Department of Mathematics and Statistics

On the Fourier expansion method for highly accurate computation of the Voigt/complex error function in a rapid algorithm

2013-05-08T17:56:16Z

On the Fourier expansion method for highly accurate computation of the Voigt/complex error function in a rapid algorithm S. M. Abrarov, B. M. Quine In our recent publication [1] we presented an exponential series approximation suitable for highly accurate computation of the complex error function in a rapid algorithm. In this Short Communication we describe how a simplified representation of the proposed complex error function approximation makes possible further algorithmic optimization resulting in a considerable computational acceleration without compromise on accuracy.

Efficient algorithmic implementation of the Voigt/complex error function based on exponential series approximation

2013-05-08T17:03:27Z

Efficient algorithmic implementation of the Voigt/complex error function based on exponential series approximation Abrarov, S. M.; Quine, B. M. We show that a Fourier expansion of the exponential multiplier yields an exponential series that can compute high-accuracy values of the complex error function in a rapid algorithm. Numerical error analysis and computational test reveal that with essentially higher accuracy it is as fast as FFT-based Weideman’s algorithm at a regular size of the input array and considerably faster at an extended size of the input array. As this exponential series approximation is based only on elementary functions, the algorithm can be implemented utilizing freely available functions from the standard libraries of most programming languages. Due to its simplicity, rapidness, high-accuracy and coverage of the entire complex plane, the algorithm is efficient and practically convenient in numerical methods related to the spectral line broadening and other applications requiring error-function evaluation over extended input arrays.

Nonparametric tests for differential gene expression and interaction effects in multi-factorial microarray experiments

2013-05-08T15:52:00Z

Nonparametric tests for differential gene expression and interaction effects in multi-factorial microarray experiments Gao, Xin; Song, Peter XK Background Numerous nonparametric approaches have been proposed in literature to detect differential gene expression in the setting of two user-defined groups. However, there is a lack of nonparametric procedures to analyze microarray data with multiple factors attributing to the gene expression. Furthermore, incorporating interaction effects in the analysis of microarray data has long been of great interest to biological scientists, little of which has been investigated in the nonparametric framework. Results In this paper, we propose a set of nonparametric tests to detect treatment effects, clinical covariate effects, and interaction effects for multifactorial microarray data. When the distribution of expression data is skewed or heavy-tailed, the rank tests are substantially more powerful than the competing parametric F tests. On the other hand, in the case of light or medium-tailed distributions, the rank tests appear to be marginally less powerful than the parametric competitors. Conclusion The proposed rank tests enable us to detect differential gene expression and establish interaction effects for microarray data with various non-normally distributed expression measurements across genome. In the presence of outliers, they are advantageous alternative approaches to the existing parametric F tests due to the robustness feature.

The Courant-Herrmann conjecture

2013-05-08T15:34:17Z

The Courant-Herrmann conjecture Gladwell1, Graham M. L.; Zhu, Hongmei The Courant-Herrmann Conjecture (CHC) concerns the sign properties of combinations of the Dirichlet eigenfunctions of elliptic pde’s, the most important of which is the Helmholtz equation Δu + λρu = 0 for D ∈ R↑N. If the eigenvalues are ordered increasingly, CHC states that the nodal set of a combination v = SIGMA ciui (1=