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

Atmospheric NLTE-models for the spectroscopic analysis of blue stars with winds. IV. Porosity in physical and velocity space

[Abridged] Clumping in the radiation-driven winds of hot, massive stars affects the derivation of synthetic observables across the electromagnetic spectrum. We implement a formalism for treating wind clumping - in particular the light-leakage effects associated with a medium that is porous in physical and velocity space - into the global (photosphere+wind) NLTE model atmosphere code FASTWIND. We assume a stochastic, two-component wind consisting of a mixture of optically thick and thin clumps embedded in a rarefied inter-clump medium. We account fully for the reductions in opacity associated with porosity in physical and velocity-space, and for the well-known effect that opacities depending on rho^2 are higher in clumpy winds than in smooth ones of equal mass-loss rate. By formulating our method in terms of suitable mean and effective opacities for the clumpy wind, we are able to compute models with the same speed (~15 min. on a modern laptop) as in previous code-generations. Some first, generic results of the new models include: i) Confirming earlier results that velocity-space porosity is critical for analysis of UV wind lines in O-stars; ii) for the optical Halpha line, optically thick clumping effects are small for O-stars, but potentially very important for late B and A-supergiants; iii) spatial porosity is a marginal effect for absorption of high-energy X-rays in O-stars, as long as the mean-free path between clumps are kept at realistic values; iv) porosity is negligible at typical O-star radio-photosphere radii; v) regarding the wind ionization balance, a general trend is that increased rates of recombination in simulations with optically thin clumps lead to overall lower degrees of ionization than in corresponding smooth models, but that this effect now is counteracted by the increased levels of light-leakage associated with porosity in physical and velocity space.Comment: 12 pages, 5 figures, accepted for publication in Astronomy & Astrophysic

2D wind clumping in hot, massive stars from hydrodynamical line-driven instability simulations using a pseudo-planar approach

Context: Clumping in the radiation-driven winds of hot, massive stars arises naturally due to the strong, intrinsic instability of line-driving (the `LDI'). But LDI wind models have so far mostly been limited to 1D, mainly because of severe computational challenges regarding calculation of the multi-dimensional radiation force. Aims: To simulate and examine the dynamics and multi-dimensional nature of wind structure resulting from the LDI. Methods: We introduce a `pseudo-planar', `box-in-a-wind' method that allows us to efficiently compute the line-force in the radial and lateral directions, and then use this approach to carry out 2D radiation-hydrodynamical simulations of the time-dependent wind. Results: Our 2D simulations show that the LDI first manifests itself by mimicking the typical shell-structure seen in 1D models, but how these shells then quickly break up into complex 2D density and velocity structures, characterized by small-scale density `clumps' embedded in larger regions of fast and rarefied gas. Key results of the simulations are that density-variations in the well-developed wind statistically are quite isotropic and that characteristic length-scales are small; a typical clump size is ~0.01R at 2R, thus resulting also in rather low typical clump-masses ~10^17 g. Overall, our results agree well with the theoretical expectation that the characteristic scale for LDI-generated wind-structure is of order the Sobolev length. We further confirm some earlier results that lateral `filling-in' of radially compressed gas leads to somewhat lower clumping factors in 2D simulations than in comparable 1D models. We conclude by discussing an extension of our method toward rotating LDI wind models that exhibit an intriguing combination of large- and small-scale structure extending down to the wind base.Comment: 9 pages, 7 figures + 1 Appendix with 1 figure. Recommended for publication in A&

Mass loss from inhomogeneous hot star winds III. An effective-opacity formalism for line radiative transfer in accelerating, clumped two-component media, and first results on theory and diagnostics

[Abridged] We develop and benchmark a fast and easy-to-use effective-opacity formalism for line and continuum radiative transfer in an accelerating two-component clumpy medium. The formalism bridges the limits of optically thin and thick clumps, and is here used to i) design a simple vorosity-modified Sobolev with exact integration (vmSEI) method for analyzing UV wind resonance lines in hot, massive stars, and ii) derive simple correction factors to the line force driving the outflows of such stars. We show that (for a given ionization factor) UV resonance doublets may be used to analytically predict the upward corrections in empirically inferred mass-loss rates associated with porosity in velocity space (a.k.a. velocity-porosity, or vorosity), but that severe solution degeneracies exist. For an inter-clump density set to 1 % of the mean density, we for O and B supergiants derive upward empirical mass-loss corrections of typically factors of either ~5 or ~50, depending on which of the two applicable solutions is chosen. Overall, our results indicate this solution dichotomy severely limits the use of UV resonance lines as direct mass-loss indicators of clumped hot stellar winds. We next apply the effective-opacity formalism to the standard CAK theory of line-driven winds. By analytic and numerical hydrodynamics calculations, we show that in cases where vorosity is important at the critical point setting the mass-loss rate, the reduced line-force leads to a lower theoretical mass loss, by a factor scaling with the normalized velocity filling factor fvel. On the other hand, if vorosity is important only above this critical point, the predicted mass loss is not affected, but the wind terminal speed is reduced. This shows that porosity in velocity space can have a significant impact not only on the diagnostics, but also on the dynamics and theory of radiatively driven winds.Comment: 13 pages, 8 figures, accepted for publication in Astronomy and Astrophysic