Symmetry

This paper studies a generalized class of linear operators acting on spaces of analytic functions, defined by , where and . This formulation encompasses several classical operators, including composition, weighted composition, differentiation–composition, and the Stević–Sharma operator. We focus on the action of $P_n^{\psi ,\phi }$ from $BMOA$ and analytic Besov spaces ${\mathcal{B}}_p$ into the Bloch space $\mathcal{B}$, and provide necessary and sufficient conditions for boundedness and compactness. These results unify and extend many previously known characterizations and demonstrate the flexibility of the $P_n^{\psi ,\phi }$ framework in the context of analytic operator theory.

The flow of a two-dimensional incompressible hybrid nanofluid over a stretching cylinder containing microorganisms with parallel effect of inclined magnetohydrodynamic was examined in the current study in relation to chemical reactions, heat source effect, nonlinear heat radiation, and multiple convective boundaries. The main objective of this research is the optimization of heat transfer with inclined MHD and variation in different physical parameters. The governing partial differential equations are transformed into a set of ordinary differential equations by applying the appropriate similarity transformations. The Runge–Kutta method is recognized for using shooting as a technique. Surface plots, graphs, and tables have been used to illustrate how various parameters affect the local Nusselt number, mass transfer, and heat transmission. It is demonstrated that when the chemical reaction parameter rises, the concentration and motile concentration profiles drop. The least responsive is the rate of heat transfer to changes in the inclined magnetic field and most associated with changes in the Biot number and radiation parameter shown in contour plot. The streamline graph illustrates the way fluid flow is affected simultaneously by the magnetic parameter M and an angled magnetic field. Local Nusselt number and local skin friction are improved by the curvature parameter and mixed convection parameter. The contours highlight the intricate interactions between restricted magnetic field, significant radiation, and substantial convective condition factors by displaying the best heat transfer. The three-dimensional surface, scattered graph, pie chart, and residual plotting demonstrate the statistical analysis of the heat transfer. The results support their use in sophisticated energy, healthcare, and industrial systems and enhance our theoretical knowledge of hybrid nanofluid dynamics.

Chronic kidney disease (CKD) impacts over 850 million people globally, representing a critical public health issue, yet existing risk assessment methodologies inadequately address the complexity of disease progression trajectories. Traditional machine learning approaches encounter critical limitations including inefficient hyperparameter selection and lack of clinical transparency, hindering their deployment in healthcare settings. This study introduces an innovative computational framework that integrates adaptive Multi-Armed Bandit (MAB) strategies with BorderlineSMOTE sampling techniques to improve CKD risk assessment. The proposed methodology leverages XGBoost within an ensemble learning paradigm enhanced by Upper Confidence Bound exploration strategy, coupled with a comprehensive interpretability system incorporating SHAP and LIME analytical tools to ensure model transparency. To address the challenge of algorithmic interpretability while maintaining clinical utility, a four-level risk categorization framework was developed, employing cross-validated stratification methods and balanced performance evaluation metrics, thereby ensuring fair predictive accuracy across diverse patient populations and minimizing bias toward dominant risk categories. Through rigorous empirical evaluation on clinical datasets, we performed extensive comparative analysis against sixteen established algorithms using paired statistical testing with Bonferroni correction. The MAB-optimized framework achieved superior predictive performance with accuracy of 91.8%, F1-score of 91.0%, and ROC-AUC of 97.8%, demonstrating superior performance within the evaluated cohort of reference algorithms (p-value < 0.001). Remarkably, our optimized framework delivered nearly ten-fold computational efficiency gains relative to conventional grid search methods while preserving robust classification performance. Feature importance analysis identified albumin-to-creatinine ratio, eGFR measurements, and CKD staging as dominant prognostic factors, demonstrating concordance with established clinical nephrology practice. This research addresses three core limitations in healthcare artificial intelligence: optimization computational cost, model interpretability, and consistent performance across heterogeneous clinical populations, offering a practical solution for improved CKD risk stratification in clinical practice.