Continuity of condenser capacity under holomorphic motions

We show that condenser capacity varies continuously under holomorphic motions, and the corresponding family of the equilibrium measures of the condensers is continuous with respect to the weak-star convergence. We also study the behavior of uniformly perfect sets under holomorphic motions.

The convergence properties of orthogonal rational functions on the extended real line and analytic on the upper half plane

We study the orthogonal rational functions with a given sequence of poles on the half plane. We give the weak-star convergence results for the rational measures which relate to the Poisson kernel and reproducing kernel. Those rational measures represent the same inner product as the normalized Nevanlinna measure in [Formula: see text]. Moreover, we consider the convergence properties of the interpolants which interpolate the Carathéodory function [Formula: see text].

The accelerating rotation of the magnetic He-weak star HD 142990

ABSTRACT HD 142990 (V 913 Sco; B5 V) is a He-weak star with a strong surface magnetic field and a short rotation period (Prot ∼ 1 d). Whilst it is clearly a rapid rotator, recent determinations of Prot are in formal disagreement. In this paper, we collect magnetic and photometric data with a combined 40-yr baseline in order to re-evaluate Prot and examine its stability. Both period analysis of individual data sets and O − C analysis of the photometric data demonstrate that Prot has decreased over the past 30 yr, violating expectations from magnetospheric braking models, but consistent with behaviour reported for 2 other hot, rapidly rotating magnetic stars, CU Vir and HD 37776. The available magnetic and photometric time series for HD 142990 can be coherently phased assuming a spin-up rate $\dot{P}$ of approximately −0.6 s yr−1, although there is some indication that $\dot{P}$ may have slowed in recent years, possibly indicating an irregular or cyclic rotational evolution.

Octahedrality in Lipschitz-free Banach spaces

The aim of this note is to study octahedrality in vector-valued Lipschitz-free Banach spaces on a metric space, under topological hypotheses on it, by analysing the weak-star strong diameter 2 property in Lipschitz function spaces. Also, we show an example that proves that our results are optimal and that octahedrality in vector-valued Lipschitz-free Banach spaces actually relies on the underlying metric space as well as on the Banach one.

Concentration analysis in Banach spaces

The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach–Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of [Formula: see text]-convergence by Lim [Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179–182] instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and [Formula: see text]-spaces, but not in [Formula: see text], [Formula: see text]. [Formula: see text]-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies the connection of [Formula: see text]-convergence with the Brezis–Lieb lemma and gives a version of the latter without an assumption of convergence a.e.