers of the second kind. Explicit expressions and related identities are derived, including special cases where the random variable follows a gamma or Bernoulli distribution. Furthermore, probabilistic degenerate poly-Lah-Bell polynomials are introduced and their explicit expressions are obtained.

This study presents a comparative evaluation of three nonlinear state estimation filters, the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), and Particle Filter (PF), for the task of 3D facial landmark tracking. Using a publicly available dataset, we assess each filter's performance under both deterministic (noise-free) and stochastic (noisy) conditions. Metrics such as mean squared error (MSE), convergence rates of state and covariance estimates, and consistency over time are used to quantify tracking performance. Results show that the EKF consistently outperforms the UKF and PF, achieving faster convergence and lower estimation error, particularly in scenarios characterized by mild nonlinearity. Heatmap analyses under varying noise conditions further highlight the EKF's robustness and accuracy, especially in low-noise regimes, while PF performance deteriorates with increased process noise. Our findings suggest that while UKF and PF offer advantages in highly nonlinear or non-Gaussian environments, the EKF provides the best trade-off between computational efficiency and estimation accuracy for the facial tracking task studied in mild nonlinearity.

This study investigates boundary layer stagnation-point flow and heat transfer over an exponentially stretching/shrinking sheet embedded in a porous medium, with emphasis on the effects of suction, blowing, thermal radiation, and heat generation. The governing nonlinear boundary layer equations are numerically solved using the Taylor wavelet method. The results show that suction enhances flow stability by reducing the boundary layer thickness and promoting heat transfer, whereas blowing destabilizes the flow, leading to an increase in boundary layer thickness and a reduction in heat transfer. Thermal radiation raises the temperature within the boundary layer, and heat generation further amplifies this effect, causing additional thickening of the thermal boundary layer. The porous medium introduces resistance, which reduces fluid velocity and intensifies the temperature gradient near the surface. The effectiveness of the Taylor wavelet method in solving nonlinear boundary layer problems is demonstrated. The findings provide valuable insights for optimizing heat and mass transfer in engineering applications such as cooling technologies, material processing, and energy systems.

In this study, a famous Brusselator non-linear model arising in enzymatic and triple collision is considered to analyse its various physical features. These features include positivity, stability, and convergence. To attain these features, a novel positivity-preserving non-standard finite difference (NSFD) operator splitting scheme is proposed for the investigation of a numerical solution of non-linear Brusselator model, which describes the dynamical behaviour of autocatalytic chemical reactions and a well-known minimal mathematical model for chemical oscillations. The results show that the proposed scheme is implicit in nature and efficient in time compared to explicit finite difference (FD) schemes. It also converges towards a true equilibrium point at all step sizes under a certain condition.

The beta regression model (BRM) is a widely applied modeling approach for data bounded within the open interval (0, 1), and it is extensively used in fields such as chemistry, environmental science, medicine, and biology. Parameter estimation in these models is conventionally performed using the maximum likelihood estimator (MLE). However, the MLE is known to be sensitive to multicollinearity among predictors and the presence of outliers, which can bias coefficient estimates, inflate their variance, and increase the mean squared error (MSE), potentially leading to erroneous inferences. To address these issues, this paper introduces new robust biased estimators for BRM that are named robust ridge-type estimators. These estimators are designed to handle the adverse effects of multicollinearity and outliers. We used a theoretical comparison to compare the proposed estimators against the MLE and existing robust ridge estimators. Furthermore, a simulation study was performed under different conditions to evaluate the performance of the proposed estimators. Both the theoretical comparison and simulation results demonstrate the superior performance of the proposed estimators in the presence of multicollinearity and outliers. The practical utility of the methods was further validated through an application to a real-world dataset on breast cancer data. The results confirm that the proposed estimators provide greater reliability and robustness compared to existing methods. These results emphasize the importance of using robust biased estimation techniques to enhance the accuracy and reliability of regression models, especially in empirical research involving multicollinear and outlier data.

ABSTRACT This paper investigates the Painlevé-Kuratowski convergence of solution sets in set optimization problems under perturbations. By introducing the notion of u - E -Strictly quasiconvexity for set-valued mappings, we establish sufficient conditions ensuring Painlevé-Kuratowski convergence results of (weak) E - u -solutions sets when both the improvement set and the feasible set are subject to perturbations. The theoretical results are illustrated with concrete examples to verify their validity and applicability. These findings contribute to the stability analysis of set optimization and provide a theoretical foundation for handling solution sensitivity in variational and equilibrium problems involving set-valued structures.

A numerical solution of the interior Dirichlet problem for the homogeneous conductivity equation is considered. After introducing certain assumptions and discretization of the domain, the boundary value problem for a second-order elliptic equation with variable coefficients is reduced to a set of coupled problems with the Helmholtz-type equation defined on multiple disjoint subdomains. The coefficients in the transmission coupled problems are numerically approximated using the Gauss-Legendre and the trapezoid quadrature rules. The coupled problems are simultaneously solved using the method of fundamental solutions, in which the unknown functions are approximated by linear combinations of fundamental solutions, and the coefficients are determined using the collocation method. The applicability and efficiency of the proposed approach are confirmed by the results of numerical experiments.

ABSTRACT This paper extends the existing two-factor Schwartz model into a comprehensive three-factor framework that incorporates stochastic volatility through the Heston model, enabling more precise pricing of European options on crude oil futures. In this enhanced model, we account for key variables including market volatility and yield changes, which significantly influence futures pricing dynamics. We present an analytical formula for the pricing of futures contracts, facilitating robust calibration against actual market data. To solve the associated partial differential equations for option pricing, we implement the Deep Galerkin Method (DGM), a novel approach leveraging deep neural networks that proves to be efficient and accurate, particularly in high-dimensional settings. The obtained results demonstrate that the DGM method outperforms traditional numerical techniques, offering substantial improvements in both accuracy and computational efficiency.

ABSTRACT In recent years, power grids have undergone rapid integration of converter-based power sources. Modern switching converters are replacing conventional synchronous machines. Owing to their inherent inertial property, virtual inertial (VI)-based virtual synchronous machines (VSMs) are increasingly being used in grid-connected converter control strategies With the availability of high-resolution measurement data from hardware measurements or simulations, the use of data-driven methods (DDM) for modelling system dynamics has become more popular than conventional modelling methods. This study employs a data-driven approach to model the dynamics of grid-connected VSM (GCVSM)-based converters. A simulation of the system’s detail switching model is used to create the DDM model, done by using Optimized Dynamic Mode Decomposition (ODMD) and sparse identification of nonlinear dynamical systems (SINDy). The results are compared with those of the actual switching model, the average nonlinear model, and the linearized model of the GCVSM. The ODMD performed very well, with a goodness of fit of approximately 89%, which is very close to that provided by the nonlinear model, which is approximately 94% for the 50 kW active power setpoint scenario. The data-driven model developed in this study is a reduced-order model that can be useful for understanding the transient and steady-state characteristics of systems and for developing control strategies.