Weighted composition operators on the predual and dual Banach spaces of Beurling algebras on Z

Let vv be a weight sequence on ZZ and let ψ,φψ,φ be complex-valued functions on ZZ such that φ(Z)⊂Zφ(Z)⊂Z. In this paper we study the boundedness, compactness and weak compactness of weighted composition operators Cψ,φCψ,φ on predual Banach spaces c0(Z,1/v)c0(Z,1/v) and dual Banach spaces ℓ∞(Z,1/v)ℓ∞(Z,1/v) of Beurling algebras ℓ1(Z,v)ℓ1(Z,v).

Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l ∞(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel–Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system’s data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Cánovas et al. (SIAM J. Optim. 20, 1504–1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system’s coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.

Local convergence of quasi-Newton methods under metric regularity

We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis–Moré condition for q-superlinear convergence. Simple numerical examples illustrate the results.

Effect of water vapour condensation on the radon content in subsurface air in a hypogeal inactive-volcanic environment in Galdar cave, Spain

Fluctuations of trace gas activity as a response to variations in weather and microclimate conditions were monitored over a year in a shallow volcanic cave (Painted Cave, Galdar, Canary Islands, Spain). 222Rn concentration was used due to its greater sensitivity to hygrothermal variations than CO2 concentration. Radon concentration in the cave increases as effective vapour condensation within the porous system of the rock surfaces inside the cave increases due to humidity levels of more than 70%. Condensed water content in pores was assessed and linked to a reduction in the direct passage of trace gases. Fluctuations in radon activity as a response to variations in weather and microclimate conditions were statistically identified by clustering entropy changes on the radon signal and parameterised to predict radon concentration anomalies. This raises important implications for other research fields, including the surveillance of shallow volcanic and seismic activity, preventive conservation of cultural heritage in indoor spaces, indoor air quality control and studies to improve understanding of the role of subterranean terrestrial ecosystems as reservoirs and/or temporary sources of trace gases.

Radon measurements in the cave houses of Crevillente (Spain)

This paper focuses on the cave houses of Crevillente (Spain) as a traditional housing experience which takes advantage of local environmental conditions through simple architectural proposals, paying particular attention to the presence of radon gas inside these underground constructions. Our aim is twofold: first, to analyse the architectural conditions of the different excavated typologies found in the municipality and second, to relate them to the existing radon gas levels after checking internal concentration by means of E-PERM® long-term devices placed inside the cave houses in 2011. The measurements corresponding to the main typologies in normal use conditions show that the highest values are 881.9 Bq/m3 in the cave typology, 484.1 Bq/m3 in the cave + attached constructions typology and 373.4 Bq/m3 in the cave + house typology, with geometric mean values of 572.1, 114.0 and 75.5 Bq/m3, respectively. It can be inferred from these results that cave house levels sometimes exceed those included in the 90/143/Euratom European Commission Recommendation on the protection of the public against indoor exposure to radon. The reason why cave houses are more susceptible to radon accumulation in their spaces lies in their direct and permanent contact with the ground where they are located.

On Gruenhage spaces, separating σ-isolated families, and their relatives

We prove that, given a topological space X, the following conditions are equivalent. (α) X is a Gruenhage space. (β) X has a countable cover by sets of small local diameter (property SLD) by F∩G sets. (γ) X has a separating σ-isolated family M⊂F∩G. (δ) X has a one-to-one continuous map into a metric space which has a σ-isolated base of F∩G sets. Besides, we provide an example which shows Fragmentability ⇏ property SLD ⇏ the space to be Gruenhage.