The authors develop an algorithm for the numerical evaluation of Cauchy principal value integrals of oscillatory functions. The method is based on an interpolatory procedure at the zeros of the orthogonal polynomials with respect to a Jacobi weight. A numerically stable procedure is obtained and the corresponding algorithm can be implemented in a fast way yielding satisfactory numerical results. Bounds of the error and of the amplification factor are also proved.

A method for the numerical inversion of the Laplace transform of a continuous positive function $f(t)$ is proposed. Random matrices distributed according to a Gibbs law whose energy $V(x)$ is a function of $f(t)$ are considered as well as random polynomials orthogonal with respect to $w(x)=e^{-V(x)}$. The equation relating $w(x)$ to the reproducing kernel and to the condensed density of the roots of the random orthogonal polynomials is exploited. Basic results from the theories of orthogonal polynomials, random matrices and random polynomials are revisited in order to provide a unified and almost self--contained context. The qualitative behavior of the solutions provided by the proposed method is illustrated by numerical examples and discussed by using logarithmic potentials with external fields that give insight into the asymptotic behavior of the condensed density when the number of data points goes to infinity.

Space-born measuring devices require an accurate determination of the satellite rest frame. This frame consists of a clock and a triad of orthonormal axes which provide a local Cartesian reference system. The aim of this paper is to find the mathematical representation of this triad in two cases which may correspond to actual satellite attitudes. First we construct a Fermi frame which can be operationally fixed by a set of three mutually orthogonal gyroscopes, then we find a frame which corresponds to the expected attitude of the satellite GAIA which was ESA approved to fly not later than 2012. In the latter case, we were able to find an analytical solution accurate to (v/c)3. In order to exploit this solution in the treatment of GAIA's astrometrical observations, we illustrate all the steps needed to deduce the components of this triad of vectors.

Relativistic astrometry has recently become an active field of research owing to new observational technologies which allow for accuracies of a microarcsecond. To assure this accuracy in data analysis, one has to perform ray tracing in a general relativistic framework including terms of the order of (v/c)3 in the weak field treatment of Einstein equations applied to the solar system. Basic to the solution of a ray tracing problem are the boundary conditions that one has to fix from the observational data. In this paper we solve this problem to (v/c)3 in a fully analytical way

We study the motion of test particles and electromagnetic waves in the Kerr–Newman–Taub–NUT spacetime in order to elucidate some of the effects associated with the gravitomagnetic monopole moment of the source. In particular, we determine in the linear approximation the contribution of this monopole to the gravitational time delay and the rotation of the plane of the polarization of electromagnetic waves. Moreover, we consider 'spherical' orbits of uncharged test particles in the Kerr–Taub–NUT spacetime and discuss the modification of the Wilkins orbits due to the presence of the gravitomagnetic monopole

Test particle geodesic motion is analysed in detail for the background spacetimes of the degenerate Ferrari–Ibañez colliding gravitational wave solutions. Killing vectors have been used to reduce the equations of motion to a first-order system of differential equations which have been integrated numerically. The associated constants of motion have also been used to match the geodesics as they cross over the boundary between the single plane wave and interaction zones

Accelerated circular orbits in the equatorial plane of the Taub-NUT spacetime are analyzed to investigate the effects of its gravitomagnetic monopole source. The effect of a small gravitomagnetic monopole on these orbits is compared to the corresponding orbits pushed slightly off the equatorial plane in the absense of the monopole.

The de Rham Laplacian $\Delta_{\rm (dR)}$ for differential forms is a geometric generalization of the usual covariant Laplacian $\Delta$, and it may be extended naturally to tensor-valued $p$-forms using the exterior covariant derivative associated with a metric connection. Using it the wave equation satisfied by the curvature tensors in general relativity takes its most compact form. This wave equation leads to the Teukolsky equations describing integral spin perturbations of black hole spacetimes.

The behaviour of a massless Dirac field on a general spacetime background representing two colliding gravitational plane waves is discussed in the Newman-Penrose formalism. The geometrical properties of the neutrino current are analysed and explicit results are given for the special Ferrari-Iba\~nez solution.

