Optimal Control of a Chemical Vapor Deposition Reactor

We study a simple model of chemical vapor deposition on a silicon wafer. The control is the flux of chemical species, and the objective is to grow the semiconductor film so that its surface attains a prescribed profile as nearly as possible. The surface is spatially fast oscillating due to the small feature scale, and therefore the problem is formulated in terms of its homogenized approximation. We prove that the optimal control is bang-bang, and we use this information to develop a numerical scheme for computing the optimal control.     

Anatomy of the thyroarytenoid branch of the recurrent laryngeal nerve.

OBJECTIVE: To determine the position and anatomic variability of the thyroarytenoid (TA) branch of the recurrent laryngeal nerve (RLN). METHODS: The RLN on 13 human cadaver specimens (24 sides) was dissected to the termination of the TA branch in the TA muscle. A pin was placed at the inferior aspect of the thyroid cartilage at the inferior tubercle. Using a caliper, the distance from the pin to the TA branch of the RLN was measured. The direction of the nerve and number of branches were recorded. RESULTS: The average distance to the TA branch of the RLN is 4.23 mm with a standard deviation of 2.86 mm. The median distance is 3.75 mm. Most of the specimens fell in a range of 1 to 4 mm. Overall, 54% of the nerves traveled in a horizontal direction, but vertical and oblique orientations were observed. About 20% of specimens demonstrated branching of the TA nerve. CONCLUSION: Measuring 4 mm from the inferior tubercle along a perpendicular line from the thyroid tubercle on the inferior border of the thyroid cartilage provides a good estimate of the location of the TA branch of the RLN. This information is useful when creating a posterior thyrotomy for TA neurectomy for patients with adductor spasmodic dysphonia. Knowledge of the course and possible branching of the nerve, will aid in localizing the nerve as well as ensuring adequate resection.     

without sharps

We show that the supremum of the lengths of prewellorderings of the reals can be , with inaccessible to reals, assuming only the consistency of an inaccessible.