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

On Meaningful Assessment of \u27Motivation\u27 : Steps Toward More Detailed Causal Modeling and Measurement

This study was a continuance of prior research and theory on the nature and assessment of motivation. The focus was on the academic performance situation. Existing empirical and theoretical work was utilized to formulate a complex causal model which identifies constituent elements related to motivation and performance, and which explicates their respective relationships. The theoretical model was employed to create an operational model for measurement and prediction of collegiate grade performance, a cumulative index of performance. Three main fields of theory and inquiry were incorporated in the model: Attribution theory, self-system theory and metacognition theory. Literature dealing with pertinent knowledge in these areas and their interfacings was discussed in the process of outlining the theoretical model. Measures were selected on this basis to form the operational model. Measures included indices of effort, ability, locus of control, self-esteem and study style. Regression analyses were used to determine which elements were useful as collective predictors of grade-point average. Reliability and validity were investigated for the individual measures. Finally, the operational model was investigated using the LISREL-VI program. Results indicated that some twenty-five percent of the variance in grade-point average was accounted for by the model. Most useful predictors were ability, effort and study methodology, respectively. Reliability and validity estimates were concordant with known characteristics in the literature. The operational model was found to have been mismapped onto the theoretical model initially, with the result being a model which could not be analyzed by the LISREL program due, in part, to very poor fit with the data. After reformulation, without statistical aid, the model succeeded in accounting for about ninety-one percent of the total variance in the data. The fit of the model to the data was good. Overall, while the model was well- specified in terms of internal relationships, there is need to specify additional parameters in future studies. Possibilities were discussed. Results were generally encouraging, despite observable weaknesses. These weaknesses and means of coping with them were discussed. Also, this study was placed in reference to other research and directions for future study were considered

Viscoelasticity and metastability limit in supercooled liquids

A supercooled liquid is said to have a kinetic spinodal if a temperature Tsp exists below which the liquid relaxation time exceeds the crystal nucleation time. We revisit classical nucleation theory taking into account the viscoelastic response of the liquid to the formation of crystal nuclei and find that the kinetic spinodal is strongly influenced by elastic effects. We introduce a dimensionless parameter \lambda, which is essentially the ratio between the infinite frequency shear modulus and the enthalpy of fusion of the crystal. In systems where \lambda is larger than a critical value \lambda_c the metastability limit is totally suppressed, independently of the surface tension. On the other hand, if \lambda < \lambda_c a kinetic spinodal is present and the time needed to experimentally observe it scales as exp[\omega/(\lambda_c-\lambda)^2], where \omega is roughly the ratio between surface tension and enthalpy of fusion

A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators

We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called phi-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the Matlab function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.Comment: 20 pages, 3 colour figures, code available from http://www.maths.leeds.ac.uk/~jitse/software.html . v2: Various changes to improve presentation as suggested by the refere

Homogeneous bubble nucleation limit of mercury under the normal working conditions of the planned European Spallation Source

In spallation neutron sources, liquid mercury is the subject of big thermal and pressure shocks, upon adsorbing the proton beam. These changes can cause unstable bubbles in the liquid, which can damage the structural material. While there are methods to deal with the pressure shock, the local temperature shock cannot be avoided. In our paper we calculated the work of the critical cluster formation (i.e. for mercury micro-bubbles) together with the rate of their formation (nucleation rate). It is shown that the homogeneous nucleation rates are very low even after adsorbing several proton pulses, therefore the probability of temperature induced homogeneous bubble nucleation is negligible.Comment: 22 Pages, 11 figures, one of them is colour, we plan to publish it in Eur. Phys. J.

Capillary pressure of van der Waals liquid nanodrops

The dependence of the surface tension on a nanodrop radius is important for the new-phase formation process. It is demonstrated that the famous Tolman formula is not unique and the size-dependence of the surface tension can distinct for different systems. The analysis is based on a relationship between the surface tension and disjoining pressure in nanodrops. It is shown that the van der Waals interactions do not affect the new-phase formation thermodynamics since the effect of the disjoining pressure and size-dependent component of the surface tension cancel each other.Comment: The paper is dedicated to the 80th anniversary of A.I. Rusano

Constrained Dynamics of Universally Coupled Massive Spin 2-spin 0 Gravities

The 2-parameter family of massive variants of Einstein's gravity (on a Minkowski background) found by Ogievetsky and Polubarinov by excluding lower spins can also be derived using universal coupling. A Dirac-Bergmann constrained dynamics analysis seems not to have been presented for these theories, the Freund-Maheshwari-Schonberg special case, or any other massive gravity beyond the linear level treated by Marzban, Whiting and van Dam. Here the Dirac-Bergmann apparatus is applied to these theories. A few remarks are made on the question of positive energy. Being bimetric, massive gravities have a causality puzzle, but it appears soluble by the introduction and judicious use of gauge freedom.Comment: 6 pages; Talk given at QG05, Cala Gonone (Italy), September 200

Actin3 promoter reveals undulating F-actin bundles at shanks and dynamic F-actin meshworks at tips of tip-growing pollen tubes

The dynamic actin cytoskeleton of pollen tubes is both the driver of the tip growth and the organizer of cell polarity. In order to understand this fast re-arranging cytoskeletal system, we need reliable constructs expressed under relevant promoters. Here we are reporting that the Lifeact reporter, expressed under the pollen-specific Actin3 promoter, visualizes very dynamic F-actin elements both in germinating pollen grains and tip-growing pollen tubes. Importantly, we have documented very active actin polymerization at the cell periphery, especially in the bulging area during pollen germination and in the apical clear zone. Expression of the Lifeact reporter under control of the pollen-specific Actin3 promoter revealed 2 new aspects: (i) long F-actin bundles in pollen tube shanks are dynamic, showing undulating movements, (ii) subapical ‘actin collars’ or ‘fringes’ are absent