Lower bound for the Perron-Frobenius degrees of Perron numbers

Using an idea of Doug Lind, we give a lower bound for the Perron-Frobenius degree of a Perron number that is not totally-real. As an application, we prove that there are cubic Perron numbers whose Perron-Frobenius degrees are arbitrary large; a result known to Lind, McMullen and Thurston. A similar result is proved for biPerron numbers.Comment: To appear in Ergodic Theory and Dynamical Systems, 15 pages, 4 figure

On Thurston's Euler class one conjecture

In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a negative answer to Thurston's conjecture. Here counterexamples have been constructed conditional on the fully marked surface theorem. In the second paper, joint with David Gabai, a proof of the fully marked surface theorem is given.Comment: 42 pages, 21 figures. The paper is split into two parts, and the appendix is appearing as a separate article joint with David Gabai. The results on taut foliations on sutured solid tori are generalised. A section on relative Euler class is added to address a possible oversight in the literature. Exposition is improved, and new open questions are raised. Final version to appear in Acta Mathematic

High-altitude reconnaissance aircraft

At the equator the ozone layer ranges from 65,000 to 130,000+ ft, which is beyond the capabilities of the ER-2, NASA's current high-altitude reconnaissance aircraft. This project is geared to designing an aircraft that can study the ozone layer. The aircraft must be able to satisfy four mission profiles. The first is a polar mission that ranges from Chile to the South Pole and back to Chile, a total range of 6000 n.m. at 100,000 ft with a 2500-lb payload. The second mission is also a polar mission with a decreased altitude and an increased payload. For the third mission, the aircraft will take off at NASA Ames, cruise at 100,000 ft, and land in Chile. The final mission requires the aircraft to make an excursion to 120,000 ft. All four missions require that a subsonic Mach number be maintained because of constraints imposed by the air sampling equipment. Three aircraft configurations were determined to be the most suitable for meeting the requirements. The performance of each is analyzed to investigate the feasibility of the mission requirements

Outage Analysis of Uplink Two-tier Networks

Employing multi-tier networks is among the most promising approaches to address the rapid growth of the data demand in cellular networks. In this paper, we study a two-tier uplink cellular network consisting of femtocells and a macrocell. Femto base stations, and femto and macro users are assumed to be spatially deployed based on independent Poisson point processes. We consider an open access assignment policy, where each macro user based on the ratio between its distances from its nearest femto access point (FAP) and from the macro base station (MBS) is assigned to either of them. By tuning the threshold, this policy allows controlling the coverage areas of FAPs. For a fixed threshold, femtocells coverage areas depend on their distances from the MBS; Those closest to the fringes will have the largest coverage areas. Under this open-access policy, ignoring the additive noise, we derive analytical upper and lower bounds on the outage probabilities of femto users and macro users that are subject to fading and path loss. We also study the effect of the distance from the MBS on the outage probability experienced by the users of a femtocell. In all cases, our simulation results comply with our analytical bounds

Towards Spectral Geometry for Causal Sets

We show that the Feynman propagator (or the d'Alembertian) of a causal set contains the complete information about the causal set. Intuitively, this is because the Feynman propagator, being a correlator that decays with distance, provides a measure for the invariant distance between pairs of events. Further, we show that even the spectra alone (of the self-adjoint and anti-self-adjoint parts) of the propagator(s) and d'Alembertian already carry large amounts of geometric information about their causal set. This geometric information is basis independent and also gauge invariant in the sense that it is relabeling invariant (which is analogue to diffeomorphism invariance). We provide numerical evidence that the associated spectral distance between causal sets can serve as a measure for the geometric similarity between causal sets.Comment: 15 pages, 8 figures. v2: Minor edits and additions, references added, discussion added on distinguishing manifoldlike causal sets from non-manifoldlike causal sets, comments added on the extension of results to 4D and on spectral dimensio