We establish Diophantine inequalities for the fractional parts of generalized polynomials, in particular for sequences $\unicode[STIX]{x1D708}(n)=\lfloor n^{c}\rfloor +n^{k}$ with $c>1$ a non-integral real number and $k\in \mathbb{N}$, as well as for $\unicode[STIX]{x1D708}(p)$ where $p$ runs through all prime numbers. This is related to classical work of Heilbronn and to recent results of Bergelson et al.

Crystallite size determined by X-ray line profile analysis is often smaller than the grain or subgrain size obtained by transmission electron microscopy, especially when the material has been produced by plastic deformation. It is shown that besides differences in orientation between grains or subgrains, dipolar dislocation walls without differences in orientation also break down coherency of X-rays scattering. This means that the coherently scattering domain size provided by X-ray line profile analysis provides subgrain or cell size bounded by dislocation boundaries or dipolar walls.

Using the dislocation model of strain anisotropy in X-ray diffraction peak profile analysis it is shown that in nanocrystalline copper produced by inert gas condensation dislocations are present, at least, down to average grain sizes of the order of 20 nm. Based on the analysis of the dislocation contrast factors it is suggested that with decreasing grain size the proportion of Lomer-Cottrell type dislocations increases.

For a wide class of one-parameter families of Thue equations of arbitrary degree n[ges ]3 all solutions are determined if the parameter is sufficiently large. The result is based on the Lang–Waldschmidt conjecture, on the primitivity of the associated number fields and on an index bound, which does not depend on the coefficients. By applying the theory of Hilbertian fields and results on thin sets, primitivity is proved for almost all choices (in the sense of density) of the parameters.

A rapid increase of the yield stress with increasing temperature, often observed in L12 ordered alloys, is commonly called the “anomalous flow behavior”. This phenomenon is believed to result from the thermally activated transformation of the core of 1/2<110> screw dislocations from a glissile form to a sessile form at high temperatures. It is shown here that another class of L12 alloys exists in which these two forms of the screw dislocation core are not available. These are the alloys in which the APB on {111} planes is not stable and the atomistic studies of screw dislocations in such alloys show that their cores are always sessile. The yield stress of these alloys then increases with decreasing temperature and no increase at high temperatures occurs. Such behavior has been observed, for example, in Pt3Al. This “normal” behavior is analogous to that of b.c.c. metals and a theory of the temperature dependence of the yield stress has been developed along the same lines as in the case of b.c.c. metals. Comparison of this theory with measurements on Pt3Al single crystals shows a good agreement.