A Zariski Topology for Modules

Given a duo module $M$ over an associative (not necessarily commutative) ring $R,$ a Zariski topology is defined on the spectrum $\mathrm{Spec}^{\mathrm{fp}}(M)$ of {\it fully prime} $R$-submodules of $M$. We investigate, in particular, the interplay between the properties of this space and the algebraic properties of the module under consideration.Comment: 22 pages; submitte

Proportions of r-regular elements in finite classical groups

For a prime $r$, we obtain lower bounds on the proportion of $r$-regular elements in classical groups and show that these lower bounds are the best possible lower bounds that do not depend on the order of the defining field. Along the way, we also provide new upper bounds and answer some open questions of the first author, P\'{a}lfy and Saxl.Comment: 22 page

Asymptotic properties of the solutions of a differential equation appearing in QCD

We establish the asymptotic behaviour of the ratio $h^\prime(0)/h(0)$ for $\lambda\rightarrow\infty$, where $h(r)$ is a solution, vanishing at infinity, of the differential equation $h^{\prime\prime}(r) = i\lambda \omega (r) h(r)$ on the domain $0 \leq r <\infty$ and $\omega (r) = (1-\sqrt{r} K_1(\sqrt{r}))/r$. Some results are valid for more general $\omega$'s.Comment: 6 pages, late

Robust nonparametric quantification of clustering density of molecules in single-molecule localization microscopy

We report a robust nonparametric descriptor,&nbsp;J&prime;(r), for quantifying the density of clustering molecules in single-molecule localization microscopy.&nbsp;J&prime;(r), based on nearest neighbor distribution functions, does not require any parameter as an input for analyzing point patterns. We show that&nbsp;J&prime;(r) displays a valley shape in the presence of clusters of molecules, and the characteristics of the valley reliably report the clustering features in the data. Most importantly, the position of the&nbsp;J&prime;(r) valley () depends exclusively on the density of clustering molecules (&rho;c). Therefore, it is ideal for direct estimation of the clustering density of molecules in single-molecule localization microscopy. As an example, this descriptor was applied to estimate the clustering density of&nbsp;ptsG&nbsp;mRNA in&nbsp;E. coli&nbsp;bacteria.</p