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A simple algorithm for Horn's problem and two results on discrepancy

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Title
A simple algorithm for Horn's problem and two results on discrepancy
Name (type = personal)
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Franks
NamePart (type = given)
William Cole
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1991-
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William Cole Franks
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author
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Saks
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Michael E
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Michael E Saks
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Advisory Committee
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chair
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Saraf
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Shubhangi
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Shubhangi Saraf
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Kahn
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Jeffry
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Jeffry Kahn
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internal member
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Gurvits
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Leonid
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Leonid Gurvits
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Advisory Committee
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outside member
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Rutgers University
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degree grantor
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School of Graduate Studies
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theses
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2019
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2019-05
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English
Abstract (type = abstract)
In the second chapter we consider the discrepancy of permutation families. A k--permutation family on n vertices is a set-system consisting of the intervals of k permutations of [n]. Both 1- and 2-permutation families have discrepancy 1, that is, one can color the vertices red and blue such that the number of reds and blues in each edge differs by at most one. That is, their discrepancy is bounded by one. Beck conjectured that the discrepancy of for 3-permutation families is also O(1), but Newman and Nikolov disproved this conjecture in 2011. We give a simpler proof that Newman and Nikolov's sequence of 3-permutation families has discrepancy at least logarithmic in n. We also exhibit a new, tight lower bound for the related notion of root-mean-squared discrepancy of a 6-permutation family, and show new upper bounds on the root--mean--squared discrepancy of the union of set--systems.
In the third chapter we study the discrepancy of random matrices with m rows and n >> m independent columns drawn from a bounded lattice random variable, a model motivated by the Koml'os conjecture. We prove that, with high probability, the discrepancy is at most twice the l∞-covering radius of the lattice. As a consequence, the discrepancy of a m x n random t-sparse matrix is at most 1 with high probability for n ≥ m3 log2m, an exponential improvement over Ezra and Lovett (Ezra and Lovett, emph{Approx+Random}, 2015). More generally, we show polynomial bounds on the size of n required for the discrepancy to become at most twice the covering radius of the lattice with high probability.
In the fourth chapter, we obtain a simple algorithm to solve a class of linear algebraic problems. This class includes Horn's problem, the problem of finding Hermitian matrices that sum to zero with prescribed spectra. Other problems in this class arise in algebraic complexity, analysis, communication complexity, and quantum information theory. Our algorithm generalizes the work of (Garg et. al., 2016) and (Gurvits, 2004).
Subject (authority = local)
Topic
Horn's problem
Subject (authority = RUETD)
Topic
Mathematics
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Rutgers University Electronic Theses and Dissertations
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ETD_9735
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1 online resource (x, 115 pages)
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Ph.D.
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Includes bibliographical references
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School of Graduate Studies Electronic Theses and Dissertations
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rucore10001600001
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Identifier (type = doi)
doi:10.7282/t3-1k1c-8h60
Genre (authority = ExL-Esploro)
ETD doctoral
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The author owns the copyright to this work.
RightsHolder (type = personal)
Name
FamilyName
Franks
GivenName
William
MiddleName
Cole
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Permission or license
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2019-04-09 12:41:20
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William Franks
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Rutgers University. School of Graduate Studies
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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