Michael Bolt

A local geometric characterization of the Bochner-Martinelli kernel

In this paper it is shown that a connected smooth local hyper-surface in C n for which the skew-hermitian part of the Bochner-Martinelli kernel has a weak singularity must lie on a surface having one of the following forms: S 2m+1 x C n-m-1 for some 1 < m < n, or C x C n-1 where C is a one-dimensional curve. This strengthens results of Boas about the Bochner-Martinelli kernel and it generalizes a result of Kerzman and Stein about the Cauchy kernel.

van der Pauw's Theorem on Sheet Resistance.

Abstract The sheet resistance of a conducting material of uniform thickness is analogous to the resistivity of a solid material and provides a measure of electrical resistance. In 1958, L. J. van der Pauw found an effective method for computing sheet resistance that requires taking two electrical measurements from four points on the edge of a simply connected sample of material. In this article we give a statement and proof of the van der Pauw theorem using ideas from complex variables. We also include details of a demonstration of the method for finding the sheet resistance of a stainless steel shim using typical equipment in a physics laboratory.

THE KERZMAN-STEIN OPERATOR FOR PIECEWISE CONTINUOUSLY DIFFERENTIABLE REGIONS

The Kerzman–Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on a rectifiable curve. If the curve is continuously differentiable, the Kerzman–Stein operator is compact on the Hilbert space of square integrable functions; when there is a corner, the operator is noncompact. Here, we give a complete description of the spectrum for a finite symmetric wedge and we show how this reveals the essential spectrum for curves that are piecewise continuously differentiable. We also give an explicit construction for a smooth curve whose Kerzman–Stein operator has large norm.

Szegő kernel transformation law for proper holomorphic mappings

Let Ω1,Ω2 be smoothly bounded doubly connected regions in the complex plane. We establish a transformation law for the Szegő kernel under proper holomorphic mappings. This extends known results concerning biholomorphic mappings between multiply connected regions as well as proper holomorphic mappings from multiply connected regions to simply connected regions.

The Fastest Way Not to Run a Four-Minute Mile

Summary In this manuscript we present the mathematics that is needed to answer three counterintuitive problems related to the averaging of functions. The problems are manifestations of the question, “Is the average rate of change on a given interval determined by the average rate of change on subintervals of a fixed length?” We also ask questions in higher dimensions that may have interesting geometric significance.