All Questions

Let $\mathcal{S}$ denote the set of all unit complex-valued $d$-dimensional vectors, i.e., $$ \mathcal{S} \triangleq \left\{ \mathbf{s}\in \mathbb{C}^{d} \mid \mathbf{s}^{\mathrm{H}}\mathbf{s}=1 \...

I have a point cloud (2D for now) of $N$ normally distributed points (with a certain $\sigma$). My first question would be how the pairwise distance distribution looks (just by chance I discovered a ...

Let $\mathcal{B} \subseteq \mathbb{R}^n$ be an $n$-dimensional solid body. Assume that we sample $N$ points, say $S = \{ x_1, ..., x_N \}$, from $\mathcal{B}$ uniformly at random. Consider the ...

Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$. Let $P_{\...

If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? ...

Edit (June 2015): Addressing this problem is a brief project report from the Illinois Geometry Lab (University of Illinois at Urbana-Champaign), dated May 2015, that appears here along with a foot-...