Higher-order Sobolev embeddings and isoperimetric inequalities

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev type inequalities of any order, involving arbitrary rearrangement-invariant norms, on open sets in Rⁿ${\mathbb{R}}^n$, possibly endowed with a measure density, are reduced to much simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a consequence, the optimal target space in the relevant Sobolev embeddings can be determined both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, our results are applied to any-order Sobolev embeddings in regular (John) domains of the Euclidean space, in Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and in families of product probability spaces, of which the Gauss space is a classical instance.     

Stage-I fatigue crack studies in order to validate the dislocation-free zone model of fracture for bulk materials

Stage-I fatigue cracks are commonly described by the model of Bilby, Cottrell and Swinden (BCS model). However, since several experimental investigations have shown a dislocation-free zone (DFZ) in front of crack-tips, it is necessary to validate the new DFZ model and to examine the deviations to the BCS model. Therefore, the dislocation density distribution is derived from height profiles of slip lines in front of stage-I fatigue cracks in CMSX4® single crystals measured by contact-mode atomic force microscopy. This is possible, because the cracks are initiated at notches milled by focused ion beam technique directly on slip planes with a high Schmid factor. Consequently, the directions of the Burgers vectors are well known; it is possible to calculate the dislocation density distributions from the height profiles. The measured distributions are compared to the calculated distribution function of the DFZ model proposed by Chang et al. The additionally measured microscopic friction stress of the dislocations is then used to calculate the influence of grain boundaries on the dislocation density distribution in front of stage-I cracks. The calculation is done by the extended DFZ model of Shiue et al. and compared with the measured distribution function in polycrystalline specimens. Finally, the crack-tip sliding displacement as a measure for the crack propagation rate is compared for the DFZ model and the BCS model with the experimentally revealed values. The important result: the often used BCS model does not reflect the experimental measurements. On the contrary, the DFZ model reflects the measurements at stage-I cracks qualitatively and quantitatively.     

2,4-Dimethoxycarbonyl-1,1,3,3-tetrakis(dimethylamino)-1λ5,3λ5-diphosphete

1,1,3,3 - Tetrakis(dimethylamino) - 1λ⁵,3λ⁵ - diphosphete, 1 , reacts with cyanoformic acid methyl ester to form two isomers of the title compound 2 . Properties, NMR, mass, and IR spectra of 2 , as well as the molecular and crystal structure of the E isomer of 2 , are described and discussed.     

Non-markovian effects in NMR relaxation processes: An application to molecular dynamics in branched and cross-linked solid polymers

The dynamic of restricted rotatranslation of mobile structural defects along a polymer chain are treated with an equation of motion with a non-Markovian, stochastic force. The MORI/KUBO/ZWANZIG representation gives a stochastic process with a memory where the loss of correlation is dependend on the restriction of diffusional motion of defects. The influence of crosslinks and branchings of the polymer chains on the molecular dynamics are discussed. The analytical spectral densities of ω^−-α-type characterized by the fractal dimensions of the relevant processes in the different polymeric systems, e. g. cis-polybutadiene, cis-polyisoprene, styrene-butadiene-rubber, high density and linear low density polyethylenes, are compared with the observed NMR data.