Date of Award
3-2024
Document Type
Thesis
Degree Name
Master of Science
Department
Department of Mathematics and Statistics
First Advisor
John D. Jasper, PhD
Abstract
An equichordal tight fusion frame (ECTFF) is an example of an optimal packing of subspaces. In particular, an ECTFF is an optimal packing of points in the Grassmannian with respect to chordal distance. Equivalently, every ECTFF is an arrangement of subspaces that meet certain criteria; tightness and equichordality. The existence of an ECTFF can be rephrased as, a certain polynomial mapping as a root. Hence, one can prove the existence of an ECTFF by applying Newton–Kantorovich theorem to this polynomial mapping, given a close enough approximation of one. Newton–Kantorovich requires checking an inequality within a neighborhood of the approximate root, but a sufficient condition can be adapted that only involves checking one inequality at the approximate root. This inequality can then be checked using computer software (such as MAT) to provide an exact answer, which will determine if we have proven the existence of an ECTFF. Additionally, ECTFFs can be created using translations of cyclic groups, which can provide ECTFFs of larger size because of some redundancy in their structure. In this paper, we provide background proofs used in our methodology for completeness, along with detailed discussion about our methodology itself. We provide specific examples of ECTFFs, as well as a complete list of ECTFFs we have existence proofs for, including ECTFFs that had not previously been proven to exist.
AFIT Designator
AFIT-ENC-MS-24-M-001
DTIC Accession Number
AD1318930
Recommended Citation
Davis, Staci R., "Proving the Existence of Equichordal Tight Fusion Frames using the Newton–Kantorovich Theorem" (2024). Theses and Dissertations. 7791.
https://scholar.afit.edu/etd/7791
Comments
A 12-month embargo was observed for posting this work on AFIT Scholar.
Distribution Statement A, Approved for Public Release. PA case number on file.