28 research outputs found
A Dense Packing of Regular Tetrahedra
We construct a dense packing of regular tetrahedra, with packing density .Comment: full color versio
Continuous Rankin Bound for Hilbert and Banach Spaces
Let be a measure space and be a normalized continuous Bessel family for a real Hilbert space
. If the diagonal is measurable in the measure space , then we
show that \begin{align} (1) \quad\quad\quad\quad \sup _{\alpha, \beta \in
\Omega, \alpha\neq \beta}\langle \tau_\alpha, \tau_\beta\rangle \geq
\frac{-(\mu\times\mu)(\Delta)}{(\mu\times\mu)((\Omega\times\Omega)\setminus\Delta)}.
\end{align} We call Inequality (1) as continuous Rankin bound. It improves 76
years old result of Rankin [\textit{Ann. of Math., 1947}]. It also answers one
of the questions asked by K. M. Krishna in the paper [Continuous Welch bounds
with applications, \textit{Commun. Korean Math. Soc., 2023}]. We also derive
Banach space version of Inequality (1).Comment: 6 Pages, 0 Figure
Archimedes' principle for Brownian liquid
We consider a family of hard core objects moving as independent Brownian
motions confined to a vessel by reflection. These are subject to gravitational
forces modeled by drifts. The stationary distribution for the process has many
interesting implications, including an illustration of the Archimedes'
principle. The analysis rests on constructing reflecting Brownian motion with
drift in a general open connected domain and studying its stationary
distribution. In dimension two we utilize known results about sphere packing.Comment: Published in at http://dx.doi.org/10.1214/11-AAP765 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org