Purdue University Graduate School
Leah McNabb - Dissertation.pdf (456.18 kB)

Applications of One-Point Quadrature Domains

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posted on 2024-04-16, 16:29 authored by Leah Elaine McNabbLeah Elaine McNabb

This thesis presents applications of one-point quadrature domains to encryption and decryption as well as a method for estimating average temperature. In addition, it investigates methods for finding explicit formulas for certain functions and introduces results regarding quadrature domains for harmonic functions and for double quadrature domains. We use properties of quadrature domains to encrypt and decrypt locations in two dimensions. Results by Bell, Gustafsson, and Sylvan are used to encrypt a planar location as a point in a quadrature domain. A decryption method using properties of quadrature domains is then presented to uncover the location. We further demonstrate how to use data from the decryption algorithm to find an explicit formula for the Schwarz function for a one-point area quadrature domain. Given a double quadrature domain, we show that the fixed points within the area and arc length quadrature identities must be the same, but that the orders at each point may differ between these identities. In the realm of harmonic functions, we demonstrate how to uncover a one-point quadrature identity for harmonic functions from the quadrature identity for a simply-connected one-point quadrature domain for holomorphic functions. We use this result to state theorems for the density of one-point quadrature domains for harmonic functions in the realm of smooth domains with $C^{\infty}$-smooth boundary. These density theorems then lead us to discuss applications of quadrature domains for harmonic functions to estimating average temperature. We end by illustrating examples of the encryption process and discussing the building blocks for future work.


NSF grant DMS 1764167


Degree Type

  • Doctor of Philosophy


  • Mathematics

Campus location

  • West Lafayette

Advisor/Supervisor/Committee Chair

Dr. Steven R. Bell

Additional Committee Member 2

Dr. Johnny Brown

Additional Committee Member 3

Dr. László Lempert

Additional Committee Member 4

Dr. Sai-Kee Yeung