On Strassen's law of the iterated logarithm

Summary Kolmogorov's law of the iterated logarithm has been sharpened by Strassen who proved a more refined theorem by using tools from functional analysis. The present paper gives a classical proof of Strassen's theorem, using a method along the lines of Kolmogorov's original approach. At the same time the result proved here is more general since a) the random variables involved need not have the same distributions, b) the condition of independence is weakened and c) instead of Kolmogorov's growth condition on the random variables, only a mild restriction on their moments of order l3 is needed.     

Random walk in an inhomogeneous medium with local impurities

In spite of Sinai's result that the decay of the velocity autocorrelation function for a random walk on ^d (d=2) can drastically change if local impurities are present, it is shown that local impurities can not abolish weak convergence to the Brownian motion if d2.     

Non-conventional Behavior of a 2,1-Benzazaphosphole: Heterodiene or Hidden Phosphinidene?

Invited for the cover of this issue are Zoltán Benkő, Libor Dostál and co-workers at the University of Pardubice and the Budapest University of Technology and Economics. The image depicts signs for the two different pathways representing the two differing reaction types which were clearly observed for 2,1-benzazaphosphole. Read the full text of the article at 10.1002/chem.202101686 .     

An analytical method for the determination of stress and strain concentrations in textile-reinforced GF/PP composites with elliptical cutout and a finite outer boundary and its numerical verification

Stress concentrations in the vicinity of cutouts can often be regarded to be the limiting factor for a whole structure. As a further development of prior research at the Institute of Lightweight Engineering and Polymer Technology, an analytical method for the determination of the whole stress-strain fields in the vicinity of holes in multilayered textile-reinforced composites has been developed, which takes into consideration the influences of a finite outer boundary of the specimen. The analytical method is based on the classical laminate theory and the use of complex-valued potential functions. To account for the shape of the specimen, the method of conformal mappings is applied for the inner boundary, while a combination of boundary collocation and least squares method is used for the outer boundary. The method allows a layer-by-layer analysis of stress concentrations. For the verification of the developed calculation model, extensive experimental and numerical finite-element (FE) studies have been carried out on multilayered GF/PP plates with different laminate layups, notches, and specimen dimensions. The comparison of the experimentally or numerically determined results with the analytically calculated ones shows a very good correlation, of which the numerical studies are presented here for the first time. In a second step, the applicable boundary conditions on the outer boundary have been extended in such a way that varying stress and moment resultants can be applied, so that the calculation method can be used as an analytical sub-model in combination with FE techniques.     

Furazan-containing bromoarenes in the Suzuki-Miyaura reaction

The palladium-catalyzed cross-coupling reactions of 3-[bromo(het)aryl]furazans and bromobenzofurazans with arylboronic acids afford target biaryls in good yields. 3-Bromo-4-phenylfurazan containing a bromine atom in the furazan ring undergoes decomposition under the reaction conditions.     

The role of the solvation shell decomposition of alkali metal ions in their selective complexation by resorcinarene and its cavitand

2-Methylresorcinarene and its methylene-bridged cavitand derivative as host compounds were investigated in selective complexation of alkali metal ions as guests in methanol media by photoluminescence measurements. These host molecules possess either flexible (2-methylresorcinarene) or rigid (cavitand) molecular skeleton. The Benesi–Hildebrand method and the van't Hoff theory have been applied to determine the stability constants and the thermodynamic parameters, respectively. Considerable interactions between 2-methylresorcinarene and Li⁺ or Na⁺ ions have been observed while the rigid cavitand derivative can interact only with K⁺ or Cs⁺ ions. Neither the complexes of 2-methylresorcinarene with K⁺ or Cs⁺ nor those of the cavitand derivative with Li⁺ or Na⁺ ions are stable at room temperature in methanol media. Quantum-chemical investigations justified that only solvated Li⁺ and Na⁺ ions can form stable complexes with 2-methylresorcinarene while unsolvated K⁺ and Cs⁺ ions form stable complexes with the methylene-bridged cavitand. These results highlight that the stability of the guest solvation shell and its size could play a key role in the selectivity behaviour of host molecules.