Colorings of spaces,and random graphs

This work deals with some problems on the embeddings of finite geometric graphs into the random ones. In particular, we study here applications of the random graph theory to the Nelson-Erdös-Hadwiger problem on coloring spaces.     

Chromatic numbers of spaces with forbidden monochromatic triangles

New lower estimates for chromatic numbers of Euclidean spaces with forbidden monochromatic isosceles triangles are obtained.     

Synthesis and studies of benzoxazinesiloxanes

A synthesis of organosilicon 1,3-benzoxazines (BZ-Si) was accomplished by the reaction of organic phenols, paraformaldehyde and 3-aminopropyl(trialkoxy)silanes, as well as organosilicon phenols of the general formula R´[SiMe₂O]_n[SiMeRO]_mSiMe₂R´ (R´ is 4-hydroxy-3methoxyphenylpropyl-, R is methylor 4-hydroxy-3-methoxyphenylpropyl), paraformaldehyde, and aniline (or monoethanolamine). The structure of BZ-Si was confirmed by NMR and IR spectroscopy. The studies of BZ-Si by DSC showed that the benzoxazine ring opening takes place in the temperature range 130—265 °C. The temperature of 5% mass loss of BZ-Si in air is within 220—270 °C (TGA).     

On the chromatic numbers of low-dimensional spaces

New lower bounds are found for the minimum number of colors needed to color all points of a Euclidean space in such a way that any two points at a distance of 1 have different colors.     

On independence numbers of distance graphs with vertices in {-1,0,1}<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>: estimates,conjectures, and applications to the Nelson–Erdős–Hadwiger problem and the Borsuk problem

The paper states and studies a problem that is closely related to the problems mentioned in the title.     

On a series of problems related to the Borsuk and Nelson-Erdős-Hadwiger problems

In the present paper, a series of problems connecting the Borsuk and Nelson-Erd?s-Hadwiger classical problems in combinatorial geometry is considered. The problem has to do with finding the number χ(n, a, d) equal to the minimal number of colors needed to color an arbitrary set of diameter d in n-dimensional Euclidean space in such a way that the distance between points of the same color cannot be equal to a. Some new lower bounds for the quantity χ(n, a, d) are obtained.     

Synthesis of new polycarbonate-polysiloxanes based on oligomeric organosilicon bisphenols

New linear polycarbonate-polysiloxanes are synthesized through the heterophase polycondensation of α,ω-bis[3-(4-hydroxy-3-methoxyphenyl)propyl]oligoorganosiloxanes (P_Si-bisphenols) with α,ω-bis(chloroformato)oligocarbonates (method I), the phosgenation of P_Si-bisphenol-diphenylolpropane mixtures (method II), the interaction of the same bisphenols with bis(4-chloroformatophenyl)propane (method III), and the polycondensation of the latter with P_Si-bisphenol-α,ω-dihydroxyoligocarbonate mixtures (method IV). The highest molecular masses (as high as 130 × 10³ at a degree of multiblockiness of 11–14 block pairs) are inherent in poly-carbonate-polysiloxanes synthesized by methods II and III; moreover, the same copolymers have the highest mechanical characteristics (σ_br and ?_br are as high as 48 MPa and 300%, respectively).     

Small subgraphs in preferential attachment networks

In this paper, we give an overview of old and new results on the global and average local clustering coefficients as well as on the distribution of small subgraphs in the Bollobás–Riordan and Buckley–Osthus models of web graphs.