Large-scale computation of pseudospectra using ARPACK and eigs

ARPACK and its MATLAB counterpart, eigs, are software packages that calculate some eigenvalues of a large non-symmetric matrix by Arnoldi iteration with implicit restarts. We show that at a small additional cost, which diminishes relatively as the matrix dimension increases, good estimates of pseudospectra in addition to eigenvalues can be obtained as a by-product. Thus in large-scale eigenvalue calculations it is feasible to obtain routinely not just eigenvalue approximations, but also information as to whether or not the eigenvalues are likely to be physically significant. Examples are presented for matrices with dimension up to 200,000

Computing Lyapunov constants for random recurrences with smooth coefficients

In recent years, there has been much interest in the growth and decay rates (Lyapunov constants) of solutions to random recurrences such as the random Fibonacci sequence $x_{n+1} = \pm x_{n} \pm x_{n-1}$. Many of these problems involve non-smooth dynamics (nondifferentiable invariant measures), making computations hard. Here, however, we consider recurrences with smooth random coefficients and smooth invariant measures. By computing discretised invariant measures and applying Richardson extrapolation, we can compute Lyapunov constants to ten digits of accuracy. In particular, solutions to the recurrence $x_{n+1} = x_{n} + c_{n+1} x_{n-1}$, where the $\{c_{n}\}$ are independent standard normal variables, increase exponentially (almost surely) at the asymptotic rate $(1.0574735537...)^{n}$. Solutions to the related recurrences $x_{n+1} = c_{n+1}x_{n} + x_{n-1}$, and $x_{n+1} = c_{n+1}x_{n} + d_{n+1}x_{n-1}$ (where the $\{d_{n}\}$ are also independent standard normal variables) increase (decrease) at the rates $(1.1149200917...)^{n}$ and $(0.9949018837...)^(n)$ respectively

Five limit cycles for a simple cubic system

We resolve the centre-focus problem for a specific class of cubic systems and determine the number of limit cycles which can bifurcate from a fine focus. We also describe the methods which we have developed to investigate these questions in general. These involve extensive use of Computer Algebra; we have chosen to use REDUCE

Eigenvalues and Pseudospectra of Rectangular Matrices

Pseudospectra of rectangular matrices vary continuously with the matrix entries, a feature that eigenvalues of rectangular matrices do not have. Some properties of eigenvalues and pseudospectra of rectangular matrices are explored, and an efficient algorithm for the computation of pseudospectra is given. Applications are given in (square) eigenvalue computation (Lanczos and Arnoldi iteration), square pseudospectra approximation, control theory (nearest uncontrollable system) and game theory

High Angular Resolution Stellar Imaging with Occultations from the Cassini Spacecraft II: Kronocyclic Tomography

We present an advance in the use of Cassini observations of stellar occultations by the rings of Saturn for stellar studies. Stewart et al. (2013) demonstrated the potential use of such observations for measuring stellar angular diameters. Here, we use these same observations, and tomographic imaging reconstruction techniques, to produce two dimensional images of complex stellar systems. We detail the determination of the basic observational reference frame. A technique for recovering model-independent brightness profiles for data from each occulting edge is discussed, along with the tomographic combination of these profiles to build an image of the source star. Finally we demonstrate the technique with recovered images of the {\alpha} Centauri binary system and the circumstellar environment of the evolved late-type giant star, Mira.Comment: 8 pages, 8 figures, Accepted by MNRA

The Chemical Compositions of the Type II Cepheids -- The BL Her and W Vir Variables

Abundance analyses from high-resolution optical spectra are presented for 19 Type II Cepheids in the Galactic field. The sample includes both short-period (BL Her) and long-period (W Vir) stars. This is the first extensive abundance analysis of these variables. The C, N, and O abundances with similar spreads for the BL Her and W Vir show evidence for an atmosphere contaminated with $3\alpha$-process and CN-cycling products. A notable anomaly of the BL Her stars is an overabundance of Na by a factor of about five relative to their presumed initial abundances. This overabundance is not seen in the W Vir stars. The abundance anomalies running from mild to extreme in W Vir stars but not seen in the BL Her stars are attributed to dust-gas separation that provides an atmosphere deficient in elements of high condensation temperature, notably Al, Ca, Sc, Ti, and $s$-process elements. Such anomalies have previously been seen among RV Tau stars which represent a long-period extension of the variability enjoyed by the Type II Cepheids. Comments are offered on how the contrasting abundance anomalies of BL Her and W Vir stars may be explained in terms of the stars' evolution from the blue horizontal branch.Comment: 41 pages including 11 figures and 4 tables; Accepted for publication in Ap

Enhanced spectral histology in the colon using high-magnification benchtop FTIR imaging

