My mathematical research has focused on two areas: classical analysis and geometry. In classical analysis, I focused on singular integrals and Fourier analysis. In geometry, I focused on Dirichlet tessellations. I am no longer actively involved in mathematical research. Instead, my interest in mathematics has shifted to thinking about the best way to teach this most useful and beautiful discipline.
The following annotated bibliography of my research papers gives a flavor of my mathematical interests. The J. M. Ash who was a co-author in two of my papers is my brother, Marshall Ash, a professor at DePaul University.
A transformation on a Hilbert space is called an isometry if it preserves the norm of all functions in the space. The important class of unitary operators is a subset of the class of isometries, and in fact the hardest part of proving an operator is unitary is showing that it is an isometry. The gist of the paper is contained in two results:
This paper contains most of the results of my thesis, which was written to answer a question posed to me by Antoni Zygmund, together with related results from later research. Singular integrals are a generalization of the Hilbert transform, so important in Fourier analysis, to Rn where n > 1. They are useful in the study of partial differential equations. The main idea of my research was to replace an ordinary singular integral, which is based on a kernel K homogeneous of degree -n by a similar, but better-behaved, kernel Ka (a > 0) homogeneous of degree -n-ia. I was able to prove that subject to some mild conditions on the kernel, Kaf converges to Kf in norm as a approaches 0.
The highlight of the paper was Charles Fefferman's surprising discovery that Kaf does not necessarily converge to Kf pointwise almost everywhere.
A two-dimensional Dirichlet tessellation is a diagram generated by a set of n points in a plane. Imagine the plane is a map of a city, and the points are polling places. Imagine election law requires that each citizen vote at the polling place closest to their home. Then the resulting n election districts form the Dirichlet tessellation for the polling places.
Dirichlet tessellation have applications in many areas of science and technology, and not surprisingly they have been rediscovered many times and given different names. They are also known as Voronoi diagrams (the preferred name today among geometers and computer scientists), Thiessen polygons (geography), Wigner-Seitz regions (physics), area-of-influence polygons (mining), and plant polygons (ecology).
For more information on the Voronoi diagram, including a picture, see my page The Voronoi Diagram Package.
The purpose of this paper was to provide a method for determining when a subdivision of the plane into n convex regions was in fact a Dirichlet tessellation based on some n points.
Available online. Click here.
In this paper, Ethan Bolker and I examined generalizations of the Dirichlet tessellation concept, deducing the properties of the circular Dirichlet tessellation (aka Voronoi diagram with multiplicative weights), hyperbolic Dirichlet tessellation (Voronoi diagrams with additive weights), and sectional Dirichlet tessellation (two-dimensional section of a three-dimensional Dirichlet tessellation). We also found and studied a generalization of the Dirichlet tessellation that includes all of the above types as special cases.
The most important and surprising result in the paper is that a diagram represents a sectional Dirichlet tessellation if and only if it has a reciprocal figure. Tessellations with reciprocal figures have been extensively studied in the field of graphical statics, a topic of study which flourished a century ago. For example, graphical statics together with our result implies that the class of sectional Dirichlet tessellations is precisely the set of physically possible configurations for spider-webs.
Available online. Click here.
| Hyperbolic | Circular |
|---|---|
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| Numbers indicate weight of sources.
Source: A. Okabe, B. Boots and K. Sugihara, |
Sources and weights suppressed.
Diagram created with a C++ program I wrote. |
This chapter contains a popularization of the results on sectional Dirichlet tessellations contained in the paper mentioned above.