Through a geometric capacitary analysis based on space dualities, this paper addresses several fundamental aspects of functional analysis and potential theory for the Morrey spaces in harmonic analysis over the Euclidean spaces.

Let Ω be a Greenian domain in ℝ^d, d≥2, or—more generally—let Ω be a connected $\mathcal{P}$-Brelot space satisfying axiom D, and let u be a numerical function on Ω, $u\not\equiv\infty$, which is locally bounded from below. A short proof yields the following result: The function u is the infimum of its superharmonic majorants if and only if each set {x: u(x)> t}, t∈ℝ, differs from an analytic set only by a polar set and $\int u\,d\mu_{x}^{V}\le u(x)$, whenever V is a relatively compact open set in Ω and x∈V.

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.

In this paper we consider various aspects of generalized invertibility of the operator matrix $M=\left( {A\atop 0}{C\atop B}\right)$ acting on a Banach space X⊕Y.

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).

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.

The topological center of the spectrum of the Weyl algebra W, i.e. the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i\in\mathbb {N}\}$, is characterized in a recent paper by Jabbari and Namioka (Ellis group and the topological center of the flow generated by the map $n\mapsto \lambda^{n^{k}}$, to appear in Milan J. Math.). By the techniques essentially used in the cited paper, the topological center of the spectrum of the subalgebra W_k, the norm closure of the algebra generated by the set of functions $\{n\mapsto\lambda^{n^{i}};\lambda\in\mathbb{T}\mbox{ and }i=0,1,2,\ldots,k\}$, will be characterized, for all k∈ℕ. Also an example of a non-minimal dynamical system, with the enveloping semigroup Σ, for which the set of all continuous elements of Σ is not equal to the topological center of Σ, is given.

A fundamental and very well studied region of the Erdős–Rényi process is the phase transition at m∼n/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine the Bohman–Frieze process in which edges between isolated vertices are more likely. While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class. In particular, the growth of the giant component in the barely supercritical region is linear in all cases.

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.

On montre que la fonction maximale centrée de Hardy–Littlewood, M, sur les espaces hyperboliques réels $\mathbb{H}^{n} = \mathbb{R}^{{\mathchoice {\raise .17ex\hbox {$\scriptstyle +$}}{\raise .17ex\hbox {$\scriptstyle +$}}{\raise .1ex\hbox {$\scriptscriptstyle +$}}{\scriptscriptstyle +}}} \times \mathbb{R}^{n - 1}$, satisfait l’inégalité de type faible $\| M f \|_{L^{1, \infty}} \leq A (n \log {n}) \| f\|_{1}$ pour toute f∈L¹(ℍⁿ), où A>0 est une constante indépendante de la dimension n.

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.

We show that the symmetrized bidisc may be exhausted by strongly linearly convex domains. It shows in particular the existence of a strongly linearly convex domain that cannot be exhausted by domains biholomorphic to convex ones.