Simple and efficient representations for the fundamental solutions of Stokes flow in a half-space

We derive new formulas for the fundamental solutions of slow, viscous flow, governed by the Stokes equations, in a half-space. They are simpler than the classical representations obtained by Blake and collaborators, and can be efficiently implemented using existing fast solvers libraries. We show, for example, that the velocity field induced by a Stokeslet can be annihilated on the boundary (to establish a zero slip condition) using a single reflected Stokeslet combined with a single Papkovich-Neuber potential that involves only a scalar harmonic function. The new representation has a physically intuitive interpretation

The Decoupled Potential Integral Equation for Time-Harmonic Electromagnetic Scattering

We present a new formulation for the problem of electromagnetic scattering from perfect electric conductors. While our representation for the electric and magnetic fields is based on the standard vector and scalar potentials ${\bf A},\phi$ in the Lorenz gauge, we establish boundary conditions on the potentials themselves, rather than on the field quantities. This permits the development of a well-conditioned second kind Fredholm integral equation which has no spurious resonances, avoids low frequency breakdown, and is insensitive to the genus of the scatterer. The equations for the vector and scalar potentials are decoupled. That is, the unknown scalar potential defining the scattered field, $\phi^{Sc}$, is determined entirely by the incident scalar potential $\phi^{In}$. Likewise, the unknown vector potential defining the scattered field, ${\bf A}^{Sc}$, is determined entirely by the incident vector potential ${\bf A}^{In}$. This decoupled formulation is valid not only in the static limit but for arbitrary $\omega\ge 0$.Comment: 33 pages, 7 figure

The effects of using mini whiteboards on the academic performance and engagement of students in a tenth grade resource English/Language Arts classroom

The purpose of this study was to: (a) examine the effectiveness of mini whiteboards in increasing engagement, (b) examine the effectiveness of mini whiteboards in increasing academic achievement, and (c) determine if students in a tenth grade English/Language Arts resource center classroom are satisfied with the use of mini whiteboards. The research was conducted using an ABAB single-subject design methodology. Student achievement was evaluated through weekly assessments, while daily engagement was evaluated using interval recording in 5-minute increments. Results suggest that the use of mini whiteboards may help increase the engagement and academic achievement of students in a tenth grade ELA resource center classroom. Mini whiteboards were found to increase the weekly mean engagement score for 7 out of 10 students, and the weekly academic achievement score for 8 out of 10 students. Results also show that most students felt comfortable using the mini whiteboards and some felt that it helped them academically. Implications for educating students in a resource center setting include the recommendation to utilize active responding techniques such as mini whiteboards in the classroom

A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions

AbstractWe present a numerical algorithm for the construction of efficient, high-order quadratures in two and higher dimensions. Quadrature rules constructed via this algorithm possess positive weights and interior nodes, resembling the Gaussian quadratures in one dimension. In addition, rules can be generated with varying degrees of symmetry, adaptable to individual domains. We illustrate the performance of our method with numerical examples, and report quadrature rules for polynomials on triangles, squares, and cubes, up to degree 50. These formulae are near optimal in the number of nodes used, and many of them appear to be new