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Combinatorial and Algebraic Conditions that Preclude SAPpiness Garnett, Colin
Description
It is well known that a complex zero-nonzero pattern cannot be spectrally arbitrary if its digraph doesn’t have at least two loops and at least one two cycle, or at least three loops. This talk focuses on several other combinatorial conditions on the digraph that preclude it from being spectrally arbitrary. In particular we are sometimes able to reduce the number of unknown entries to be below the threshold of $2n−1$. Furthermore there are several algebraic conditions on the coefficients of the characteristic polynomial that can be exploited to show that a pattern is not spectrally arbitrary over any field. Using Sage we were able to show that no zero-nonzero pattern with $2n−1$ nonzero entries will be spectrally arbitrary over $C$ where $n≤6$. When $n=7$ we find two zero-nonzero patterns that do not satisfy our algebraic conditions precluding them from being spectrally arbitrary.
Item Metadata
Title |
Combinatorial and Algebraic Conditions that Preclude SAPpiness
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2017-07-09T09:32
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Description |
It is well known that a complex zero-nonzero pattern cannot be spectrally arbitrary if its digraph doesn’t have at least two loops and at least one two cycle, or at least three loops. This talk focuses on several other combinatorial conditions on the digraph that preclude it from being spectrally arbitrary. In particular we are sometimes able to reduce the number of unknown entries to be below the threshold of $2n−1$. Furthermore there are several algebraic conditions on the coefficients of the characteristic polynomial that can be exploited to show that a pattern is not spectrally arbitrary over any field. Using Sage we were able to show that no zero-nonzero pattern with $2n−1$ nonzero entries will be spectrally arbitrary over $C$ where $n≤6$. When $n=7$ we find two zero-nonzero patterns that do not satisfy our algebraic conditions precluding them from being spectrally arbitrary.
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Extent |
27 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Black Hills State University
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Series | |
Date Available |
2018-01-06
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0362885
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International