Laplace operators on metric graphs are considered. It is proven that for compact graphs the spectrum of the Laplace operator determines the total length, the number of connected components, and the Euler characteristic. For a class of non-compact graphs the same characteristics are determined by the scattering data consisting of the scattering matrix and the discrete eigenvalues.

Let$X$be a smooth$n$-dimensional projective variety embedded in some projective space ℙ^{$N$}over the field ℂ of the complex numbers. Associated with the general projection of$X$to a space ℙ^{$N$-$m$}($N$-$m$>$n$+1) one defines an extended Gauss map $\overline{\gamma}\colon\overline{X}\rightarrow\text{Gr}(n;N-m)$ (in case$N$-$m$>2$n$-1 this is the Gauss map of the image of$X$under the projection). We prove that $\overline{X}$ is smooth. In case any two different points of$X$do have disjoint tangent spaces then we prove that $\overline{\gamma}$ is injective.

We present a method of finding weighted Koppelman formulas for ($p$,$q$)-forms on$n$-dimensional complex manifolds$X$which admit a vector bundle of rank$n$over$X$×$X$, such that the diagonal of$X$×$X$has a defining section. We apply the method to ℙ^{$n$}and find weighted Koppelman formulas for ($p$,$q$)-forms with values in a line bundle over ℙ^{$n$}. As an application, we look at the cohomology groups of ($p$,$q$)-forms over ℙ^{$n$}with values in various line bundles, and find explicit solutions to the $\overline{\partial}$ -equation in some of the trivial groups. We also look at cohomology groups of (0,$q$)-forms over ℙ^{$n$}×ℙ^{$m$}with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.

Let$X$be a rationally convex compact subset of the unit sphere$S$in ℂ^{2}, of three-dimensional measure zero. Denote by$R$($X$) the uniform closure on$X$of the space of functions$P$/$Q$, where$P$and$Q$are polynomials and$Q$≠0 on$X$. When does$R$($X$)=$C$($X$)?

Let Ω be an open subset of$R$^{$d$},$d$≥2, and let$x$∈Ω. A$Jensen measure$for$x$on Ω is a Borel probability measure μ, supported on a compact subset of Ω, such that ∫$u$$d$μ≤$u$($x$) for every superharmonic function$u$on Ω. Denote by$J$_{$x$}(Ω) the family of Jensen measures for$x$on Ω. We present two characterizations of ext($J$_{$x$}(Ω)), the set of extreme elements of$J$_{$x$}(Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains.