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Cwikel's bound is extended to an operator-valued setting. One application of this result is a semi-classical bound for the number of negative bound states for Schrödinger operators with operator-valued potentials. We recover Cwikel's bound for the Lieb-Thirring constant$L$_{0,3}which is far worse than the best available by Lieb (for scalar potentials). However, it leads to a uniform bound (in the dimension$d$≥3) for the quotient$L$_{0,d}/L^{cl}_{0,d}is the so-called classical constant. This gives some improvement in large dimensions.

The harmonic Bergman kernel$Q$_{Ω}for a simply, connected planar domain Ω can be expanded in terms of powers of the Friedrichs operator$F$_{Ω}║$F$_{Ω}║<1 in operator norm. Suppose that Ω is the image of a univalent analytic function ø in the unit disk with ø' ($z$)=1+ψ ($z$) where ψ(0)=0. We show that if the function ψ belongs to a space$D$_{$s$}($D$),$s$>0, of Dirichlet type, then provided that ║ψ║∞<1 the series for$Q$_{Ω}also converges pointwise in $$\bar \Omega \times \bar \Omega \backslash \Delta (\partial \Omega )$$ , and the rate of convergence can be estimated. The proof uses the eigenfunctions of the Friedrichs operator as well as a formula due to Lenard on projections in Hilbert spaces. As an application, we show that for every$s$>0 there exists a constant$C$_{$s$}>0 such that if ║ψ║_{$D$}_{s}($D$)≤$C$_{$s$}, then the biharmonic Green function for Ω=ø ($D$) is positive.

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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.

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

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.

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 prove a theorem on non-existence of lacunas of the fundamental solution for hyperbolic differential operators with constant coefficients.

We study the dynamics of polynomial automorphisms of$C$^{$k$}. To an algebraically stable automorphism we associate positive closed currents which are invariant under$f$, considering$f$as a rational map on$P$^{$k$}. These currents give information on the dynamics and allow us to construct a canonical invariant measure which is shown to be mixing.

We investigate a problem of approximation of a large class of nonlinear expressions$f$($x, u$, ∇$u$), including polyconvex functions. Here$u$: Ω→$R$^{$m$}, Ω⊂$R$^{$n$}, is a mapping from the Sobolev space$W$^{$1,p$}. In particular, when$p=n$, we obtain the approximation by mappings which are continuous, differentiable a.e. and, if in addition$n=m$, satisfy the Luzin condition. From the point of view of applications such mappings are almost as good as Lipschitz mappings. As far as we know, for the nonlinear problems that we consider, no natural approximation results were known so far. The results about the approximation of$f$($x, u$, ∇$u$) are consequences of the main result of the paper, Theorem 1.3, on a very strong approximation of Sobolev functions by locally weakly monotone functions.