Fourier Transform of the Stretched Exponential Function: Analytic Error Bounds, Double Exponential Transform, and Open-Source Implementation libkww

The C library \texttt{libkww} provides functions to compute the Kohlrausch-Williams-Watts function, i.e.\ the Laplace-Fourier transform of the stretched (or compressed) exponential function $\exp(-t^\beta)$ for exponents $\beta$ between 0.1 and 1.9 with sixteen-digits accuracy. Analytic error bounds are derived for the low and high frequency series expansions. For intermediate frequencies the numeric integration is enormously accelerated by using the Ooura-Mori double exponential transformation. The source code is available from the project home page \url{http://apps.jcns.fz-juelich.de/doku/sc/kww}.Comment: Version 3. 11 pages, 4 figures. Describes software version 2.

Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation

A Panel Data Study of the Determinants of Life Expectancy in Low Income Countries

This study attempts to determine the impact of several socioeconomic determinants of life expectancy for 34 low income countries using ordinary least squares linear regression. Most explanatory variables were statistically significant, implying that the socioeconomic variables of interest, including government health expenditures, access to basic sanitation facilities, HIV prevalence, urbanization, education, and sex, are important measures in influencing life expectancy. Foreign aid, corruption, and undernourishment, were determined insignificant when determining life expectancy. Based on the analysis results, it has been suggested that these developing countries implement appropriate policies and programs to increase HIV education and preventative measures, increase women’s rights and labor force participation, and specifically direct foreign aid inflows, in order to increase the life expectancy of people in the country

Probabilistic load flow in systems with high wind power penetration

This paper proposes a method for solving a probabilistic load flows that takes into account the uncertainties of wind generation, but also of load and conventional systems. The method uses a combination of methods including cumulant, point estimate and convolution. Cornish Fisher expansion series are also used to find the CDF. The method is of especial application to estimate active power flows through lines

Research Achievements Review Series no. 20 - Mathematics and computation research

Computational mathematics, perturbed orbit three-body problem, and periodic trajectories solutions through computer method

Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Poschl-Teller-Ginocchio potential wave functions

The fast computation of the Gauss hypergeometric function 2F1 with all its parameters complex is a difficult task. Although the 2F1 function verifies numerous analytical properties involving power series expansions whose implementation is apparently immediate, their use is thwarted by instabilities induced by cancellations between very large terms. Furthermore, small areas of the complex plane are inaccessible using only 2F1 power series formulas, thus rendering 2F1 evaluations impossible on a purely analytical basis. In order to solve these problems, a generalization of R.C. Forrey's transformation theory has been developed. The latter has been successful in treating the 2F1 function with real parameters. As in real case transformation theory, the large canceling terms occurring in 2F1 analytical formulas are rigorously dealt with, but by way of a new method, directly applicable to the complex plane. Taylor series expansions are employed to enter complex areas outside the domain of validity of power series analytical formulas. The proposed algorithm, however, becomes unstable in general when |a|,|b|,|c| are moderate or large. As a physical application, the calculation of the wave functions of the analytical Poschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.Comment: 29 pages; accepted in Computer Physics Communication