Graphs of bounded degree and the pp-harmonic boundary

Let $p$ be a real number greater than one and let $G$ be a connected graph of bounded degree. In this paper we introduce the $p$-harmonic boundary of $G$. We use this boundary to characterize the graphs $G$ for which the constant functions are the only $p$-harmonic functions on $G$. It is shown that any continuous function on the $p$-harmonic boundary of $G$ can be extended to a function that is $p$-harmonic on $G$. Some properties of this boundary that are preserved under rough-isometries are also given. Now let $\Gamma$ be a finitely generated group. As an application of our results we characterize the vanishing of the first reduced $\ell^p$-cohomology of $\Gamma$ in terms of the cardinality of its $p$-harmonic boundary. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on $\Gamma$, the $p$-harmonic boundary of $\Gamma$ with the first reduced $\ell^p$-cohomology of $\Gamma$.Comment: Give a new proof for theorem 4.7. Change the style of the text in the first two section

The first LpL^p-cohomology of some finitely generated groups and pp-harmonic functions

Let $G$ be a finitely generated infinite group and let $p > 1$. In this paper we make a connection between the first $L^p$-cohomology space of $G$ and $p$-harmonic functions on $G$. We also describe the elements in the first $L^p$-cohomology space of groups with polynomial growth, and we give an inclusion result for nonamenable groups

Carina OB Stars: X-ray Signatures of Wind Shocks and Magnetic Fields

The Chandra Carina Complex contains 200 known O- and B type stars. The Chandra survey detected 68 of the 70 O stars and 61 of 127 known B0-B3 stars. We have assembled a publicly available optical/X-ray database to identify OB stars that depart from the canonical Lx/Lbol relation, or whose average X-ray temperatures exceed 1 keV. Among the single O stars with high kT we identify two candidate magnetically confined wind shock sources: Tr16-22, O8.5 V, and LS 1865, O8.5 V((f)). The O4 III(fc) star HD 93250 exhibits strong, hard, variable X-rays, suggesting it may be a massive binary with a period of >30 days. The visual O2 If* binary HD 93129A shows soft 0.6 keV and hard 1.9 keV emission components, suggesting embedded wind shocks close to the O2 If* Aa primary, and colliding wind shocks between Aa and Ab. Of the 11 known O-type spectroscopic binaries, the long orbital-period systems HD 93343, HD 93403 and QZ Car have higher shock temperatures than short-period systems such as HD 93205 and FO 15. Although the X-rays from most B stars may be produced in the coronae of unseen, low-mass pre-main-sequence companions, a dozen B stars with high Lx cannot be explained by a distribution of unseen companions. One of these, SS73 24 in the Treasure Chest cluster, is a new candidate Herbig Be star.Comment: To be published in a special issue of the Astrophysical Journal Supplement on the Chandra Carina Complex Projec