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

The influence of athlete fear avoidance on acute concussive symptoms

There are millions of concussions each year and the number of symptoms or severity of a sport related concussion vary significantly. Recently there is more evidence for fear avoidance being associated with the number of symptoms reported, in the chronic stage. The purpose of this thesis was to assess acute concussions in athletes using the SCAT5 as well as fear avoidance, catastrophizing, depression, and anxiety. Main Outcomes and Measures: We assessed 34 athletes’ concussions using the SCAT5, overall general health using the Short Form General Health Survey (SF36), pain catastrophizing using the Pain Catastrophizing Scale, fear avoidance using the Athlete Fear Avoidance Questionnaire (AFAQ), and anxiety and depression using the Hospital Anxiety and Depression Questionnaire. Results: The participants had an average of 7.4 5.1 total number of symptoms and 16.3 17.0 symptom severity score on the SCAT5. The total number of symptoms reported on the SCAT5 was associated with the AFAQ score (r=0.493). The symptom severity score was associated with the AFAQ score (r=0.481). The total number of symptoms reported on the SCAT5 was associated with the HADS score (r=0.686). The symptom severity score was associated with the HADS score (r=0.602). The AFAQ score, HADS depression and HADS anxiety scores model was a significant predictor of the total number of symptoms reported on the SCAT5, accounting for 50.4% of the variance (p>0.001), and a significant predictor of the severity of symptoms reported on the SCAT5, accounting for of the 41% of the variance (p=0.001). Discussion: Our study identified a significant relationship between athlete fear avoidance and the number of concussion symptoms and the severity of the symptoms. A higher fear avoidance means that patients report more symptoms, and this relationship could explain why there is variability in the reporting of concussion symptoms

Aspects of the reproduction of an invasive crab, Hemigrapsus sanguineus, in northern and southern New England

Populations of the invasive shore crab, Hemigrapsus sanguineus , were studied in northern and southern New England to determine if crabs differ in reproductive behavior or characteristics between these regions. Additionally, effects of temperature on reproductive activity were quantified through laboratory experimentation. Number of broods per season increased with temperature, but the seasonal total was limited to three broods in laboratory experiments. Broods experienced limited success at the lowest temperature, 10°C. The reproductive season was longer at lower latitudes, and females at this site had smaller average ovigerous size. Patterns of ovigery varied between the regions, suggesting the production of one brood per season in New Hampshire, compared to two to three broods per season in Rhode Island. Overall, temperature may limit the possibility and degree of reproductive output by females, which may slow the spread or limit establishment of this species in northern latitudes

Nevanlinna-Pick interpolation on distinguished varieties in the bidisk

This article treats Nevanlinna-Pick interpolation in the setting of a special class of algebraic curves called distinguished varieties. An interpolation theorem, along with additional operator theoretic results, is given using a family of reproducing kernels naturally associated to the variety. The examples of the Neil parabola and doubly connected domains are discussed.Comment: 31 pages. The question left open at the end of version 1 has been answered in the affirmative; see Theorem 1.12 and Corollary 1.13 in version