Riemannian metric and geometric mean for positive semidefinite matrices of fixed rank

This paper introduces a new metric and mean on the set of positive semidefinite matrices of fixed-rank. The proposed metric is derived from a well-chosen Riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. The resulting Riemannian space has strong geometrical properties: it is geodesically complete, and the metric is invariant with respect to all transformations that preserve angles (orthogonal transformations, scalings, and pseudoinversion). A meaningful approximation of the associated Riemannian distance is proposed, that can be efficiently numerically computed via a simple algorithm based on SVD. The induced mean preserves the rank, possesses the most desirable characteristics of a geometric mean, and is easy to compute. © 2009 Society for Industrial and Applied Mathematics.

Updated correlation between aircraft smoke number and black carbon concentration

Aircraft emissions of black carbon (BC) contribute to anthropogenic climate forcing and degrade air quality. The smoke number (SN) is the current regulatory measure of aircraft particulate matter emissions and quantifies exhaust plume visibility. Several correlations between SN and the exhaust mass concentration of BC (CBC) have been developed, based on measurements relevant to older aircraft engines. These form the basis of the current standard method used to estimate aircraft BC emissions (First Order Approximation version 3 [FOA3]) for the purposes of environmental impact analyses. In this study, BC with a geometric mean diameter (GMD) of 20, 30, and 60 nm and filter diameters of 19 and 35 mm are used to investigate the effect of particle size and sampling variability on SN measurements. For BC with 20 and 30 nm GMD, corresponding to BC emitted by modern aircraft engines, a smaller SN results from a given CBC than is the case for BC with 60 nm GMD, which is more typical of older engines. An updated correlation between CBC and SNthat accounts for typical size of BC emitted by modern aircraft is proposed. An uncertainty of ±25% accounts for variation in GMD in the range 20-30 nm and for the range of filter diameters. The SN-CBC correlation currently used in FOA3 underestimates by a factor of 2.5-3 for SN <15, implying that current estimates of aircraft BC emissions derived from SN are underestimated by the same factor. Copyright © American Association for Aerosol Research.

Rank-preserving geometric means of positive semi-definite matrices

The generalization of the geometric mean of positive scalars to positive definite matrices has attracted considerable attention since the seminal work of Ando. The paper generalizes this framework of matrix means by proposing the definition of a rank-preserving mean for two or an arbitrary number of positive semi-definite matrices of fixed rank. The proposed mean is shown to be geometric in that it satisfies all the expected properties of a rank-preserving geometric mean. The work is motivated by operations on low-rank approximations of positive definite matrices in high-dimensional spaces.© 2012 Elsevier Inc. All rights reserved.

Diameter and wall number control of carbon nanotubes by chemical vapor deposition

We analyze the relationship between the average wall number (N) and the diameter (d) for carbon nanotubes (CNTs) grown by chemical vapour deposition. It is found that N depends linearly on d for diameters in the range of 2.5-10 nm, while single wall nanotubes predominate for diameters under about 2.1 nm. The linear relationship is found to depend somewhat on the growth conditions. It is also verified that the mean diameter depends on the diameter of the originating catalyst nanoparticle, and thus on the initial catalyst thickness where a thin film catalyst is used. This simplifies the characterisation of CNTs by electron microscopy. We also find a linear relationship between nanotube diameter and initial catalyst film thickness. © 2013 AIP Publishing LLC.

THE SENSITIVITY OF 2D COMPRESSOR INCIDENCE RANGE TO IN-SERVICE GEOMETRIC VARIATION

The loss mechanisms which control 2D incidence range are discussed with an emphasis on determining which real in-service geometric variations will have the largest impact. For the majority of engine compressor blades (Minlet>0.55) both the negative and positive incidence limits are controlled by supersonic patches. It is shown that these patches are highly sensitive to the geometric variations close to, and around the leading edge. The variations used in this study were measured from newly manufactured as well as ex-service blades. Over most the high pressure compressor considered, it was shown that manufacture variations dominated. The first part of the paper shows that, despite large geometric variations (~10% of leading edge thickness), the incidence range responded in a linear way. The result of this is that the geometric variations have little effect on the mean incidence range of a row of blades. In the second part of the paper a region of the design space is identified where non-linear behavior can result in a 10% reduction in positive incidence range. The mechanism for this is reported and design guidelines for its avoidance offered. In the final part of the paper, the linear behavior at negative incidence and the transonic nature of the flow is exploited to design a robust asymmetric leading edge with a 5% increase in incidence range.