An intermediate Baum-Katz theorem

We extend the classical Hsu-Robbins-Erd?s theorem to the case when all moments exist, but the moment generating function does not, viz., we assume that Eexp{(log⁺|X|)^α}<∞ for some α>1. We also present multi-index versions of the same and of a related result due to Lanzinger in which the assumption is that Eexp{|X|^α}<∞ for some α∈(0,1).     

Flowing maps to minimal surfaces: Existence and uniqueness of solutions

We study the new geometric flow that was introduced in the paper [12] of Topping and the author that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (weak) solution exists for as long as the metrics remain in a bounded region of moduli space, i.e. as long as the flow does not collapse a closed geodesic in the domain manifold to a point. Furthermore, we prove that this solution is unique in the class of all weak solutions with non-increasing energy. This work complements the paper of Topping and the author [12] where the flow was introduced and its asymptotic convergence to branched minimal immersions is discussed.     

The density of discriminants of -sextic number fields

We prove an asymptotic formula for the number of sextic number fields with Galois group and absolute discriminant . In addition, we give an interpretation of the constant in the formula in terms of the asymptotic densities of given local completions among these sextic fields. Our proof gives analogous results when we count -sextic extensions of any number field, and also when finitely many local completions have been specified for the sextic extensions.

     

Comproportionation of CO2 and Cellulose to Formate Using a Floating Semiconductor-Enzyme Photoreforming Catalyst

Formate production via both CO₂ reduction and cellulose oxidation in a solar-driven process is achieved by a semi-artificial biohybrid photocatalyst consisting of immobilized formate dehydrogenase on titanium dioxide (TiO₂| FDH ) producing up to 1.16±0.04 mmol_formate g ⁻¹ in 24 hours at 30 °C and 101 kPa under anaerobic conditions. Isotopic labeling experiments with ¹³C-labeled substrates support the mechanism of stoichiometric formate formation through both redox half-reactions. TiO₂| FDH was further immobilized on hollow glass microspheres to perform more practical floating photoreforming allowing vertical solar light illumination with optimal light exposure of the photocatalyst to real sunlight. Enzymatic cellulose depolymerization coupled to the floating photoreforming catalyst generates 0.36±0.04 mmol_formate per m² irradiation area after 24 hours. This work demonstrates the synergistic solar-driven valorization of solid and gaseous waste streams using a biohybrid photoreforming catalyst in aqueous solution and will thus provide inspiration for the development of future semi-artificial waste-to-chemical conversion strategies.