We find some necessary conditions for a real Banach space to be an almost CL-space. We also discuss the stability of CL-spaces and almost CL-spaces by$c$_{0}- and$l$_{1}-sums. Finally, we address the question if a space of vector-valued continuous functions can be a CL-space or an almost CL-space.

We prove that the electronic densities of atomic and molecular eigenfunctions are real analytic in$R$^{3}away from the nuclei.

For any proper polynomial map$f: C$^{$k$}→$C$^{$k$}define the function α as $$\alpha (z): = \mathop {\lim \sup }\limits_{n \to \infty } \frac{{\log ^ + \log ^ + \left| {f^n (z)} \right|}}{n},where log^ + : = \max \{ \log , 0\} .$$ Let$f=(P$_{1},...,$P$_{$k$}) be a proper polynomial map. We define a notion of$s$-regularity using the extension of$f$to P^{k}. When$f$is (maximally) regular we show that the function α is lower semicontinuous and takes only finitely many values: 0 and$d$_{1},...,$d$_{$k$}, where$d$_{$i$}:=deg$P$_{$i$}. We then describe dynamically the sets {α≤$d$_{$i$}}. We give a concrete description of regular maps. If$d$_{$i$}>1, this allows us to construct the equilibrium measure μ associated with$f$as a generalized intersection of positive currents. We then give an estimate of the Hausdorff dimension of μ. We extend the approach to the larger class of (π,$s$)-regular maps. This gives an understanding of the largest values of α. The results can be applied to construct dynamically interesting measures for automorphisms.

Let$M$be a non-compact connected Riemann surface of a finite type, and$R$⋐$M$be a relatively compact domain such that$H$_{1}($M$,$Z$)=$H$_{1}($R$,$Z$). Let $$\tilde R \to R$$ be a covering. We study the algebra$H$^{∞}($U$) of bounded holomorphic functions defined in certain subdomains $$U \subset \tilde R$$ . Our main result is a Forelli type theorem on projections in$H$^{∞}($D$).

We study Rademacher chaos indexed by a sparse set which has a fractional combinatorial dimension. We obtain tail estimates for finite sums and a normal limit theorem as the size tends to infinity. The tails for finite sums may be much larger than the tails of the limit.