Motivated by new examples of functional Banach spaces over the unit disk, arising as the symbol spaces in the study of random analytic functions, for which the monomials $\{z^n\}_{n\geq 0}$ exhibit features of an unconditional basis yet they often don’t even form a Schauder basis, we introduce a notion called solid basis for Banach spaces and p-Banach spaces and study its properties. Besides justifying the rich existence of solid bases, we study their relationship with unconditional bases, the weak-star convergence of Taylor polynomials, the problem of a solid span and the curious roles played by c0. The two features of this work are as follows: (1) during the process, we are led to revisit the axioms satisfied by a typical Banach space of analytic functions over the unit disk, leading to a notion of $\mathcal{X}^\mathrm{max}$ (and $\mathcal{X}^\mathrm{min}$), as well as a number of related functorial constructions, which are of independent interests; (2) the main interests of solid basis lie in the case of non-separable (p-)Banach spaces, such as BMOA and the Bloch space instead of VMOA and the little Bloch space.

Industrial policy is back on the agenda of policymakers at a global level. In the EU, industrial policy has a very controversial history: it has been not only conditioned by the multilevel competencies shared between the EU and the Member States, by the different national priorities and the varieties of economic cultures, but also by the crucial problem of resources. Based on archival sources and relevant multidisciplinary literature, the paper traces the emergence of European initiatives in industrial policy from the 1960s until the launch of the Europe 2020 strategy (2010). Following this, the role of the Juncker Plan (2014) in promoting EU industrial policy is examined, highlighting the potential contradiction between ambitious goals and limited resources. Subsequently, an analysis is conducted to determine whether the launch of the new EU industrial strategy in 2020, coupled with the adoption of the Next Generation EU, could offer a blueprint for establishing an ambitious EU industrial policy and represent a potential Hamiltonian moment for the Union.

It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus.

Early-life family meal participation has been associated with several aspects of nutritional health, but longitudinal associations with linear growth have not yet been investigated. The aim of this study was to investigate whether family meal participation at 12 months of age associates with anthropometric measures 3 years later. We used follow-up data from children born to mothers in the Norwegian Fit for Delivery trial (NFFD) and included 368 first-borns with dietary and anthropometric data at 12 months and 4 years of age. We treated the sample as a cohort and conducted subgroup analyses by randomization status. A family meal participation score was used as exposure, and weight, height, and body mass index (BMI) as outcomes in crude and multivariable linear regression models adjusted for maternal education, randomization status, and child sex.

Higher family meal participation score at 12 months was positively associated with length at 12 months (B = 0.198, 95% CI 0.028, 0.367, p = 0.022) and 4 years (B = 0.283, 95% CI 0.011, 0.555, p = 0.042) in multivariable models. After additional adjustment for maternal height the associations attenuated and were no longer significant. An inverse association with BMI at 4 years of age was observed in children born to mothers that had been exposed to the NFFD intervention (B = −0.144, 95% CI −0.275, −0.014, p = 0.030), but attenuated after adjustment for maternal BMI.

The longitudinal association observed between early family meal participation and child height was largely explained by maternal height. The relationship with BMI differed according to maternal participation in a lifestyle intervention trial during pregnancy.

Airborne particles, such as dust and volcanic ash, pose a serious hazard to aircraft in flight due to their potential to cause erosion damage to engine components. It is crucial to anticipate and address the impact of erosion wear on engine performance and safety. This study aims to enhance our understanding of how volcanic ash particles behave when ingested through a high bypass turbofan engine (HBTFE) and assess the development of erosion wear in the front components. The effects of four different ash samples are assessed in various scenarios of encountering volcanic ash during cruise flight conditions. First, the flow solution is obtained for all front components, including the Pitot intake, spinner, fan, inlet guide vanes (IGVs), outlet guide vanes (OGVs), and connecting ducts. Based on the flow data, the particle motion equations are solved step by step using an in-house trajectory and erosion code. This latter adopts the Lagrangian approach, which incorporates a particle-eddy interaction model and includes probabilistic descriptions for the release positions of particles, sizes, and restitution factors. The finite element method (FEM) is used to track particles through the computational cells and determine impact positions and conditions. As a result, the Pitot intake design seems to prevent many ash particles from reaching the fan blade beyond 80% of the span. The fan blade leading edge (LE) exhibits extreme erosion on both sides. The blade’s pressure side (PS) displays erosion spreading practically on the entirety of the surface, especially near the trailing edge (TE). In contrast, the suction side (SS) has scattered erosion at lower rates. Furthermore, the rotor’s hub presents almost uniform erosion patterns, whereas the shroud depicts scattered erosion. This large fan appears to function as a separator, expelling a significant amount of ash particles through the secondary duct, thereby reducing the engine core’s susceptibility to erosion. Out of the four volcanic ash samples, those from the Kelud and Etna volcanoes appear to cause the highest hourly eroded mass, about twice as much as the samples from the Chaiten and Eyjafjallajokull volcanoes.

We present a streamlined proof of a result essentially presented by the author in [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys. 28(4) (2008), 1291–1322], namely that for every set $S = \{s_1, s_2, \ldots \} \subset \mathbb {N}$ of zero Banach density and finite set A, there exists a minimal zero-entropy subshift $(X, \sigma )$ so that for every sequence $u \in A^{\mathbb {Z}}$, there is $x_u \in X$ with $x_u(s_n) = u(n)$ for all $n \in \mathbb {N}$. Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the polynomial Sarnak conjecture reported by Eisner [A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris 353(7) (2015), 569–572] which are significantly more general than some recently provided by Kanigowski, Lemańczyk and Radziwiłł [Prime number theorem for analytic skew products. Ann. of Math. (2) 199 (2024), 591–705] and by Lian and Shi [A counter-example for polynomial version of Sarnak’s conjecture. Adv. Math. 384 (2021), Paper no. 107765] and shows that no similar result can hold under only the assumptions of minimality and zero entropy.

Via an investigation of the collaboration between banks and foreign offices, this article demonstrates that the restructuring of China’s state debt before and after the revolution of 1911 was not simply a part of Chinese globalisation undertaken under the benign stewardship of foreign banks. Rather, the contest for Chinese loans reflected a power struggle for regional hegemony between expansionist states that hinged on both competition and cooperation, as well as Chinese attempts to preserve their economic and territorial sovereignty. It further shows that the notion that financial institutions operated independently of the state does not reflect the clear record of close coordination between states and financial institutions during loan negotiations.