We introduce the notion of compactly locally reflexive Banach spaces and show that a Banach space$X$is compactly locally reflexive if and only if $\mathcal{K}(Y,X^{**})\subseteq\mathcal{K}(Y,X)^{**}$ for all reflexive Banach spaces$Y$. We show that$X$^{*}has the approximation property if and only if$X$has the approximation property and is compactly locally reflexive. The weak metric approximation property was recently introduced by Lima and Oja. We study two natural weak compact versions of this property. If$X$is compactly locally reflexive then these two properties coincide. We also show how these properties are related to the compact approximation property and the compact approximation property with conjugate operators for dual spaces.

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$R$($X$)=$C$($X$)?