A single master equation is given describing spin s <~ 2 test fields that are gauge- and tetrad-invariant perturbations of the Kerr-Taub-NUT (Newman-Unti-Tamburino) spacetime representing a source with a mass M, gravitomagnetic monopole moment -l, and gravitomagnetic dipole moment (angular momentum) per unit mass a. This equation can be separated into its radial and angular parts. The behavior of the radial functions at infinity and near the horizon is studied and used to examine the influence of l on the phenomenon of superradiance, while the angular equation leads to spin-weighted spheroidal harmonic solutions generalizing those of the Kerr spacetime. Finally, the coupling between the spin of the perturbing field and the gravitomagnetic monopole moment is discussed.

Segmentation (tissue classification) of medical images obtained from a magnetic resonance (MR) system is a primary step in most applications of medical image post-processing. This paper describes nonparametric discriminant analysis methods to segment multispectral MR images of the brain. Starting from routinely available spin-lattice relaxation time, spin-spin relaxation time, and proton density weighted images (T1w, T2w, PDw) the proposed family of statistical methods is based on: i) a transform of the images into components that are statistically independent from each other; ii) a nonparametric estimate of probability density functions of each tissue starting from a training set; iii) a classic Bayes 0-1 classification rule. Experiments based on a computer built brain phantom (brainweb) and eight real patient data set are shown. A comparison with parametric discriminant analysis is also reported. The capability of nonparametric discriminant analysis in improving brain tissue classification of parametric methods is demonstrated. Finally, an assessment of the role of multispectrality in classifying brain tissues is discussed.

The historical origins of Fermi-Walker transport and Fermi coordinates and the construction of Fermi-Walker transported frames in black hole spacetimes are reviewed. For geodesics this transport reduces to parallel transport and these frames can be explicitly constructed using a Killing-Yano tensor as shown by Marck. For accelerated or geodesic circular orbits in such spacetimes, both parallel and Fermi-Walker transported frames can be given, and allow one to study circular holonomy and related clock and spin transport effects. In particular the total angle of rotation that a spin vector undergoes around a closed loop can be expressed in a factored form, where each factor is due to a different relativistic effect, in contrast with the usual sum of terms decomposition. Finally the Thomas precession frequency is shown to be a special case of the simple relationship between the parallel transport and Fermi-Walker transport frequencies for stationary circular orbits.

We introduce a nonparametric method for discriminant analysis based on the search of independent components in a signal (ICDA). Keypoints of the method are reformulation of the classification problem in terms of transform matrices; use of Independent Component Analysis (ICA) to choose a transform matrix so that transformed components are as independent as possible; nonparametric estimation of the density function for each independent component; application of a Bayes rule for class assignment. Convergence of the method is proved and its performance is illustrated on simulated and real data examples.

We show that a nonparametric estimator of a regression function, obtained as solution of a specific regularization problem is the bestlinear unbiased predictor in some nonparametric mixed effect model. Since this estimator is intractable from a numerical point of view, we propose a tight approximation of it easy and fast to implement. This second estimator achieves the usual optimal rate of convergence of the mean integrated squared error over a Sobolev class both for equispaced and non equispaced design. Numerical experiments are presented both on simulated and ERP real data.

The conceptual framework for modeling the inertial subrange is strongly influenced by the Kolmogorov cascade phenomena, which is now the subject of significant reinterpretation. It has been argued that the effects of boundary conditions influence large-scale motion and direct interaction between large and small scales is possible by means other than passing sequentially through the full cascade. Using longitudinal (u) and vertical (w) velocity and temperature (T) time series measurements collected in the atmospheric surface layer (ASL), we evaluate whether the inertial subrange multifractral function (f(alpha)) of all three flow variables is influenced byatmospheric stability (xi), which is a bulk measure of the effect of boundary conditions on large scale flow properties for ASL turbulence. This study is the first to demonstrate that xi significantly influences f(alpha) for all three flow variables. Here, statistical significance is evaluated using a novel wavelet-based Functional Analysis of Variance (FANOVA) approach that explicitly considers different classes of xi, the flow variable type, and possible interactions between xi and the three flow variables.

In the field of resource constrained scheduling, the papers in the literature are mainly focused on minimizing the maximum completion time of a set of tasks to be carried out, paying attention to respecting the maximum simultaneous availability of each resource type in the system. This work, instead, considers the issues of balancing the resource usage and minimizing the peak of the resources allocated each time in the schedule, while keeping the makespan low. To this aim we propose a local search algorithm which acts as a multi start greedy heuristic. Extensive experiments on various randomly generated test instances are provided. Furthermore, we present a comparison with lower bounds and known heuristics.