The Kink Phenomenon in Fejér and Clenshaw-Curtis Quadrature

The Fejér and Clenshaw-Curtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N, (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critical value, the error behaves like $O(\varrho^{-2N})$, where $\varrho$ is a constant greater than 1. For these values of N the accuracy of both the Fejér and Clenshaw-Curtis rules is almost indistinguishable from that of the more celebrated Gauss-Legendre quadrature rule. For larger N, however, the error decreases at the rate $O(\varrho^{-N})$, i.e., only half as fast as before. Convergence curves typically display a kink where the convergence rate cuts in half. In this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon.\ud \ud This work was supported by the Royal Society of the UK and the National Research Foundation of South Africa under the South Africa-UK Science Network Scheme. The first author also acknowledges grant FA2005032300018 of the NRF

X-Ray Absorption from the Milky Way Halo and the Local Group

Million degree gas is present at near-zero redshift and is due either to a gaseous Galactic Halo or a more diffuse but very massive Local Group medium. We can discriminate between these models because the column densities should depend on location in the sky, either relative to the Galaxy bulge or to the M31-Milky Way axis. To search for these signatures, we measured the OVII Kalpha absorption line strength toward 25 bright AGNs, plus LMC X-3, using XMM-Newton RGS archival data. The data are in conflict with a purely Local Group model, but support the Galactic Halo model. The strongest correlation is between the OVII equivalent widths and the ROSAT background emission measurement in the R45 band (0.4-1 keV), for which OVII emission makes the largest single contribution. This suggests that much of the OVII emission and absorption are cospatial, from which the radius of a uniform halo appears to lie the range 15-110 kpc. The present data do not constrain the type of halo gas model and an equally good fit is obtained in a model where the gas density decreases as a power-law, such as r^(-3/2). For a uniform halo with a radius of 20 kpc, the electron density would be 9E-4 cm^(-3), and the gas mass is 4E8 Msolar. The redshift of the four highest S/N OVII measurements is consistent with a Milky Way origin rather than a Local Group origin.Comment: 32 pages (14 figures); ApJ, in pres

Parabolic and Hyperbolic Contours for Computing the Bromwich Integral

Some of the most effective methods for the numerical inversion of the Laplace transform are based on the approximation of the Bromwich contour integral. The accuracy of these methods often hinges on a good choice of contour, and several such contours have been proposed in the literature. Here we analyze two recently proposed contours, namely a parabola and a hyperbola. Using a representative model problem, we determine estimates for the optimal parameters that define these contours. An application to a fractional diffusion equation is presented.\ud \ud JACW was supported by the National Research Foundation in South Africa under grant FA200503230001

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

Talbot quadratures and rational approximations

Many computational problems can be solved with the aid of contour integrals containing $e^z$ in the the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reaction-diffusion equations. One approach to the numerical quadrature of such integrals is to apply the trapezoid rule on a Hankel contour defined by a suitable change of variables. Optimal parameters for three classes of such contours have recently been derived: (a) parabolas, (b) hyperbolas, and (c) cotangent contours, following Talbot in 1979. The convergence rates for these optimized quadrature formulas are very fast: roughly $O(3^{-N})$, where $N$ is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, $O(9.28903^{-N})$, can be achieved by using a different approach: best supremum-norm rational approximants to $e^z$ for $z\in (-\infty,0]$, following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of self-adjoint operators or real integrands.) It is shown that the quadrature formulas can be interpreted as rational approximations and the rational approximations as quadrature formulas, and the strengths and weaknesses of the different approaches are discussed in the light of these connections. A MATLAB function is provided for computing Cody--Meinardus--Varga approximants by the method of Carathèodory-Fejèr approximation

