Distribution of volumes and coordination number in jammed matter: mesoscopic ensemble

We investigate the distribution of the volume and coordination number associated to each particle in a jammed packing of monodisperse hard sphere using the mesoscopic ensemble developed in Nature 453, 606 (2008). Theory predicts an exponential distribution of the orientational volumes for random close packings and random loose packings. A comparison with computer generated packings reveals deviations from the theoretical prediction in the volume distribution, which can be better modeled by a compressed exponential function. On the other hand, the average of the volumes is well reproduced by the theory leading to good predictions of the limiting densities of RCP and RLP. We discuss a more exact theory to capture the volume distribution in its entire range. The available data suggests a plausible order/disorder transition defining random close packings. Finally, we consider an extended ensemble to calculate the coordination number distribution which is shown to be of an exponential and inverse exponential form for coordinations larger and smaller than the average, respectively, in reasonable agreement with the simulated data.Comment: 20 pages, 6 figures, accepted by JSTA

The uniqueness of vertex pairs in π\pi-separable groups

Let $G$ be a finite $\pi$-separable group, where $\pi$ is a set of primes, and let $\chi$ be an irreducible complex character that is a $\pi$-lift of some $\pi$-partial character of $G$.It was proved by Cossey and Lewis that all of the vertex pairs for $\chi$ are linear and conjugate in $G$ if $2\in\pi$, but the result can fail for $2\notin\pi$. In this paper we introduce the notion of the twisted vertices in the case where $2\notin\pi$, and establish the uniqueness for linear twisted vertices under the conditions that either $\chi$ is an $\mathcal N$-lift for a $\pi$-chain $\mathcal N$ of $G$ or it has a linear Navarro vertex, thus answering a question proposed by them

Lifts of Brauer characters in characteristic two

A conjecture raised by Cossey in 2007 asserts that if $G$ is a finite $p$-solvable group and $\varphi$ is an irreducible $p$-Brauer character of $G$ with vertex $Q$, then the number of lifts of $\varphi$ is at most $|Q:Q'|$. This conjecture is now known to be true in several situations for $p$ odd, but there has been little progress for $p$ even. The main obstacle appeared in characteristic two is that all the vertex pairs of a lift are neither linear nor conjugate. In this paper we show that if $\chi$ is a lift of an irreducible $2$-Brauer character in a solvable group, then $\chi$ has a linear Navarro vertex if and only if all the vertex pairs of $\chi$ are linear, and in that case all of the twisted vertices of $\chi$ are conjugate. Our result can also be used to study other lifting problems of Brauer characters in characteristic two. As an application, we prove a weaker form of Cossey's conjecture for $p=2$ "one vertex at a time"

Design of a multiple bloom filter for distributed navigation routing

Unmanned navigation of vehicles and mobile robots can be greatly simplified by providing environmental intelligence with dispersed wireless sensors. The wireless sensors can work as active landmarks for vehicle localization and routing. However, wireless sensors are often resource scarce and require a resource-saving design. In this paper, a multiple Bloom-filter scheme is proposed to compress a global routing table for a wireless sensor. It is used as a lookup table for routing a vehicle to any destination but requires significantly less memory space and search effort. An error-expectation-based design for a multiple Bloom filter is proposed as an improvement to the conventional false-positive-rate-based design. The new design is shown to provide an equal relative error expectation for all branched paths, which ensures a better network load balance and uses less memory space. The scheme is implemented in a project for wheelchair navigation using wireless camera motes. © 2013 IEEE