On the anomaly of the specific heat of real KH2PO4 crystals

The splitting of the anomaly revealed in the specific heat of real crystals of potassium dihydrogen phosphate KH₂PO₄ is discussed. It is demonstrated that the observed effect is characteristic of first-order phase transitions occurring near the tricritical point and originates from the specific features inherent in the defect structure of real crystals. A method is proposed for describing the experimentally observed splitting of the specific heat anomaly in the framework of the Landau theory with due regard for the possible formation of metastable states characterized by the polarization aligned antiparallel to the applied electric field in the polar phase.     

Reconstruction of the order parameter of oriented liquid crystal droplets

An optical method is proposed to reconstruct the order parameter for liquid crystal droplets with rigidly fixed poles. The method is based on measurements of the coherent transmission coefficient of a oneatom layer (monolayer) of polymer-dispersed liquid crystal droplets and on comparison with the results of the solution of the direct problem. In solving the direct problem for determining the coherent transmission coefficient of the polymer-dispersed liquid crystal monolayer, the anomalous diffraction approximation and the approximation of the effective refractive indices for liquid crystal droplets were used. The capabilities of the method are illustrated by the reconstruction of one component of the diagonal tensor of the droplet order parameter.     

Classification of ultrahigh-energy extensive air showers in terms of the muon content

Ultrahigh-energy extensive air showers detected at the Yakutsk array are analyzed. Showers different in the muon content are revealed. Four classes of showers are separated.     

Universal entire functions with gap power series

Let be the family of all compact sets in which have connected complement. For K ε M we denote by A(K) the set of all functions which are continuous on K and holomorphic in its interior.Suppose that {z_n} is any unbounded sequence of complex numbers and let Q be a given sub-sequence of ₀.If Q has density Δ(Q) = 1 then there exists a universal entire function with lacunary power series

1. (1) (z) = ε^∞_{v = 0 v}Z^v, _v = 0 for v Q, which has for all K ε M the following properties simultaneously

2. (2) the sequence {(Z + Z_n)} is dense in A(K)

3. (3) the sequence { (ZZ_n)} is dense in A(K) if 0 K.

Also a converse result is proved: If is an entire function of the form (1) which satisfies (3), then Q must have maximal density Δ_max(Q) = 1.     

Specific features in the vibronic fluorescence spectra of analogues of 1,4-bis(5-phenyl-2-oxazolyl)benzene with oxadiazole and furan rings

The fine-structure fluorescence and fluorescence excitation spectra of conjugated chain compounds, namely, 2,5-bis(5-phenyl-1,3,4-oxadiazol-2-yl)furan (PDFDP) and 2,5-bis[5-(2,4-dimethylphenyl)-1,3,4-oxadiazol-2-yl]furan (XDFDX), were obtained by the Shpolskii method in an n-octane matrix at a temperature of 4.2 K. These spectra were simulated by representing the band of each of the vibronic transitions as the sum of a zero-phonon line and a phonon wing with the corresponding parameters, such as the half-widths of the spectral lines and the Debye-Waller factors. The results obtained made it possible to estimate the relative intensities of the vibronic transitions between the S ₀ and S*₁ states. The anharmonicity revealed in the conjugate spectra of fluorescence and fluorescence excitation of the PDFDP and XDFDX compounds was explained in terms of the interference of the Franck-Condon and Herzberg-Teller interactions occurring in the molecules under investigation. The influence of the substitution of the furan heterocycle (F) for the central benzene ring (P) in 1,4-bis-(5-phenyl-2-oxadiazolyl)benzene (PDPDP) on the parameters of the intramolecular interactions responsible for the formation of the vibronic spectra was considered.     

Sub-Riemannian distance on the Lie group SL(2)

We find the distances between arbitrary elements of the Lie group SL(2) for the left invariant sub-Riemannian metric also invariant with respect to the right shifts by elements of the Lie subgroup SO(2) ? SL(2), in other words, the invariant sub-Riemannian metric on the weakly symmetric space (SL(2) × SO(2))/ SO(2) of Selberg.     

Electronic structure and stability of fullerene C82 isolated-pentagon-rule isomers

All nine isolated-pentagon-rule isomers of fullerene C(82) were investigated by the DFT method with the B3LYP functional at the 6-31G, 6-31G*, and 6-31+G* levels. The distribution of single, double, and delocalized π-bonds in the molecules of these isomers is shown for the first time. The obtained results are fully supported by DFT quantum-chemical calculations of electronic and geometrical structures of these isomers. The molecules of isomers 7 (C(3v)), 8 (C(3v)), and 9 (C(2v)) contain some radical substructures (such as the phenalenyl-radical substructure), which indicates that they are unstable and cannot be obtained as empty molecules. Thus, there is a possibility of obtaining them only as endohedral metallofullerenes or exohedral derivatives. Isomers 1 (C(2)), 2 (C(s)), 4 (C(s)), 5 (C(2)), and 6 (C(s)) with closed electronic shell are supposed to be stable, resembling isomer 3 (C(2)), which has just been extracted experimentally as an empty fullerene. We assume they can be produced as empty molecules.