The Real-Quaternionic Indicator of Irreducible Self-Conjugate Representations of Real Reductive Algebraic Groups and A Comment on the Local Langlands Correspondence of GL(2, F )

The real-quaternionic indicator, also called the $\delta$ indicator, indicates if a self-conjugate representation is of real or quaternionic type. It is closely related to the Frobenius-Schur indicator, which we call the $\varepsilon$ indicator. The Frobenius-Schur indicator $\varepsilon(\pi)$ is known to be given by a particular value of the central character. We would like a similar result for the $\delta$ indicator. When $G$ is compact, $\delta(\pi)$ and $\varepsilon(\pi)$ coincide. In general, they are not necessarily the same. In this thesis, we will give a relation between the two indicators when $G$ is a real reductive algebraic group. This relation also leads to a formula for $\delta(\pi)$ in terms of the central character. For the second part, we consider the construction of the local Langlands correspondence of $GL(2,F)$ when $F$ is a non-Archimedean local field with odd residual characteristics. By re-examining the construction, we provide new proofs to some important properties of the correspondence. Namely, the construction is independent of the choice of additive character in the theta correspondence.

The Application of an Exogenous Linear and Radial Electrical Field to an In Vitro Chronic Diabetic Ulcer Model for Evaluation as a Potential Treatment

Chronic diabetic ulcers affect approximately 15% of patients with diabetes worldwide. Currently, applied electric fields are being investigated as a reliable and cost-effective treatment. This in vitro study aimed to determine the effects of a constant and spatially variable electric field on three factors: endothelial cell migration, proliferation, and angiogenic gene expression. Results for a constant electric field of 0.01 V demonstrated that migration at short time points increased 20-fold and proliferation at long time points increased by a factor of 1.40. Results for a spatially variable electric field did not increase directional migration, but increased proliferation by a factor of 1.39 and by a factor of 1.55 after application of 1.00 V and 0.01 V, respectively. Both constant and spatially variable applied fields increased angiogenic gene expression. Future research that explores a narrower range of intensity levels may more clearly identify the optimal design specifications of a spatially variable electric field.