On the bridge number of knot diagrams with minimal crossings

Given a diagram $D$ of a knot $K$, we consider the number $c(D)$ of crossings and the number $b(D)$ of overpasses of $D$. We show that, if $D$ is a diagram of a nontrivial knot $K$ whose number $c(D)$ of crossings is minimal, then $1+\sqrt{1+c(D)} \leq b(D)\leq c(D)$. These inequalities are shape in the sense that the upper bound of $b(D)$ is achieved by alternating knots and the lower bound of $b(D)$ is achieved by torus knots. The second inequality becomes an equality only when the knot is an alternating knot. We prove that the first inequality becomes an equality only when the knot is a torus knot.Comment: 18 pages, 7 figure

Silicon nanowire devices

Transport measurements were carried out on 15–35 nm diameter silicon nanowires grown using SiH4 chemical vapor deposition via Au or Zn particle-nucleated vapor-liquid-solid growth at 440°C. Both Al and Ti/Au contacts to the wires were investigated. The wires, as produced, were essentially intrinsic, although Au nucleated wires exhibited a slightly higher conductance. Thermal treatment of the fabricated devices resulted in better electrical contacts, as well as diffusion of dopant atoms into the nanowires, and increased the nanowire conductance by as much as 10^4. Three terminal devices indicate that the doping of the wires is p type

A Proposed Scale-Dependent Cosmology for the Inhomogeneous Cosmology

We propose a scale-dependent cosmology in which the Robertson--Walker metric and the Einstein equation are modified in such a way that $\Omega_0$, $H_0$ and the age of the Universe all become scale-dependent. Its implications on the observational cosmology and possible modifications of the standard Friedmann cosmology are discussed. For example, the age of the Universe in this model is longer than that of the standard model.Comment: 28 pages, RevTex, no figures, to be run twic