arXiv subjects
Publications and source records attributed to Stephen Zhou.
We find that partisan mis\`ere quotients can have any finite cardinality other than 3, answering a question of Allen. This contrasts with impartial mis\`ere quotients, which must have even cardinality.
A game $G$ is said to have the evil twin property if there exists $G^* \in \{G,G+*\}$ such that $o^+(G) = o^-(G^*)$ and $o^+(G^*) = o^-(G)$. We study sums of wildflowers, games of form $G:H$. We find that a large closed set of sums of wildflowers has the evil twin property, extending work of McKay--Milley--Nowakowski and Lo. Our argument partially generalizes the mis\`ere genus theory of Conway to partizan games, and requires proving several general theorems on ways to extend sets with the evil twin property. Many sums of mutant flowers of the form $\{*x_1,\dots,*x_n\}:a$, where $a$ is a number, also have the evil twin property. We also prove that this set of mutant flowers is the largest such closed set with the evil twin property, and that it is $\mathsf{NP\text{-}hard}$ to compute the outcome class of a sum of mutant flowers under either play convention via a reduction from \textsc{3-Sat}. Previous work on this topic was done by McKay, Milley, and Nowakowski, and later Lo.
We study positive singular solutions of the Loewner-Nirenberg problem on conical domains and establish the existence of solutions that admit prescribed asymptotic expansions near vertices, valid to arbitrarily high order of approximation.