Hypersingular integral equations over a disc: Convergence of a spectral method and connection with Tranter's method

Two-dimensional hypersingular equations over a disc are considered. A spectral method is developed, using Fourier series in the azimuthal direction and orthogonal polynomials in the radial direction. The method is proved to be convergent. Then, Tranter's method is discussed. This method was devised in the 1950s to solve certain pairs of dual integral equations. It is shown that this method is also convergent because it leads to the same algebraic system as the spectral method

The airfoil equation on near disjoint intervals: Approximate models and polynomial solutions

The airfoil equation is considered over two disjoint intervals. Assuming the distance between the intervals is small an approximate solution is found and relationships between this approximation and the solution of the classical airfoil equation are obtained. Numerical results show the convergence of the solution of the original problem to the approximation. Polynomial solutions for an approximate model are obtained and a spectral method for the generalized airfoil equation on near disjoint intervals is proposed

Fast simulation of through-casing resistivity measurements using semi-analytical asymptotic models. Part 1: Accuracy study

When trying to obtain a better characterization of the Earth's subsurface, it is common to use borehole through-casing resistivity measurements. It is also common for the wells to be surrounded by a metal casing to protect the well and avoid possible collapses. The presence of this metal case highly complicates the numeric simulation of the problem due to the high conductivity of the casing compared to the conductivity of the rock formations. Here we present an application of some theoretical asymptotic methods in order to deal with complex borehole scenarios like cased wells. The main idea consist in replacing the part of the domain related to the casing by a transmission impedance boundary condition. The small thickness of the casing makes it ideal to apply this kind of mathematical technique. When eliminating the casing from the computational domain, the computational cost of the problems considerably decreases, while the effect of the casing does not disappear due to the impedance transmission conditions. The results show that when applying an order three impedance boundary condition for a simplified domain, it only generates a negligible approximation error, while it considerably reduces the computational cost

Strict positive definiteness on spheres via disk polynomials

We characterize complex strictly positive definite functions on spheres in two cases, the unit sphere of ℂq, q≥3, and the unit sphere of the complex ℓ2. The results depend upon the Fourier-like expansion of the functions in terms of disk polynomials and, among other things, they enlarge the classes of strictly positive definite functions on real spheres studied in many recent papers

Phytochemical investigations on Artemisia alba Turra growing in the North-East of Italy

Artemisia alba Turra (Asteraceae) is an Euro-Mediterranean plant used in Veneto (North-East of Italy) as traditional medicine for the treatment of various diseases. A. alba is a taxonomically problematic species, characterized by common polymorphism leading to a quite high variability in secondary metabolites content. Nonetheless, the phytochemical knowledge on its phytoconstituents, especially non-volatile components, is limited. In the present paper, the phytochemical composition of a tincture obtained from the aerial parts of A. alba growing in Veneto is presented. Extensive chromatographic separations led to the isolation of three new sesquiterpene derivatives, whose structures were elucidated by 1D and 2D NMR experiments and mass spectrometry. Furthermore, flavonoid composition and volatile constituents of the tincture of A. alba were preliminary studied by HPLC-MSn and GC-MS, respectivel