arXiv subjects
Publications and source records attributed to Jasha Sommer-Simpson.
Every semidirect product of groups $K\rtimes H$ has size $\left | K \right | \cdot \left | H\right |$, yet the size of such a group's automorphism group varies with the chosen action of $H$ on $K$. This paper will explore groups of the form $\text{Aut}(K\rtimes H)$, considering especially the case where $H$ and $K$ are cyclic. Only finite groups will be considered.
Given a density $f$ on the non-negative real line, Dümbgen's algorithm is a routine for finding the (unique) log-convex, non-decreasing function $\hatϕ$ such that $\int\hatϕ(x)f(x)dx=1$ and such that the likelihood $\prod_{i=1}^{n}f(x_i)\hatϕ(x_i)$ of given data $x_1,\ldots,x_n$ under density $x\mapsto \hatϕ(x)f(x)$ is maximized. We summarize Dümbgen's algorithm for finding this MLE $\hatϕ$, and we present a novel guarantee of the algorithm's termination and convergence.
Given groupoids $\mathscr G$ and $\mathscr H$ as well as an isomorphism $Ψ:\text{Sd}\,\mathscr G\cong\text{Sd}\,\mathscr H$ between subdivisions, we construct an isomorphism $P:\mathscr G\cong\mathscr H$. If $Ψ$ equals $\text{Sd} F$ for some functor $F$, then the constructed isomorphism $P$ is equal to $F$. It follows that the restriction of $\text{Sd}$ to the category of groupoids is conservative. These results do not hold for arbitrary categories.