Optimally small sumsets in finite abelian groups

Let G be a finite abelian group of order g. We determine, for all 1?r,s?g, the minimal size μ_G(r,s)=min|A+B| of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μ_G(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μ_G are recalled in the Introduction.     

Chlorine Dioxide as an Electron-Transfer Oxidant of Olefins. Preliminary Communication

The kinetics and product studies of oxidation of eight olefins 1 - 8 by ClO₂ in H₂O in the pH range 3-7 are described. The reaction is faster as the pH decreases. At pH < 4, ClO₂ reacts equimolarly with olefins to yield isomeric mixtures of chlorohydrines and 1,2-dioxygenated products, following the equation: The order of reactivity is: (E)-stilbene > indene > β-methylstyrene > acenaphthylene > α-methylstyrene > styrene > cyclohexene > allylbenzene. A multi-stage radical-cation mechanism is proposed, in which an initial reversible protonation: is followed by an electron-transfer stage (rate-determining): The cation-radical thus produced, adds rapidly an additional ClO₂ to form dioxygenated products. The chlorohydrines most likely arise from HClO additions to the olefinic double bonds, which, in turn, generate from dismutation of 2 HClO₂ into HClO + H⁺ + ClO.     

Old and new formulas for the Hopf–Stiefel and related functions

We give a list of the various formulas describing the Hopf–Stiefel function and provide direct, arithmetical proofs of their equivalence.     

Titanium Vacancies in TiO2 Nanofibers Enable Highly Efficient Photodriven Seawater Splitting

Photodriven seawater splitting is considered to be one of the most promising techniques for sustainable hydrogen production. However, the high salinity of seawater would deactivate catalysts and consume the photogenerated carriers. Metal vacancies in metal oxide semiconductors are critical to directed electron transfer and high salinity resistance; they are thus desirable but remain a challenge. We demonstrate a facile controllable calcination approach to synthesize TiO₂ nanofibers with rich Ti vacancies with excellent photo/electro performances and long-time stability in photodriven seawater splitting, including photocatalysis and photo-electrocatalysis. Experimental measurements and theoretical calculations reveal the formation of titanium vacancies, as well as unidirectional electron trap and superior H⁺ adsorption ability for efficient charge transfer and resistance to corrosion by seawater. Therefore, atomic-/nanoscale characteristics and mechanism have been proposed to clarify the generation of titanium vacancies and the corresponding interfacial electron transfer.