arXiv subjects

Christopher Donnay

Publications and source records attributed to Christopher Donnay.

4 recordsLinked to original sources

arXiv2023

Asymptotics of Redistricting the $n\times n$ Grid

Redistricting is the act of dividing a region into districts for electoral representation. Motivated by this application, we study two questions. How many ways are there to partition the $n\times n$ grid into $n$ contiguous districts of equal size? How many of these partitions are ``compact"? We give asymptotic bounds on the number of plans: a lower bound of roughly $1.41^{n^2}$ and an upper bound of roughly $3.21^{n^2}$. We then use the lower bound to show that most plans are not compact.

math.CO↗

arXiv2023

3:1 Nesting Rules in Redistricting

In legislative redistricting, most states draw their House and Senate maps separately. Ohio and Wisconsin require that their Senate districts be made with a 3:1 nesting rule, i.e., out of triplets of adjacent House districts. We seek to study the impact of this requirement on redistricting, specifically on the number of seats won by a particular political party. We compare two ensembles generated using Markov Chain Monte Carlo methods; one which uses the ReCom chain to generate Senate maps without a nesting requirement, and the other which uses a chain that generates Senate maps with a 3:1 nesting requirement. We find that requiring a 3:1 nesting rule has minimal impact on the distribution of seats won. Moreover, we study the impact the chosen House map has on the distribution of nested Senate maps, and find that an extreme seat bias at the House level does not significantly impact the distribution of seats won at the Senate level.

cs.CY↗

arXiv2019

$p$-adic quotient sets II: quadratic forms

For $A \subseteq \{1,2,\ldots\}$, we consider $R(A) = \{a/a' : a,a' \in A\}$. If $A$ is the set of nonzero values assumed by a quadratic form, when is $R(A)$ dense in the $p$-adic numbers? We show that for a binary quadratic form $Q$, $R(A)$ is dense in $\mathbb{Q}_{p}$ if and only if the discriminant of $Q$ is a nonzero square in $\mathbb{Q}_{p}$, and for a quadratic form in at least three variables, $R(A)$ is always dense in $\mathbb{Q}_{p}$. This answers a question posed by several authors in 2017.

math.NT↗

arXiv2017

Numbers Represented by a Finite Set of Binary Quadratic Forms

Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than just the "trivial" 0? We address this question in the case of integral, positive- definite, reduced, binary quadratic forms. For forms of the same discriminant, we can use the structure of the underlying class group. If, however, the forms have different discriminants, we must apply class field theory.

math.NT↗