The Emergent Use of Online Learning in Secondary Education: A Preliminary Review of the Literature

While the preponderance of online learning is geared toward adult learners, its use in the United States for secondary education students has been around for over a decade and has a growing literature. As online learning for adolescents gains popularity, questions are being raised about the quality, strategies, and effectiveness of the programs being offered. There are several delivery methods of online learning from the full-time virtual school to the ‘blended’ model. This review of the literature will provide an understanding of current theory and practice online learning for secondary students, recommendations for implementation, and where it may lead

Index statistical properties of sparse random graphs

Using the replica method, we develop an analytical approach to compute the characteristic function for the probability $\mathcal{P}_N(K,\lambda)$ that a large $N \times N$ adjacency matrix of sparse random graphs has $K$ eigenvalues below a threshold $\lambda$. The method allows to determine, in principle, all moments of $\mathcal{P}_N(K,\lambda)$, from which the typical sample to sample fluctuations can be fully characterized. For random graph models with localized eigenvectors, we show that the index variance scales linearly with $N \gg 1$ for $|\lambda| > 0$, with a model-dependent prefactor that can be exactly calculated. Explicit results are discussed for Erd\"os-R\'enyi and regular random graphs, both exhibiting a prefactor with a non-monotonic behavior as a function of $\lambda$. These results contrast with rotationally invariant random matrices, where the index variance scales only as $\ln N$, with an universal prefactor that is independent of $\lambda$. Numerical diagonalization results confirm the exactness of our approach and, in addition, strongly support the Gaussian nature of the index fluctuations.Comment: 10 pages, 5 figure