We present an elementary and concrete description of the Hilbert scheme of points on the spectrum of fraction rings$k[X]$_{$U$}of the one-variable polynomial ring over a commutative ring$k$. Our description is based on the computation of the resultant of polynomials in$k[X]$. The present paper generalizes the results of Laksov-Skjelnes [7], where the Hilbert scheme on spectrum of the local ring of a point was described.

We give a sharp upper estimate for the response of boundary current-voltage measurements to perturbations of the admittivity in a body that are localized in space and frequency. We calculate the differential of the measurement mapping and study the$Gabor symbol$of this operator.

The problem of inversion of the attenuated X-ray transformation is solved by an explicit formula. Several subsequent results are also given.

We establish sharp asymptotics for the$L$^{$p$}-norm of Hermite polynomials and prove convergence in distribution of suitably normalized Wick powers. The results are combined with numerical integration to study an extremal problem on Wiener chaos.

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