The Hilbert transform of a function g, H(g) is an important tool in many mathematical fields. Expecially its numerical evaluation is often useful in some procedures for searcing solutions of the singular integral equations. In this context an approximation of (HV^alpha,beta,f;t), |t|1, where f is a continuous function in [-1,1] and v^alpha,beta, alpha,beta>-1 is a Jacobi weight, is required. In the last decade more then one paper appeared on this subject and among others we recall [1,2,3,4,5,14,15,20]. The procedure used in these papers can be described as follows. Approximate the function f with a Lagrange interpolating polynomial and obtain a quadrature formula with coefficients depending on the singularity t. The error of the quadrature formula was estimated assuming t in a "more or less" wide neighborhood of zero since (HV^alpha,beta,f;t) is unbounded at the endpoints of (-1,1). In this paper we propose a more accurate procedure: since (HV^alpha,beta,f;t) is unbounded at the endpoints, itis more natural to consider its approximation in some functional spaces equipped with weighted uniform norms. In Section 2 (see theorems 2.1 and 2.2) we state the behaviour of (HV^alpha,beta,f;t) in [-1,1] and we give new conditions for its exstence. In sections 3 and 4 we cosntruct approximations of (HV^alpha,beta,f;t) which converge to the exact value in weighted uniform norm. The L^p convergence of the formula is also briefly treated. Finally since the considered procedures are numerically stable, these can be used for the numerical evaluation of (HV^alpha,beta,f;t) . In this sense the present paper is a short survey on the subject.

Alcuni modelli di dispersione di sostanze inquinanti prodotte dal traffico autoveicolare in un area urbana vengono presentati. In particolare si descrive il modello OMG-Volume Source che sembra essere in grado di analizzare realisticamente l'effetto "canyon", tipico di un area urbana. I parametri del modello vengono determinati tenendo soprattutto in considerazione le caratteristiche spaziali dell'insediamento urbano.

Alcuni modelli di dispersione di sostanze inquinanti prodotte dal traffico autoveicolare in un area urbana vengono presentati. In particolare si descrive il modello OMG-Volume Source che sembra essere in grado di analizzare realisticamente l'effetto "canyon", tipico di un area urbana. I parametri del modello vengono determinati tenendo soprattutto in considerazione le caratteristiche spaziali dell'insediamento urbano.

Remote sounding of earth's atmosphere from upwelling infrared radiances is highly non-linear. A linearization step is typically involved in seeking for suitable solutions. This letter discusses possible problems which arise when combining the linearization step with schemes aimed at solving the inverse problem. Possible strategies which could be able to get rid of such problems are discussed, too. Copyright 1996 by the American Geophysical Union.

In this paper we analyze replica symmetry breaking in attractor neural networks with non-monotone activation function. We study the non-monotone version of the Edinburgh model, which allows the control of the domains of attraction by the stability parameter K, and we compute, at one step of symmetry breaking, storage capacity and, for the strongly dilute model, the domains of attraction of the stable fixed points.

Collocation and quadrature methods for Cauchy singular integral equations on an interval with variable coefficients are studied. Convergence rates are proved in weighted uniform and uniform norms.

Retrieving of temperature profiles from radiance data obtained by interferograms is an important problem in remote sensing of atmosphere. The great amount of data to process and the ill-conditioning of the problem demand objective procedures able to reduce the error of the retrieval. In this paper we use generalized singular value decomposition (GSVD), which is able to deal with deficient-rank smoothing functionals in order to regularize the problem and the L-curve criterion for choosing the optimal regularization parameter and then the proper amount of smoothing. Some test problems of temperature inversion are carried out to examine the effectiveness of the methods considered; to this purpose we use some indicators based on the bias and variance of the output temperature. We show that the objective L-curve criterion does not perform fully satisfactory in estimating the optimal regularization parameter and then in reducing output error at best. In any case GSVD plus L-curve criterion prove effective in reducing output error (with respect to the ordinary least squares method). In particular, reduction of variance over troposphere and stratosphere is high for all tested cases; reduction of bias depends on the first-guess profile. An important role in the latter is played by the choice of deficient-rank smoothing functional.

progetto coordinato da D. Mansutti con 4 UU.OO. - titolo del progetto della U.O. dell'IAC, "Modelli numerici per la cristallizzazione da fuso in microgravita': modello globale, convezione naturale nel fuso e propagazione di onde nel cristallo".

We consider a class of integral equations of Volterra type with constant coefficients containing a logarithmic difference kernel. This equation can be transformed into an equivalent singular equation of Cauchy type which allows us to give the explicit formula for the solution. The numerical method proposed in this paper consists of applying the Lagrange interpolation to the inner Cauchy type singular integral in the latter formula after subtracting the singularity. For the error of this method weighted norm estimates as well as estimates on discrete subsets of knots are given. The paper concludes with some numerical examples. © 1995 Rocky Mountain Mathematics Consortium.

Reaction-diffusion systems with cross-diffusion are analyzed here for modeling the population dynamics of epidemic systems. In this paper specific attention is devoted to the numerical analysis and simulation of such systems to show that, far from possible pathologies, the qualitative behaviour of the systems may well interpret the dynamics of real systems.