On an analogue of parallel translation on Carnot–Carathéodory spaces

A new idea of constructing the parallel translation map on a Carnot–Carathéodory space is proposed, the basic properties of this map are described, and the polynomials approximating graph mappings are qualitatively described.     

Maximal Surfaces on Five-Dimensional Group Structures

For the classes of the mappings Lipschitz in the sub-Riemannian sense and taking values in the Heisenberg group we introduce some suitable notions of variation of an argument and the corresponding increment of the area functional and derive several basic properties of maximal surfaces on the five-dimensional sub-Lorentzian structures.     

Class of Maximal Graph Surfaces on Multidimensional Two-Step Sub-Lorentzian Structures

Necessary maximality conditions for graph surfaces in a class of two-step sub-Lorentzian structures are obtained. The concept of a sub-Lorentzian mean curvature is introduced, and equations for maximal surfaces are deduced.     

Synthesis of amide derivatives of chlorin <Emphasis Type="Italic">e</Emphasis><Subscript>6</Subscript>

A number of secondary and tertiary chlorin e ₆-13-amides were synthesized in high yield by the action of primary and secondary amines on methylpheophorbide a under mild conditions. Unlike the secondary amides, tertiary 13-amides were shown to exist as two stereoisomers differing by orientation of the amide group plane with respect to the macroring. The reaction of methylpheophorbide a with 2-aminoethanol gave chlorin e ₆ derivatives containing one, two, and three hydroxy groups.     

Account of the background in wavelength dispersive X-ray electron probe microanalysis based on the simulation of bremsstrahlung X-radiation

The potentials of six approximation functions described in the literature and used to simulate bremsstrahlung X-radiation in electron probe microanalysis are analyzed within a uniform approach. An algorithm based on the use of wave spectra is proposed for determining the parameters of the functions. It is found that the calculated values of some parameters of approximation functions considerably differ from those presented by the authors of the functions. It is found that the quality of approximating bremsstrahlung X-radiation is strongly affected by the method of calculating the average atomic number of the sample and the corrections for matrix effects. In the wavelength range 0.1 ≤ λ ≤ 1.2 nm for atomic numbers 10 ≤ Z ≤ 83 and accelerating voltages 15 ≤ E ₀ ≤ 25 kV, the best function ensures background estimation with a relative standard deviation of 3–6%, which is comparable with the relative standard deviation of the direct background measurement in routine electron probe microanalysis. The use of the calculation method of the background account at λ > 1.2 nm is complicated because of the significant uncertainty of the absorption factor in the long-wavelength region for the majority of elements.