Mathematical Sciences: Complex Analysis: Computation and Control.

这个项目将集中在数学分析的四个领域。本文将对二维谐波测度的估计进行研究。二维谐波测度用于势理论和测量解析函数的增长。目前已知的估计是圆盘中满足某扇区所有半径的曲线的调和测度。最大扇区的宽度还有待确定。在这个过程中，理论和数值的努力将结合起来。我们将继续研究有界全纯函数的最优化问题。这些调查已经进行了好几年。它们产生于最优控制中的某些工程问题，涉及通过定义在圆上的有界函数或连续函数找到给定函数的最佳逼近，该函数具有解析扩展到内部。这个项目的第三个，也是最近的一个目标将涉及保角映射的计算方法。有许多现有的技术用于这种计算和具有光滑边界的域的映射，通常具有快速收敛的数值近似。目前的工作是关于包含有角的域映射的近似。最近的进展是令人鼓舞的，尽管重复的近似过程在经过几次后显示出退化。将努力挑选出当前方法适用的最佳边界曲线。这些曲线很可能会落入被称为限制弯曲的弦弧曲线的类别。最后，从更理论化的角度出发，研究圆盘最大理想空间Gleason部分上的常数内函数的存在性和性质。虽然一般理论表明存在这样的内函数，但在这一类中是否存在Blaschke积却鲜为人知。与这项工作相关的是这类产品零分布的表征。除了应用于势理论之外，这项工作还增加了关于偏微分方程数值近似格式的基本知识。

This project will focus on four areas of mathematical analysis. Work will be done on estimating two-dimensional harmonic measure which is used in potential theory and in measuring growth of analytic functions. At the present time estimates are known for the harmonic measure of curves in a disc meeting all radii in some sector. The width of the maximal sector remains to be determined. Theoretical and numerical efforts will be combined in this process. Work will continue on problems of optimization using bounded holomorphic functions. These investigations have been underway for several years. They arise from certain engineering problems in optimal control and involve finding best approximations to given functions by bounded or continuous functions defined on a circle which have analytic extensions to the interior. A third, and more recent, goal of this project will involve methods for the computation of conformal maps. There are many techniques extant for such computations and mappings of domains with smooth boundaries that generally have numerical approximants which converge rapidly. The present work is concerned with approximation involving maps of domains with corners. Recent progress has been encouraging although repetitions of the approximation process shows degradation after several passes. Efforts will be made to single out the best boundary curves for which current methods are practical. It is likely that these curves will fall into the class known as chord-arc curves which have restricted bending. Finally, in a more theoretical vein, studies will be made into the existence and nature of inner functions which are constant on Gleason parts of the maximal ideal space of a disc. While general theory indicates existence of such inner functions there is little known as to whether there are Blaschke products in this class. Related to this work is a characterization of the zero-distribution of such products. In addition to applications to potential theory, this work adds to fundamental knowledge regarding numerical approximation schemes for partial differential equations.