Let$D$($μ$) be the Dirichlet space weighted by the Poisson integral of the positive measure$μ$. We give a characterization of the measures$μ$equal to a countable sum of atoms for which the Blaschke condition is a necessary and sufficient condition for a sequence to be a zero set for$D$($μ$).

We use the known convergence of loop-erased random walk to radial SLE(2) to give a new proof that the scaling limit of loop-erased random walk excursion in the upper half-plane is chordal SLE(2). Our proof relies on a version of Wilson’s algorithm for weighted graphs which is used together with a Beurling-type estimate for random walk excursion. We also establish and use the convergence of the radial SLE path to the chordal SLE path as the bulk point tends to a boundary point. In the final section we sketch how to extend our results to more general simply connected domains.

We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Most of the definitions presented here for tropical vector bundles will be contained in$Torchiani, C.$,$Line Bundles on Tropical Varieties$, Diploma thesis, Technische Universität Kaiserslautern, Kaiserslautern,2010, for the case of line bundles. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.

We introduce a technique for handling Whitney decompositions in Gaussian harmonic analysis and apply it to the study of Gaussian analogues of the classical tent spaces$T$^{1,$q$}of Coifman–Meyer–Stein.

For a positive integer$k$and a non-negative integer$t$, a class of simplicial complexes, to be denoted by$k$-CM_{$t$}, is introduced. This class generalizes two notions for simplicial complexes: being$k$-Cohen–Macaulay and$k$-Buchsbaum. In analogy with the Cohen–Macaulay and Buchsbaum complexes, we give some characterizations of CM_{$t$}(=1−CM_{$t$}) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CM_{$t$}property under the operation of join of two simplicial complexes. We show that a complex is$k$-CM_{$t$}if and only if the links of its non-empty faces are$k$-CM_{$t$−1}. We prove that for an integer$s$≤$d$, the ($d$−$s$−1)-skeleton of a ($d$−1)-dimensional$k$-CM_{$t$}complex is ($k$+$s$)-CM_{$t$}. This result generalizes Hibi’s result for Cohen–Macaulay complexes and Miyazaki’s result for Buchsbaum complexes.