ArticleLabel-free imaging of cells and tissues is a promising tool to study the molecular alterations for improved cancer diagnosis. In this regard, the vibrational spectroscopic method of FTIR imaging has been employed to study the histo-pathological features and the alterations were correlated to a disease state. The aim of this study was to see if higher magnification images resulting in pixel sizes smaller than the diffraction limit of mid-IR wavelengths with two NA objectives could provide cellular and subcellular resolution of key diagnostic features. To this effect, FTIR spectroscopic imaging has been carried out directly on paraffinized colon tissue sections using a benchtop imaging system consisting of two different IR objectives with a NA of 0.81 and 0.62. The high NA objective (0.81) at a high magnification (63×) provided a pixel size of 0.63 × 0.63 μm2 in comparison to the standard magnification (12×) where the pixel size is 3.3 × 3.3 μm2. The second objective (0.62 NA) at a high magnification (36×) provided a pixel size of 1.1 × 1.1 μm2 in comparison to the standard magnification (7×) pixel size of 5.5 × 5.5 μm2. The spectral images were corrected for paraffin and other spectral interferences using a modified EMSC algorithm and subjected to cluster analysis in order to appreciate the histopathological details. Comparison of the IR cluster analysis results to adjacent haematoxylin and eosin stained tissue sections, which were used as the morphological controls, revealed various levels of histological and cellular organization. Together with large scale features such as the glandular and connective tissue regions, small scale features like goblet cells, interfaces between tissue types especially the peri-cryptal fibroblastic sheath were observed. Noticeable differences between the intracellular mucin and secreted mucin were also observed. However, it appears that high power mid-IR radiation sources would be beneficial, especially when measuring in rapid clinically relevant timescales.This work is part of the MINERVA project (http://www.minerva.eu) which is supported by the European Union’s Seventh Framework Programme (FP7) for research, technological development and demonstration under grant agreement no. 317803

Opposed Jet Burner Extinction Limits: Simple Mixed Hydrocarbon Scramjet Fuels vs Air

Opposed Jet Burner tools have been used extensively by the authors to measure Flame Strength (FS) of laminar non-premixed H2 air and simple hydrocarbon (HC) air counterflow diffusion flames at 1-atm. FS represents a strain-induced extinction limit based on air jet velocity. This paper follows AIAA-2006-5223, and provides new HC air FSs for global testing of chemical kinetics, and for characterizing idealized flameholding potentials during early scramjet-like combustion. Previous FS data included six HCs, pure and N2-diluted; and three HC-diluted H2 fuels, where FS decayed very nonlinearly as HC was added to H2, due to H-atom scavenging. This study presents FSs on mixtures of (candidate surrogate) HCs, some with very high FS ethylene. Included are four binary gaseous systems at 300 K, and a hot ternary system at approx. 600 K. The binaries are methane + ethylene, ethane + ethylene, methane + ethane, and methane + propylene. The first three also form two ternary systems. The hot ternary includes both 10.8 and 21.3 mole % vaporized n-heptane and full ranges of methane + ethylene. Normalized FS data provide accurate means of (1) validating, globally, chemical kinetics for extinction of non-premixed flames, and (2) estimating (scaling by HC) the loss of incipient flameholding in scramjet combustors. The n-heptane is part of a proposed baseline simulant (10 mole % with 30% methane + 60% ethylene) that mimics the ignition of endothermically cracked JP-7 like kerosene fuel, as suggested by Colket and Spadaccini in 2001 in their shock tube Scramjet Fuels Autoignition Study. Presently, we use FS to gauge idealized flameholding, and define HC surrogates. First, FS was characterized for hot nheptane + methane + ethylene; then a hot 36 mole % methane + 64% ethylene surrogate was defined that mimics FS of the baseline simulant system. A similar hot ethane + ethylene surrogate can also be defined, but it has lower vapor pressure at 300 K, and thus exhibits reduced gaseous capacity. The new FS results refine our earlier idealized reactivity scale that shows wide ranging (50 x) diameter-normalized FSs for various HCs. These range from JP-10 and methane to H2 air, which produces an exceptionally strong flame that agrees within approx. 1% of recent 2-D numerically simulations. Finally, we continue advocating the FS approach as more direct and fundamental, for assessing idealized scramjet flameholding potentials, than measurements of unstrained laminar burning velocity or blowout in a Perfectly Stirred Reactor

Exploring the utility of grids for analysing long term population change

This paper details an innovative approach which enables the analysis of small area population change across four decades. Population surfaces are generated using small area data (enumeration districts or output areas) for each Census from 1971 to 2011 inclusive. The paper details the methods used in the creation of these surfaces, and discusses the rationale behind this approach, arguing that grids represent the most appropriate model for assessing population distributions. Methods for grid creation are tested using pre-existing population grids for Northern Ireland as a benchmark. The method developed is then applied to create population grids for the rest of the UK for 1971, 1981, 1991, 2001 and 2011. The changing population structures of small areas across these five time points are explored here to illustrate the value of this approach. The publically-available data resource – the final product of the ‘PopChange’ project – will facilitate exploration of long-term changes in populations over small areas. The paper argues that maximum advantage could be taken of the ‘big data revolution’ if such data were gridded in a similar way, allowing them to be placed in a longer-term historical context, using tools made available through the PopChange project