The Dual JL Transforms and Superfast Matrix Algorithms

We call a matrix algorithm superfast (aka running at sublinear cost) if it involves much fewer flops and memory cells than the matrix has entries. Using such algorithms is highly desired or even imperative in computations for Big Data, which involve immense matrices and are quite typically reduced to solving linear least squares problem and/or computation of low rank approximation of an input matrix. The known algorithms for these problems are not superfast, but we prove that their certain superfast modifications output reasonable or even nearly optimal solutions for large input classes. We also propose, analyze, and test a novel superfast algorithm for iterative refinement of any crude but sufficiently close low rank approximation of a matrix. The results of our numerical tests are in good accordance with our formal study.Comment: 36.1 pages, 5 figures, and 1 table. arXiv admin note: text overlap with arXiv:1710.07946, arXiv:1906.0411

A unified approach to exact solutions of time-dependent Lie-algebraic quantum systems

By using the Lewis-Riesenfeld theory and the invariant-related unitary transformation formulation, the exact solutions of the {\it time-dependent} Schr\"{o}dinger equations which govern the various Lie-algebraic quantum systems in atomic physics, quantum optics, nuclear physics and laser physics are obtained. It is shown that the {\it explicit} solutions may also be obtained by working in a sub-Hilbert-space corresponding to a particular eigenvalue of the conserved generator ({\it i. e.}, the {\it time-independent} invariant) for some quantum systems without quasi-algebraic structures. The global and topological properties of geometric phases and their adiabatic limit in time-dependent quantum systems/models are briefly discussed.Comment: 11 pages, Latex. accepted by Euro. Phys. J.

The effectiveness of roleplaying in teaching preschoolers social skills

The study was designed to evaluate the effectiveness of role-play as a means of enhancing the social skills of preschool students in a Montessori setting. A total of 37 children between the ages of two and a half to six years who participated in a five week study. The interventions were conducted once a week for four weeks, during which children took turns participating in simulated scenarios that mirrored their everyday experiences. After the role-play presentation, assessments and group discussions were conducted to evaluate the children\u27s comprehension. The research data was gathered before, during and after the interventions. The researcher used checklists and observation records to document the number and details of conflicts, as well as the children\u27s behavior and conflict resolution skills. The study revealed a decrease in both the frequency of conflicts and the level of aggression, indicating the effectiveness of role-play in cultivating social skills in children. However, the study also indicated the need for further research and extended observation to support new and younger children in adapting to the environment and to keep track of how effectively children internalize and sustain the skills they have acquired