Primitive Illumination Systems for Families of Convex Bodies in the Plane

Let F= {C₁,C₂,...,C} be a family of ndisjoint convex bodies in the plane. We say that a set Vof exterior light sources illuminates F, if for every boundary point of any member of Fthere is a point in Vsuch that is visible from ,i.e. the open line segment joining and is disjoint from F. An illumination system Vis called primitive if no proper subset of Villuminates F. Let p_max(F) denote the maximum number of points forming a primitive illumination system for F, and letp_max(n) denote the minimum of F) taken over all families Fconsisting of ndisjoint convex bodies in the plane. The aim of this paper is to investigate the quantities p_max(F) and p_max(n).     

Conformational analysis of 2-phenyl-1,3-diazabicyclo[3.1.0]hexane

The 2-phenyl-1,3-diazabicyclo[3.1.0]hexane molecule was subjected to conformational analysis within the framework of the Westheimer method. The optimum conformations of two of its possible stereoisomers (endo and exo) were found, and the equilibrium concentrations of the latter were calculated and found to be 62 and 38%, respectively. The results of the calculations are in good agreement with the PMR spectroscopic data.     

Reactions of 1-hydro-and 1-halosilatranes with metal salts

1-Hydrosilatrane reduces AgNO₃ and CuCl₂ in CH₃CN to form metallic silver and CuCl, respectively; other reaction products are the silatranyl nitrate O₂NOSi(OCH₂CH₂)₃N and the 1-chlorosilatrane ClSi(OCH₂CH₂)₃N. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 4, pp. 725–726, April, 2006.     

Synthesis and NMR spectroscopy of N,N-dialkyl-N' -(dimethyl-chlorosilylmethyl) ureas with an intramolecular Si←O bond

The reaction of N,N-dialkyl-N' -trimethylsilylureas with dimethyl (chloromethyl)-chlorosilane gave N,N-dialkyl-N'-(dimethylchlorosilylmethyl)ureas with an intramolecular SiO bond. Relatively stable N,N-dialkyl-N' -[dimethyl (chloromethyl) silyl]ureas are intermediates. The structures of the compounds obtained were demonstrated by NMR spectroscopy.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 12, pp. 2841–2843, December, 1989.     

Atranes

The kinetics of the hydrolysis of 1-(-chloroalkyl)silatranes (where R=ClCH₂ Cl₂CH, and CH₃ClCH, and n=1–3) at 25°C in neutral and acidic aqueous and aqueous alcoholic media with H₂O,²H₂O, and H₂ ¹⁸O were studied. The rate of hydrolysis in acidic media is considerably higher than in neutral media. The introduction of methyl groups in the 3, 7, and 10 position of the atrane ring and an increase in the electronegativity of the substituent attached to the silicon atom lower the rate of hydrolysis. According to the mass spectrometric data, the triethanolamine formed during hydrolysis in H₂ ¹⁸O does not contain¹⁸O, which indicates hydrolytic cleavage of the Si-O bond rather than the O-C bond.See [1] for communication LXVII.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 10, pp. 1344–1346, October, 1976.