On the Chromatic Numbers of Random Hypergraphs

Doklady Mathematics - The asymptotic behavior of the chromatic number of the binomial random hypergraph $$H(n,k,p)$$ is studied in the case when $$k \geqslant 4$$ is fixed, n tends to infinity, and...     

One-dimensional photonic crystals with a planar oriented nematic layer: Temperature and angular dependence of the spectra of defect modes

Transmission spectra of a one-dimensional photonic crystal (PC) formed by two multilayer dielectric mirrors and a planar oriented layer of 5CB nematic liquid crystal (LC) that is sandwiched between these mirrors and serves as a structure defect are investigated experimentally. Specific features of the behavior of the spectrum of defect modes as a function of the angle of incidence of light on the crystal are studied for two polarizations: parallel and perpendicular to the director of the LC; the director either lies in the plane of incidence or is perpendicular to it. It is shown that, for the configurations considered, the maxima of the defect modes shift toward the short-wavelength region as the tilt angle of incidence radiation increases; this tendency is more manifest for the parallel-polarized component, when the director lies in the plane of incidence. In the latter case, the width of the photonic band gap (PBG) appreciably decreases. The temperature dependence of the polarization components of the transmission spectra of a PC is investigated in the case of normal incidence of light. The spectral shift of defect modes due to the variation of the refractive index of the LC at the nematic-isotropic liquid phase transition point is measured. It is shown that, in real PCs, the amplitude of defect modes decreases when approaching the center of the band gap, as well as when the number of layers in the dielectric mirrors increases. Theoretical transmission spectra of the PCs calculated by the method of recurrence relations with regard to the decay of defect modes are in good agreement with experimental data.     

Study of a Composite Consisting of Metal Nanoparticles in a Dielectric Matrix and Multilayer Bandpass Filters Based on It

Doklady Physics - The frequency and concentration dependences of the effective complex permittivity of a composite consisting of silver nanoparticles in a polystyrene dielectric matrix have been...     

New binuclear complex of bis(2,4,6,8-tetramethyl-2,4,6,8- tetraazabicyclo(3.3.0)octane-3,7-dione-<Emphasis Type="Italic">O,O</Emphasis>')-diaquatetrakis(nitrato-<Emphasis Type="Italic">O,O</Emphasis>')-dimanganese(II) monohydrate: Synthesis and crystal structure

The centrosymmetric binuclear manganese(II) nitrate complex with a bicyclic bis-carbamide, namely, 2,4,6,8-tetramethyl-2,4,6,8-tetraazabicyclo(3.3.0)octane-3,7-dione, or mebicar (Mk) [Mn(C₈H₁₄N₄O₂)(H₂O)(NO₃)₂]₂ · H₂O (I), has been synthesized for the first time. The structure of complex I has been solved (CCDC no. 1435139). Crystals of complex I are monoclinic, space group P2₁/c, a = 12.8108(11) Å, _b = 10.0662(2) Å, c = 18.6367(17) Å, β = 136.512(16)°, V = 1654.0(4) ų, ρcalcd = 1.659 g/cm³, Z = 2. Each manganese atom is coordinated to the two oxygen atoms of two Mk molecules related by the symmetry codes (1–x, 2–y, 1–z) and to two bidentate nitrate anions and one water molecule. The coordination polyhedron of the manganese atom is a strongly distorted pentagonal bipyramid. The Mn···Mn distance in the complex is 8.7261(9) Å.     

Dielectric anisotropy of 5CB liquid crystal in a decimeter wavelength range

Parallel and perpendicular components of the dielectric constant for 5CB nematic liquid crystal are measured in a relaxation region. Two frequencies f ₁≈30 and f ₂≈350 MHz are determined at which the inversion of the dielectric anisotropy sign is observed. It is shown that the observed sizable high-frequency shift of f ₂ with respect to the value calculated according the Debye theory is caused by the dielectric resonance near the relaxation region.     

Mixed crystals of p-dichlorobenzene with p-bromchlorobenzene in the presence of vacancies in the structure

Polarization studies of low-frequency Raman spectra of solid solutions of p-dichlorobenzene with p-bromchlorobenzene (50% p-dichlorobenzene) were carried out. Analysis of the lattice vibration spectra of these mixed crystals shows that vacancies can be present in their structure. The presence of vacancies is responsible for the emergence of additional lines, including those in the 70 cm^?1 region. The calculation of the diffusion activation energy in a mixed crystal proves that its magnitude is determined by the spatial arrangement of p-bromchlorobenzene molecules in the lattice, their orientation relative to parahalides, and (to a lesser extent) temperature variations.     

Transformation of the radiation fluctuations of a low-coherence source in a two-arm interferometer

We consider fluctuations of the radiation of a low-coherence source transmitted through a two-arm interferometer and transformed by a square-law detector. It is assumed that the output radiation of the source is a stationary Gaussian narrow-band random process. The fluctuation process at the detector output is presented as the sum of the intensity noise and the interference-component noise. We calculate the relationships between the spectral densities of these components and the degree of their correlation for different delays between the interferometer arms. We show theoretically and confirm experimentally that the power of interference-component fluctuations is nonzero even if the delay between the interferometer arms exceeds the correlation time of the process at the source output. Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 4, pp. 350–356, September., 2000     

Metalation of toluene and cumene with alkali metal-crown ether complexes

Metalation of toluene and isopropylbenzene with alkali metal-crown ether complexes led to the corresponding α-metalated alkylbenzenes. Treatment of the latter in succession with solid carbon dioxide, water, and hydrochloric acid gave carboxylic or dicarboxylic acids in 65–78% yield. Metalation of isopropylbenzene with sodium or potassium crown ether complexes above 90°C was accompanied by cleavage of the polyether ring with formation of organometallic compounds which then reacted with isopropylbenzene to produce 2-sodio(potassio)-2-phenylpropane and open-chain oligoether.