Mitchel T. Keller

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Let P be a poset in which each point is incomparable to at most ∆ others. Tanenbaum, Trenk, and Fishburn asked in a 2001 paper if the linear discrepancy of such a poset is bounded above by b(3∆−1)/2c. This paper answers their question in the affirmative for two classes of posets. The first class is the interval orders, which are shown to have linear(More)
The linear discrepancy of a poset P is the least k such that there is a linear extension L of P such that if x and y are incomparable in P, then |hL(x) − hL(y)| ≤ k, where hL(x) is the height of x in L. Tanenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem of characterizing the(More)
Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant d such that if P is a poset with cover graph of P of pathwidth at most 2, then dim(P ) ≤ d. We answer this question in the affirmative. We also show that if P is a poset containing the standard example S5 as a subposet, then the cover graph of P has treewidth at least(More)
The linear discrepancy of a poset P is the least k for which there is a linear extension L of P such that if x and y are incomparable in P, then |hL(x) − hL(y)| ≤ k, where hL(x) is the height of x in L. In this paper, we consider linear discrepancy in an online setting and devise an online algorithm that constructs a linear extension L of a poset P so that(More)
We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal I and the Stanley depth of its compliment, S/I. Using these results we are able to prove that if S is a polynomial ring with at most 5 indeterminates and I is a square-free monomial ideal, then the Stanley depth of S/I is strictly larger than the Stanley(More)
To my parents, Jim and Karol, who believed a farm kid from North Dakota could do whatever he wanted and helped me achieve whatever I imagined and more. iii ACKNOWLEDGEMENTS There are many people without whose mentoring, guidance, support, and friendship this dissertation never would have been written. My advisor, William T. Trotter, is at the top of that(More)
This paper uses chain complexes of based, finitely-generated Z-modules to study the Laplacians of signed plane graphs. We extend a theorem of Lien and Watkins [6] regarding the Goeritz equivalence of the signed Laplacians of a signed plane graph and its dual by showing that it is possible to use only (±1)-diagonal forms instead of the (0,±1)-diagonal forms(More)