Potential Theory on Compact Sets

The primary goal of this work is to extend the notions of potential theory to compact sets. There are several equivalent ways to define continuous harmonic functions H(K) on a compact set K in [the set of real numbers]n. One may let H(K) be the uniform closure of all functions in C(K) which are restrictions of harmonic functions on a neighborhood of K, or take H(K) as the subspace of C(K) consisting of functions which are finely harmonic on the fine interior of K. In [9] it was shown that these definitions are equivalent

The Dirichlet Problem for Harmonic Functions on Compact Sets

For any compact set $K\subset \mathbb{R}^n$ we develop the theory of Jensen measures and subharmonic peak points, which form the set $\mathcal{O}_K$, to study the Dirichlet problem on $K$. Initially we consider the space $h(K)$ of functions on $K$ which can be uniformly approximated by functions harmonic in a neighborhood of $K$ as possible solutions. As in the classical theory, our Theorem 8.1 shows $C(\mathcal{O}_K)\cong h(K)$ for compact sets with $\mathcal{O}_K$ closed. However, in general a continuous solution cannot be expected even for continuous data on \rO_K as illustrated by Theorem 8.1. Consequently, we show that the solution can be found in a class of finely harmonic functions. Moreover by Theorem 8.7, in complete analogy with the classical situation, this class is isometrically isomorphic to $C_b(\mathcal{O}_K)$ for all compact sets $K$.Comment: There have been a large number of changes made from the first version. They mostly consists of shortening the article and supplying additional reference

Superconducting fluctuations in boron-doped nanocrystalline diamond films: a study of disordered granular metals using diagrammatic quantum field theory

In this thesis, we demonstrate how to generalise the diagrammatic methods of quantum field theory commonly used for calculating transport phenomena in disordered homogeneous metals, such that they may be used for disordered granular metals. The predictions of our granular model are then compared to experimental resistance versus temperature data measured in boron-doped nanocrystalline diamond films. We find semi-quantitative agreement under the assumption of a constant phase breaking rate, $\tau_{\phi}^{-1}$, and explore the possible temperature dependence of $\tau_{\phi}^{-1}$. We find that our current model of a disordered granular metal does not generate a phase breaking mechanism which is able to quantitatively match theory to experiment. We suggest different avenues to explore theoretically and experimentally to determine the origin of the contributions to the fluctuation conductivity in BNCD, so that we better understand the onset of superconductivity in disordered granular metals

Faculty Mentoring Practices in Academic Emergency Medicine

Background Mentoring is considered a fundamental component of career success and satisfaction in academic medicine. However, there is no national standard for faculty mentoring in academic emergency medicine (EM) and a paucity of literature on the subject. Objectives The objective was to conduct a descriptive study of faculty mentoring programs and practices in academic departments of EM. Methods An electronic survey instrument was sent to 135 department chairs of EM in the United States. The survey queried faculty demographics, mentoring practices, structure, training, expectations, and outcome measures. Chi-square and Wilcoxon rank-sum tests were used to compare metrics of mentoring effectiveness (i.e., number of publications and National Institutes of Health [NIH] funding) across mentoring variables of interest. Results Thirty-nine of 135 departments completed the survey, with a heterogeneous mix of faculty classifications. While only 43.6% of departments had formal mentoring programs, many augmented faculty mentoring with project or skills-based mentoring (66.7%), peer mentoring (53.8%), and mentoring committees (18%). Although the majority of departments expected faculty to participate in mentoring relationships, only half offered some form of mentoring training. The mean number of faculty publications per department per year was 52.8, and 11 departments fell within the top 35 NIH-funded EM departments. There was an association between higher levels of perceived mentoring success and both higher NIH funding (p = 0.022) and higher departmental publications rates (p = 0.022). In addition, higher NIH funding was associated with mentoring relationships that were assigned (80%), self-identified (20%), or mixed (22%; p = 0.026). Conclusions Our findings help to characterize the variability of faculty mentoring in EM, identify opportunities for improvement, and underscore the need to learn from other successful mentoring programs. This study can serve as a basis to share mentoring practices and stimulate conversation around strategies to improve faculty mentoring in EM