In this article we present an ordinary differential equation based technique to study the quadratic stability of non-linear dynamical systems. The non-linear dynamical systems are modeled with norm bounded linear differential inclusions. The proposed methodology reformulate non-linear differential inclusion to an equivalent non-linear system. Lyapunov function demonstrate the existence of a symmetric positive definite matrix to analyze the stability of non-linear dynamical systems. The proposed method allows us to construct a system of ordinary differential equations to localize the spectrum of perturbed system which guarantees the stability of non-linear dynamical system.

The relationship among eigenvalues, singular values, and quadratic forms associated with linear transforms of doubly stochastic matrices has remained an important topic since 1949. The main objective of this article is to present some useful theorems, concerning the spectral properties of doubly stochastic matrices. The computation of the bounds of structured singular values for a family of doubly stochastic matrices is presented by using low-rank ordinary differential equations-based techniques. The numerical computations illustrating the behavior of the method and the spectrum of doubly stochastic matrices is then numerically analyzed.

A full mathematical analysis of magnetohydrodynamic (MHD) hybrid nanofluid flow over a stretching/shrinking sheet with thermal radiation is presented in this paper. The hybrid nanofluid consists of copper (Cu) and alumina (Al 2 O 3 ) nanoparticles suspended in water as base fluid. By means of similarity transformations the governing partial differential equations are transformed into ordinary differential equations and are solved numerically by the Runge-Kutta-Fehlberg method with a shooting technique. The effects of the significant parameters like magnetic field parameter, radiation parameter, suction/injection parameter, stretching/shrinking parameter and nanoparticle volume fractions on the velocity and temperature profiles, skin friction coefficient and local Nusselt number are discussed . The dual solutions for the cases of shrinking sheet are obtained. The results are given in tables and figures and physical interpretation are made. The results show that hybrid nanofluids significantly enhance the heat transfer compared with mono nanofluids and the thermal radiation further increases the thermal boundary layer thickness.

Classroom engagement is an important indicator of students’ active participation in the learning process. However, the psychological processes underlying such engagement in the university setting need further investigation. Drawing on identity theory and the academic self-image perspective, this study examined whether academic self-image mediates the association between the salience of social and personal identity and classroom engagement among university students. The present study adopted a quantitative cross-sectional correlational design. The sample comprised 224 university students aged 18-30 years. Participants completed standardized questionnaires on the importance of social and personal identity, academic self-image and classroom engagement. In reliability testing, all scales showed good to excellent internal consistency. Pearson correlation analysis showed significant positive correlations between social and personal identity prominence, academic self-image, and classroom engagement. Multiple regression analysis results suggested that both social and personal identity prominence and academic self-image were significant predictors of classroom engagement. In addition, mediation analysis using PROCESS Model 4 showed that academic self-image partially mediated the relationship between the centrality of social and personal identity and classroom engagement. These results imply that students who attach more importance to their personal and social traits are more likely to have a better academic self-image, which leads to more engagement in the classroom. The present study contributes to educational psychology by integrating identity salience and academic self-image as psychological resources to enhance university students’ classroom engagement.