A novel lifetime distribution has been defined and examined in this study. The odd Lindley–Pareto (OLiP) distribution is the name we give to the new distribution. The new density function can be written as an odd Lindley-G distribution with Pareto amplification. The moment-generating function and characteristic function, entropy and asymptotic behavior, order statistics and moments, mode, variance, skewness, and kurtosis are some of the aspects of the OLiP distribution that are discovered. Seven non-Bayesian estimation techniques and Bayesian estimation utilizing Markov chain Monte Carlo were compared for performance. Additionally, when the lifetime test is truncated after a predetermined period, single acceptance sampling plans (SASPs) are created for the newly suggested, OLiP distribution. The median lifetime of the OLiP distribution with pre-specified factors is taken as the truncation time. To guarantee that the specific life test is obtained at the defined risk to the user, the minimum sample size is required. For a particular consumer’s risk, the OLiP distribution’s parameters, and the truncation time, numerical results are obtained. The new distribution is illustrated using mortality rates of COVID-19 patients in Canada and vinyl chloride data in (g/L) from ground-water monitoring wells that are located in clean-up-gradient areas.

In this study, we introduce a new class of normalized analytic and bi-univalent functions denoted by ${\mathcal{D}}_{\Sigma }(\delta ,\eta ,\lambda ,t,r)$. These functions are connected to the Bazilevič functions and the $\lambda$-pseudo-starlike functions. We employ Sakaguchi Type Functions and Horadam polynomials in our survey. We establish the Fekete-Szegö inequality for the functions in ${\mathcal{D}}_{\Sigma }(\delta ,\eta ,\lambda ,t,r)$ and derive upper bounds for the initial Taylor–Maclaurin coefficients $|a_2|$ and $|a_3|$. Additionally, we establish connections between our results and previous research papers on this topic.

We introduce a special vector field $\omega$ on a Riemannian manifold $(N^m,g)$, such that the Lie derivative of the metric g with respect to $\omega$ is equal to $\rho Ric$, where $Ric$ is the Ricci tensor of $(N^m,g)$ and $\rho$ is a smooth function on $N^m$. We call this vector field a $\rho$-Ricci vector field. We use the $\rho$-Ricci vector field on a Riemannian manifold $(N^m,g)$ and find two characterizations of the m-sphere $S^m\left(\alpha \right)$. In the first result, we show that an m-dimensional compact and connected Riemannian manifold $(N^m,g)$ with nonzero scalar curvature admits a $\rho$-Ricci vector field $\omega$ such that $\rho$ is a nonconstant function and the integral of $Ric\left(\omega ,\omega \right)$ has a suitable lower bound that is necessary and sufficient for $(N^m,g)$ to be isometric to m-sphere $S^m\left(\alpha \right)$. In the second result, we show that an m-dimensional complete and simply connected Riemannian manifold $(N^m,g)$ of positive scalar curvature admits a $\rho$-Ricci vector field $\omega$ such that $\rho$ is a nontrivial solution of the Fischer–Marsden equation and the squared length of the covariant derivative of $\omega$ has an appropriate upper bound, if and only if $(N^m,g)$ is isometric to m-sphere $S^m\left(\alpha \right)$.

This article explores the interconnections among the single-valued neutrosophic grill, single-valued neutrosophic primal and their stratification, uncovering their fundamental characteristics and correlated findings. By introducing the notion of a single-valued neutrosophic primal, a broader framework including the fuzzy primal and intuitionistic fuzzy primal is established. Additionally, the concept of a single-valued neutrosophic open local function for a single-valued neutrosophic topological space is presented. We introduce an operator based on a single-valued neutrosophic primal, illustrating that the single-valued neutrosophic primal topology is finer than the single-valued neutrosophic topology. Lastly, the concept of single-valued neutrosophic open compatibility between the single-valued neutrosophic primal and single-valued neutrosophic topologies is introduced, along with the establishment of several equivalent conditions related to this notion.

This study presents a novel enhanced exponential class of estimators for population mean under RSS by employing data on an auxiliary variable. The suggested estimators' mean square error (MSE) is calculated approximately at order one. The efficiency conditions that make the suggested enhanced exponential class of estimators superior to the traditional estimators are found. A simulation study using hypothetically drawn normal and exponential populations evaluates the execution of the suggested estimators. The findings demonstrate that the suggested estimators outperform their traditional equivalents. In addition, real data examples are examined to show how the proposed estimators can be implemented in various real life problems.

We continue studying the $\sigma$-Ricci vector field $\mathbf{u}$ on a Riemannian manifold $(N^m,g)$, which is not necessarily closed. A Riemannian manifold with Ricci operator T, a Coddazi-type tensor, is called a T-manifold. In the first result of this paper, we show that a complete and simply connected T-manifold$(N^m,g)$, $m>1$, of positive scalar curvature $\tau$, admits a closed $\sigma$-Ricci vector field $\mathbf{u}$ such that the vector $\mathbf{u}-\nabla \sigma$ is an eigenvector of T with eigenvalue $\tau m^{-1}$, if and only if it is isometric to the m-sphere $S_{\alpha }^m$. In the second result, we show that if a compact and connected T-manifold$(N^m,g)$, $m>2$, admits a $\sigma$-Ricci vector field $\mathbf{u}$ with $\sigma \ne 0$ and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ that has a suitable lower bound, then $(N^m,g)$ is isometric to the m-sphere $S_{\alpha }^m$, and the converse also holds. Finally, we show that a compact and connected Riemannian manifold $(N^m,g)$ admits a $\sigma$-Ricci vector field $\mathbf{u}$ with $\sigma$ as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature $Ric\left(\mathbf{u},\mathbf{u}\right)$ has a lower bound depending on a positive constant $\alpha$, if and only if $(N^m,g)$ is isometric to the m-sphere $S_{\alpha }^m$.