The Triebel - Lizorkin Scale of Function Spaces for the Fourier - Helgason Transform

We define and investigate the Triebel - Lizorkin scale of function spaces F, with 1< p < ∞, 1< q ≤ ∞ for the Fourier-Helgason transform on symmetric Riemannian manifolds of the noncompact type.     

Noncommutative closed Friedman universe

Heller et al. (J Math Phys 46:122501, 2005; Int J Theor Phys 46:2494, 2007) proposed a model unifying general relativity and quantum mechanics based on a noncommutative algebra defined on a groupoid having the frame bundle over space–time as its base space. The generalized Einstein equation is assumed in the form of the eigenvalue equation of the Einstein operator on a module of derivations of the algebra . No matter sources are assumed. The closed Friedman world model, when computed in this formalism, exhibits two interesting properties. First, generalized eigenvalues of the Einstein operator reproduce components of the perfect fluid energy-momentum tensor for the usual Friedman model together with the corresponding equation of state. One could say that, in this case, matter is produced out of pure (noncommutative) geometry. Second, owing to probabilistic properties of the model, in the noncommutative regime (on the Planck level) singularities are irrelevant. They emerge in the process of transition to the usual space–time geometry. These results are briefly discussed.     

Numerical Analysis of Strains and Stresses in Stretched Specimens at Microstructure Level

In the paper an incremental methodology was proposed for the determination of the states of stresses and strains in specimens while taking into account the strain history and the strain rate history during a tension stress, and a new methodology to determine mechanical characteristics of metal materials. An equation of motion and deformation of the specimen was introduced while using a variational formulation and the finite element method. The results of numerical simulations which illustrate the methods proposed, were presented. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Selection of the eddy currents frequency for conductivity measurements in two-layer structures

Conductance measurements of conductive parts can serve as a background for conclusion about structure of material and possible defects (e.g. fractures, fissures, delamination, flaking, etc.) that may occur in examined parts. When the need arises to test surfaces of parts that are made of a conductive but non-ferromagnetic material, the method of eddy currents offers unsurpassed advantages. The problem refers to application of the eddy current method to examination of conductive material coated with conductive films. The major outcome consists in development of a very simple and useful mathematical model that can be used to determine conductivity of the deep layer for the needs of non-destructive tests. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Substituent effects on 15N NMR chemical shifts in selected N-alkylthiohydroxamic acids. A comparative study

The 15N NMR spectra of three N-alkyl-delta-carbomethoxyvalerothiohydroxamic acids (2) and six synthesized N-isopropylbenzothiohydroxamic acids (3) were measured and compared with appropriate spectra of structurally similar hydroxylamines (1), benzohydroxamic acids (4), benzamides (5) and thiobenzamides (6). The analysis of the chemical shifts of the thiohydroxamic acids under investigation indicates that the inductive effect of the hydroxyl group rather than steric hindrance is responsible for non-additivity of the effect of substituents. Additionally, N-hydroxyl diminishes the effect of aromatic ring substituents on the 15N chemical shifts in the thiohydroxamic acids 3 which is approximately half that in the respective thiobenzamides 6. The chemical shift values correlate best with Brown's sigma+ parameter.     

Raman study of TiO2 role in SiO2-Al2O3-MgO-TiO2-ZnO glass crystallization

Tough glass-ceramic material of special mechanical properties with nanosize crystal phases formed by appropriately controlled crystallization was studied by Raman spectroscopy. It was obtained by TiO2 activated crystallization of Mg-aluminosilicate glass of SiO2-Al2O3-MgO-TiO2-ZnO composition. Crystallization was preceded by a change in the TiO2 structural position and state, which is manifested by a changed color of glass from yellow into blue shortly before the glass transformation (Tg) temperature. Raman spectroscopy was applied to explain the mechanism of this process and to establish the role of TiO2 in the early stage of glass crystallization that precedes a complete crystal phase formation. The starting glasses were found in almost complete disorder, since all bands were weak, broad and dominated by a Bose band at about 90 cm-1. After the sample annealing all bands turned out better resolved and the Bose band practically disappeared, both confirming the amorphous structure reorganization process. A multiplet observed in the vicinity of 150 cm-1 we assigned to the anatase and other titania structures that can be considered prime centers of crystallization. Finally, in the closest neighborhood of the Rayleigh line the low frequency mode characterizing nanoparticles was observed. According to this band theory, the mean size of initial titania crystallites is about 10nm for all samples, but the size distribution varies within factor two among them.     

Nonsmooth generalized guiding functions for periodic differential inclusions

We consider the periodic problem for differential inclusions in $$ \user2{\mathbb{R}}^{\rm N} $$ with a nonconvex-valued orientor field F(t, ζ), which is lower semicontinuous in $$ \zeta \in \user2{\mathbb{R}}^{\rm N} $$ Using the notion of a nonsmooth, locally Lipschitz generalized guiding function, we prove that the inclusion has periodic solutions. We have two such existence theorems. We also study the “convex” periodic problem and prove an existence result under upper semicontinuity hypothesis on F(t, ·) and using a nonsmooth guiding function. Our work was motivated by the recent paper of Mawhin-Ward [23] and extends the single-valued results of Mawhin [19] and the multivalued results of De Blasi-Górniewicz-Pianigiani [4], where either the guiding function is C¹ or the conditions on F are more restrictive and more difficult to verify.