Dissertation
Boundary Measures and Cubical Covers of Sets in R^n
Doctor of Philosophy (PhD), Washington State University
01/2018
Handle:
https://hdl.handle.net/2376/16417
Abstract
The art of analysis involves the subtle combination of approximation, inequalities, and geometric intuition as well as being able to work at different scales. Even the restriction to sets in R^n affords us the opportunity to hone these skills through a virtually limitless supply of examples and tools all aimed at ``taming the wild," so to speak, so as to develop techniques by which to work with arbitrary sets.
In this dissertation, we present a singular integral for measuring the level sets (i.e. the boundary of the super-level or sub-level set) of a C^{1,1} function mapping from R^n to R, that is, one whose derivative is Lipschitz continuous. We extend this to measure embedded manifolds in R^2 that are merely C^1 using the distance function.
We then present a rather playful exposition, with several open problems, related to representations of sets in R^n aimed at stimulating interest and inspiring student research in these areas. We explore finding less and less regular sets for which it is still true that the representations faithfully inform us about the original set without removing important features. While the primary focus is on cubical covers, we also briefly introduce Jones' beta numbers and varifolds from geometric measure theory, and we present a conjecture for the ratio of the measure of the boundary of the cubical cover to the measure of the C^{1,1} boundary of a compact set.
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Details
- Title
- Boundary Measures and Cubical Covers of Sets in R^n
- Creators
- Laramie Paxton
- Contributors
- Kevin R Vixie (Advisor)Bala Krishnamoorthy (Committee Member)Matthew Sottile (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Department of Mathematics and Statistics
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Number of pages
- 95
- Identifiers
- 99900581421601842
- Language
- English
- Resource Type
- Dissertation