Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues

We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain \Omega\subset\RR^n with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E > 0$ and the spectrum $\{E_j \}$ in terms of the boundary $L^2$-norm of a normalized trial solution $u$ of the Helmholtz equation $(\Delta + E)u = 0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all $E$ greater than a small constant, and improve upon the best-known bounds of Moler--Payne by a factor of the wavenumber $\sqrt{E}$. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes, of interest in its own right.Comment: 18 pages, 3 figure

Here Comes the Sunburst: Measuring and Visualizing Scholarly Impact

Our ARL institution partnered with a new service (PlumX) to track, measure, and visualize faculty scholarly impact. In a pilot project, both traditional and emerging measures of scholarly impact were collected for 32 researchers. The presenters will chronicle the data management and enhancements applied, including utilizing content from our institutional repository, importing and enriching metadata, and using an intranet to organize work and collaborate with colleagues. Results will assist faculty and those who work with them to identify strengths and weaknesses of scholarly impact and where to focus efforts to increase research visibility

Alternative Education: A Continued Examination of How States Are Addressing Alternative Education in Their Schools

In an attempt to understand the current preparation level of Arkansas for the Leave No Child Left Behind Act (NCLB) of 2002, the current project conducts a preliminary literature and web search to record what other states have done and are doing to prepare for NCLB. A brief synopsis of the history of alternative policies is also included. Results indicate that of the twelve states examined, most states are following a similar pattern with regard to preparing for their “at risk” students and passing similar alternative education policies. Three common characteristics found across the states: defining what alternative education is and who it provides for, legislation providing for each state’s plan, and finally a discussion of the various funding methods

What Does the NAEP Tell Us About Student Achievement in Arkansas?

This brief discusses the most recent and historical NAEP data in math, science, reading, and writing. Arkansas’ NAEP results are compared to national and border state averages. Based on the most recent NAEP exams, Arkansas is performing slightly below the national average in reading, math, science, and writing. There is also data to support that Arkansas students perform less well in Grade 8 than in Grade 4 when compared to the national and border state averages. Arkansas, however, is among the highest performing states with regard to change over the last decade. Since 1992, Arkansas has gained on the national and border state averages. In addition, Arkansas’ black-white and Hispanic-white achievement gaps in reading and math are lower than over half of participating states