Computing Complex Airy Functions by Numerical Quadrature

Integral representations are considered of solutions of the Airy differential equation w –zw=0 for computing Airy functions for complex values of z. In a first method contour integral representations of the Airy functions are written as non-oscillating integrals for obtaining stable representations, which are evaluated by the trapezoidal rule. In a second method an integral representation is evaluated by using generalized Gauss–Laguerre quadrature; this approach provides a fast method for computing Airy functions to a predetermined accuracy. Comparisons are made with well-known algorithms of Amos, designed for computing Bessel functions of complex argument. Several discrepancies with Amos' code are detected, and it is pointed out for which regions of the complex plane Amos' code is less accurate than the quadrature algorithms. Hints are given in order to build reliable software for complex Airy functions.     

The Problem of the Pointwise Fourier Inversion for Piecewise Smooth Functions of Several Variables

We study the pointwise convergence problem for the inverse Fourier transform of piecewise smooth functions, i.e., whether S_rD f (\bx) ? f (\bx)S_{\rho D} f (\bx) \to f (\bx) as r? ￥\rho \to \infty . r? ￥\rho \to \infty . Here for \bx,\bxi ? \Rn\bx,\bxi \in \Rn S_rDf(\bmx)=\dsf1(2p)^n/2\intli_rD [^(f)](\bxi) e^{\dst iá\bmx,\bxi?} d\bxi . S_{\rho D}f(\bm{x})=\dsf1{(2\pi)^{n/2}}\intli_{\rho D} \widehat{f}(\bxi) e^{\dst i\langle\bm{x},\bxi\rangle} d\bxi~. is the partial sum operator using a convex and open set DD containing the origin, and rD={ r\bxi:\bxi ? D }\rho D=\left\{ \rho \bxi:\bxi\in D \right\}.     

New distance measures for classifying X-ray astronomy data into stellar classes

Advances in Data Analysis and Classification - The classification of the X-ray sources into classes (such as extragalactic sources, background stars,...) is an essential task in astronomy....     

Positive Isometric Averaging Operators on {\ell^2(\mathbb{Z}, \mu)}

We show that positive isometric averaging operators on the sequence space \({\ell^2(\mathbb{Z}, \mu)}\) are determined by very subtle arithmetic conditions on \({\mu}\) (even for very simple examples), contrary to what happens in the continuous case \({L^2({\mathbb{R}}^+)}\), where any possible average value is realized by a suitable positive isometry.     

Intelligent variable orderings and re-orderings in DAC-based solvers for WCSP

Weighted constraint satisfaction problems (WCSPs) is a well-known framework for combinatorial optimization problems with several domains of application. In the last few years, several local consistencies for WCSPs have been proposed. Their main use is to embed them into a systematic search, in order to detect and prune unfeasible values as well as to anticipate the detection of deadends. Some of these consistencies rely on an order among variables but nothing is known about which orders are best. Therefore, current implementations use the lexicographic order by default. In this paper we analyze the effect of heuristic orders at three levels of increasing overhead: i) compute the order prior to search and keep it fixed during the whole solving process (we call this a static order), ii) compute the order at every search node using current subproblem information (we call this a dynamic order) and iii) compute a sequence of different orders at every search node and sequentially enforce the local consistency for each one (we call this dynamic re-ordering). We performed experiments in three different problems: Max-SAT, Max-CSP and warehouse location problems. We did not find an alternative better than the rest for all the instances. However, we found that inverse degree (static order), sum of unary weights (dynamic order) and re-ordering with the sum of unary weights are good heuristics which are always better than a random order. This research is supported by the MEC through project TIC-2002-04470-C03.     

Mixed Norm and Multidimensional Lorentz Spaces

In the last decade, the problem of characterizing the normability of the weighted Lorentz spaces has been completely solved ([16], [7]). However, the question for multidimensional Lorentz spaces is still open. In this paper, we consider weights of product type, and give necessary and sufficient conditions for the Lorentz spaces, defined with respect to the two-dimensional decreasing rearrangement, to be normable. To this end, it is also useful to study the mixed norm Lorentz spaces. Finally, we prove embeddings between all the classical, multidimensional, and mixed norm Lorentz spaces. Research partially supported by KAW 2000.0048 and STINT KU 2002-4025. Research partially supported by Grants MTM2004-02299, 2005SGR00556 and The Swedish Research Council no. 624-2003-571.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.