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In this paper we develop an adaptive algorithm for determining the optimal degree of regression in the constrained mock-Chebyshev least-squares interpolation of an analytic function to obtain quadrature formulas with high degree of exactness and accuracy from equispaced nodes. We numerically prove the effectiveness of the proposed algorithm by several examples.

Electric scalar potential estimation is a key step for non-invasive investigations of brain activity with high time resolutions. The neural sources can be reconstructed by solving a typical inverse problem based on a forward problem formulated as a set of boundary value problems coupled by interface conditions. In this paper, we propose a Shepard multinode method to numerically estimate electric scalar potentials via collocation. The method is based on a special kind of inverse distance weighting partition of unity method to increase polynomial precision, approximation order, and accuracy of the classical Shepard approximation. The barycentric form, through the use of cardinal basis function, is adopted to generate Shepard interpolants with high polynomial reproduction order. Comparisons with the analytic solution and the standard Shepard method are provided to assess the proposed approach.

A summary of the main facts about the Conference ‘Multivariate Approximation: Theory and Applications’ (MATA 2020) held at the Department of Mathematics and Computer Science of the University of Perugia (Italy), on January 16-18, 2020 and the contents of the associated Special Issue appearing on Dolomites Research Notes on Approximation, are here reported.

In vector field analysis, saddle points have two different types of invariant manifolds, namely stable ones and unstable ones. The invariant manifolds represent separatrices that partition the domain of trajectories into invariant regions of different dynamics. In this work, we analyze the basins of attraction of two different stable nodes by reconstructing the separatrices of a saddle point. To this purpose we present a computational algorithm that detects the points lying on the manifold, considering the plane generated by the two stable eigenvectors of the saddle point. Finally we reconstruct the surface by using the moving least-squares approximant method.

Functional imaging is used in the research area neurological, neurophysiology and cognitive psychology, for the diagnosis of diseases metabolic and for the detection of thin / squamous lesions (eg Alzheimer's disease) and for the development of neural interfaces (brain-computer interfaces - BCI).

Time dependent hereditary properties of complex materials are well described by power-laws with real order exponent. This experimental observation and analogous electrical experiments, yield a description of these properties by using fractional-order operators. In this paper, elasto-viscous and visco-elastic behaviors of fractional order hereditary materials are firstly described by using fractional mathematical operators, based on recent work of some of the authors. Then, electrical analogous models are introduced. Viscoelastic models have elastic and viscous components which can be obtained by combining springs and dashpots: these models can be equivalently viewed as electrical circuits, where the spring and dashpot are analogous to the capacitance and resistance, respectively. The proposed models are validated by using modal analysis. The use of electrical analogous in viscoelasticity can better reveal the real behavior of fractional hereditary materials.

Imaging techniques give a fundamental support to medical diagnostics during the pathology discovery as well as for the characterization of biological struc- tures. The imaging methods involve electromagnetic waves in a frequency range that spans from some Hz to GHz and over. Most of these methods involve scanning of wide human body areas even if only small areas need to be analyzed. In this paper, a numerical method to evaluate the shape of micro-structures for application in the medical field, with a very low in- vasiveness for the human body, is proposed. A flexible thin-wire antenna radiates the VHF waves and then, by measuring the spatial magnetic field distribution it is possible to reconstruct the micro-structures image by es- timating the location of the antenna against a sensors panel. The typical inverse problem described above is solved numerically, and first simulation results are presented in order to show the validity and the robustness of the proposed approach.