Curvatures of Stiefel manifolds with deformation metrics

We compute curvatures of a family of tractable metrics on Stiefel manifolds, introduced recently by H{\"u}per, Markina and Silva Leite, which includes the well-known embedded and canonical metrics on Stiefel manifolds as special cases. The metrics could be identified with the Cheeger deformation metrics. We identify parameter values in the family to make a Stiefel manifold an Einstein manifold and show Stiefel manifolds always carry an Einstein metric. We analyze the sectional curvature range and identify the parameter range where the manifold has non-negative sectional curvature. We provide the exact sectional curvature range when the number of columns in a Stiefel matrix is $2$, and a conjectural range for other cases. We derive the formulas from two approaches, one from a global curvature formula derived in our recent work, another using curvature formulas for left-invariant metrics. The second approach leads to curvature formulas for Cheeger deformation metrics on normal homogeneous spaces

Lagrange Multipliers and Rayleigh Quotient Iteration in Constrained Type Equations

We generalize the Rayleigh quotient iteration to a class of functions called vector Lagrangians. The convergence theorem we obtained generalizes classical and nonlinear Rayleigh quotient iterations, as well as iterations for tensor eigenpairs and constrained optimization. In the latter case, our generalized Rayleigh quotient is an estimate of the Lagrange multiplier. We discuss two methods of solving the updating equation associated with the iteration. One method leads to a generalization of Riemannian Newton method for embedded manifolds in a Euclidean space while the other leads to a generalization of the classical Rayleigh quotient formula. Applying to tensor eigenpairs, we obtain both an improvements over the state-of-the-art algorithm, and a new quadratically convergent algorithm to compute all complex eigenpairs of sizes typical in applications. We also obtain a Rayleigh-Chebyshev iteration with cubic convergence rate, and give a clear criterion for RQI to have cubic convergence rate, giving a common framework for existing algorithms

Factors influencing teaching for critical thinking in Vietnamese lower secondary schools :a mixed method study focussed on history

Phd ThesisCritical thinking (CT) has been introduced into the curriculum of numerous secondary schools worldwide. Whilst the literature on instructional and assessment approaches related to CT is abundant, few studies have been conducted to investigate factors that influence teachers’ decision-making about the development of CT in learners. In Vietnam, although education reform emphasises the importance of developing learners’ dispositions and skills for CT, traditional instructional methods continue to be prevalent. The aim of this thesis was, therefore, to identify factors affecting teaching for CT in Vietnamese lower secondary schools. A mixed methods sequential explanatory design was adopted with questionnaires, semi-structured interviews and focus groups as key data collection instruments. Data collected from 145 lower secondary history teachers in the Northern Province of Thai Binh, Vietnam indicate that assessment practice and school culture exert considerable influence on teaching for CT. Teachers understand the benefits of teaching for CT but regard it as extraneous to the requirements of tests, the criteria of teacher evaluation and the general expectation of many parents. Limited school democracy, low teacher autonomy and collegiality inhibit the application of innovative teaching techniques to enhance learners’ CT. The study suggests the influence of national culture and perceptions about teaching and learning on pedagogical practices. It challenges theories that support test-based accountability regimes by indicating that accountability pressures discourage teaching approaches that promote student interaction and critical engagement in learning. In contrast to recent research, it is found that many teachers teach against their beliefs and knowledge, because they encounter strong obstacles when attempting to change. Informed by rich empirical evidence, it is expected that this research will not only support the upcoming education reform in Vietnam but also provide useful lessons for policymakers and school leaders in countries with similar educational problems.Vietnamese government and in particular Project 165 and leaders of Thai Binh provinc