This study offers a comprehensive analysis of a modified (1 + 1)-dimensional longitudinal wave equation (LWE) tailored to explain the nonlinear dynamic structure of a magneto-electro-elastic (MEE) circular rod. The model presents the coupled interactions of electrical and magnetic fields and mechanical interactions by including physical characteristics such as stress-strain and mass density relationships, reflecting deeper insight into the propagation of longitudinal waves in MEE material. To exhibit the symmetry structure of the equation, Lie symmetry analysis approach is used, producing symmetry reductions that reduce the original partial differential equation into ordinary differential systems. Beyond the symmetry system, the dynamics of the model is methodically analyzed through different diagnostic tools, such as multistability analysis, bifurcation diagrams, power spectrum plots, heterogeneous recurrence plots, Lyapunov exponent evolution, and strange attractor representations, capturing rich chaotic dynamics and sensitive dependence on distinct initial conditions. Additionally, we create novel exact solutions of the proposing equation using two advanced symbolic techniques namely, the generalized Ansatz method and the Painlevé–Paul method to derive a wide spectrum of wave profiles, including kink, anti-kink, periodic, W-shaped, and mixed-type solutions. The importance of this work lies in its unified analysis of symmetry, chaos, and soliton theory within a physically motivated structure, inproving our understanding of nonlinear wave propagation in magneto-electro-elastic structures and providing valuable tools for modeling complex materials in applied physics and engineering contexts.

This article introduces a new flexible four parameter distribution by convolution of the exponential and Weibull distribution using the odd function transformation, which offers greater flexibility in terms of fit, its called the modified exponential-Weibull (MEW). The MEW model is designed to provide a more accurate description of failure time data resulting from a system with one or more failure modes and is characterized by a hazard rate (HR) that takes the shape of a bathtub due to its complexity. The moments properties, quantile function, and residual life are derived and discussed. We discussed the HR function and several distributional properties of the MEW model, and applied maximum likelihood and Bayesian techniques to estimate its unknown parameters. The Hamiltonian Monte Carlo (HMC) algorithm is employed to simulate the posterior distributions and verify the MEW Bayes estimators. We examined the behavior of the MEW model on two data sets with bathtub-shaped HR and compare it with five other popular bathtub-shaped methodologies. The results indicate that the MEW model provided the best description of the two failure time data sets, suggesting that the proposed model could be a viable candidate for solving various real-life problems.

This study presents a detailed analysis of the performance of the majority score clustering algorithm on three different datasets of anti-microbial evaluation, namely the minimum inhibitory concentration (MIC) of bacteria, and the antifungal activity of chemical compounds against 4 bacteria (E. coli, P. aeruginosa, S. aureus, S. pyogenes) and 2 fungi (C. albicans, As. fumigatus). Clustering is an unsupervised machine learning method used to group chemical compounds based on their similarity. In this paper, we apply the k-means clustering, Gaussian mixture model (GMM), and mixtures of multivariate t distribution to antibacterial activity datasets. To determine the optimal number of clusters and which clustering algorithm performs best, we use a variety of clustering validation indices (CVIs) which include within sum square (to be minimized), connectivity (to be minimized), Silhouette Width (to be maximized), and the Dunn Index (to be maximized). Based on the majority score clustering algorithm, we conclude that the k-means and mixture of multivariate t-distribution methods perform best in terms of the maximum CVIs, while GMM performs best in terms of the minimum CVIs. K-means clustering and mixture of multivariate t-distribution provide 3 optimal clusters for the anti-microbial evaluation of antibacterial activity dataset and 5 optimal clusters for the MIC bacteria dataset. K-means clustering, mixture of multivariate t-distribution, and GMM provide 3 optimal clusters for both the antibacterial and antifungal activity datasets. K-means clustering algorithm performs the best in terms of the majority-based clustering algorithm. This study may be useful for the pharmaceutical industry, chemists, and medical professionals in the future.

In this article, we propose a new transmuted modified power-generalized family of distributions constructed from the transmuted-generated and modified power-generated families. The proposed approach is flexible and provides a tradeoff between the two baseline families. For a prime study, we identify the main characteristics of the new transmuted modified power family, such as the asymptotic results, quantile function, series representation, and the various kinds of moment measures. By using the exponential distribution as the baseline, a new three-parameter lifetime distribution is constructed. The associated probability functions (density and hazard rate) are flexible and have a variety of asymmetric shapes, which make them attractive for statistical purposes. In particular, for the related probability density function, reversed-J, unimodal, and right-skewed shapes are observed. Measures relating to risk theory are also computed, such as the value at risk and the expected shortfall. By using both simulation analysis and the maximum likelihood approach, the estimation of the model parameters is evaluated. The effectiveness of the proposed model is demonstrated by two real-world cases (one in insurance and the other in reliability), and we show that it yields better fits when compared to other extended models connected to the exponential model.

It is agreed by industry experts that manufacturing processes are evaluated using quantitative indicators of units produced from this process. For example, the $C_{py}$ process capability index is usually unknown and therefore estimated based on a sample drawn from the requested process. In this paper, $C_{py}$ process capability index estimates were generated using two iterative methods and a Bayesian method of estimation based on stepwise controlled type II data from the Pareto model. In iterative methods, besides the traditional probability-based estimation, there are other competitive methods, known as bootstrap, which are alternative methods to the common probability method, especially in small samples. In the Bayesian method, we have applied the Gibbs sampling procedure with the help of the significant sampling technique. Moreover, the approximate and highest confidence intervals for the posterior intensity of $C_{py}$ were also obtained. Massive simulation studies have been performed to evaluate the behavior of $C_{py}$. Ultimately, application to real-life data is seen to demonstrate the proposed methodology and its applicability.