VLT observations of the asymmetric Etched Hourglass Nebula, MyCn 18

Context. The mechanisms that form extreme bipolar planetary nebulae remain unclear. Aims. The physical properties, structure, and dynamics of the bipolar planetary nebula, MyCn 18, are investigated in detail with the aim of understanding the shaping mechanism and evolutionary history of this object. Methods. VLT infrared images, VLT ISAAC infrared spectra, and long-slit optical Echelle spectra are used to investigate MyCn 18. Morpho-kinematic modelling was used to firmly constrain the structure and kinematics of the source. A timescale analysis was used to determine the kinematical age of the nebula and its main components. Results. A spectroscopic study of MyCn 18's central and offset region reveals the detailed make-up of its nebular composition. Molecular hydrogen, atomic helium, and Bracket gamma emission are detected from the central regions of MyCn 18. ISAAC spectra from a slit position along the narrow waist of the nebula demonstrate that the ionised gas resides closer to the centre of the nebula than the molecular emission. A kinematical age of the nebula and its components were obtained by the P-V arrays and timescale analysis. Conclusions. The structure and kinematics of MyCn 18 are better understood using an interactive 3-D modelling tool called shape. A dimensional and timescale analysis of MyCn 18's major components provides a possible mechanism for the nebula's asymmetry. The putative central star is somewhat offset from the geometric centre of the nebula, which is thought to be the result of a binary system. We speculate that the engulfing and destruction of an exoplanet during the AGB phase may have been a key event in shaping MyCn 18 and generating of its hypersonic knotty outflow.Comment: 15 pages, 3 tables, 13 figures. Accepted for publication by A&

Entanglement in a Valence-Bond-Solid State

We study entanglement in Valence-Bond-Solid state. It describes the ground state of Affleck, Kennedy, Lieb and Tasaki quantum spin chain. The AKLT model has a gap and open boundary conditions. We calculate an entropy of a subsystem (continuous block of spins). It quantifies the entanglement of this block with the rest of the ground state. We prove that the entanglement approaches a constant value exponentially fast as the size of the subsystem increases. Actually we proved that the density matrix of the continuous block of spins depends only on the length of the block, but not on the total size of the chain [distance to the ends also not essential]. We also study reduced density matrices of two spins both in the bulk and on the boundary. We evaluated concurrencies.Comment: 4pages, no figure

Heat of Adsorption on Charcoal of Certain Organic Vapors

Previous work on the heat of adsorption has been carried out by the use of the ice calorimeter adapted only to those liquids of appreciable vapor pressure at 0°C., and affording no investigation of the effect of temperature upon the heat of adsorption. In the present paper there is described a method for studying the heat of adsorption at different temperatures by employing a sensitive thermo-couple in a calorimeter system consisting of a known weight of low specific heat oil in a Dewar flask. With this arrangement we find that a change of 1 microvolt in the thermocouple reading corresponds to 0.06 calories per gram of charcoal

Wave packet pseudomodes of twisted Toeplitz matrices

The pseudospectra of nonsymmetric Toeplitz or circulant matrices with varying coefficients are considered. Such matrices are characterized by a symbol that depends on both position (x) and wave number (k). It is shown that when a certain winding number or twist condition is satisfied, analogous to Hörmander's commutator condition for partial differential equations, \varepsilon-pseudoeigenvectors of such matrices for exponentially small values of \varepsilon exist in the form of localized wave packets. The symbol need not be smooth, just differentiable at a point (or less)

Light-addressable liquid crystal polymer dispersed liquid crystal

Scattering-free liquid crystal polymer-dispersed liquid crystal polymer (LCPDLC) films are fabricated by combining a room temperature polymerizable liquid crystal (LC) monomer with a mesogenic photosensitive LC. The morphological and photosensitive properties of the system are analysed with polarized optical microscopy and high resolution scanning and transmission electron microscopy. A two-phase morphology comprised of oriented fibril-like polymeric structures interwoven with nanoscale domains of phase separated LC exists. The nanoscale of the structures enables an absence of scattering which allows imaging through the LCPDLC sample without optical distortion. The use of a mesogenic monomer enables much smaller phase separated domains as compared to nonmesogenic systems. All-optical experiments show that the transmitted intensity, measured through parallel polarizers, can be modulated by the low power density radiation (31 mW/cm2) of a suitable wavelength (532 nm). The reversible and repeatable transmission change is due to the photoinduced trans-cis photoisomerization process. The birefringence variation (0.01) obtained by optically pumping the LCPDLC films allow their use as an alloptical phase modulato