A topological analysis of electron density and chemical bonding in cyclophosphazenes PnNnX2n (X = H, F, Cl; n = 2, 3, 4)

The restricted Hartree-Fock method was used to determine the cycle size effects on the geometric parameters of several inorganic templates, cyclophosphazenes P_nN_nX_2n (X = H, F, Cl; n = 2, 3, 4). A topological analysis of local electronic properties at the electron density critical points of bonds allowed us to quantitatively characterize the chemical bond in cyclophosphazenes and its dependence on the cycle size and substituents at phosphorus. The calculated distributions of the electron density Laplacian and electron pair localization functions revealed the special features of the electronic structure of the nitrogen and phosphorus atoms. These results explain the nature of noncovalent interactions between the P atoms of one cyclophosphazene molecule and the N atoms of the other.     

The thermodynamic characteristics of formation of vacancies in carbon subgroup element crystals

A method for calculating the parameters of formation of vacancies in crystals formed by spherically symmetrical atoms was developed. Both quantum effects at low temperatures and the possibility of the delocalization of atoms at high temperatures were studied. The parameters of formation of vacancies in carbon subgroup element crystals C-diam, Si, Ge, α-Sn, and Pb were calculated. The inclusion of the delocalization of atoms was shown to increase the enthalpy, entropy, and volume of vacancy formation. At low temperatures, the parameters of vacancy formation were found to depend strongly on the temperature, and the entropy of vacancy formation became negative. At high temperatures, close agreement with experimental data and theoretical estimates reported by other authors was obtained. The temperature dependence of vacancy parameters was studied for diamond heated isobarically from 100 to 4500 K. The applicability scope of the Arrhenius equation with a temperature-independent activation energy is discussed. The validity of the “compensation rule” (correlation between the entropy and enthalpy of vacancy formation) was demonstrated. It was also shown that the volume and entropy of vacancy formation were correlated over the whole temperature range studied.

     

A rigid subspace of the real line whose square is a homogeneous subspace of the plane

Working in ZFC, we give an example as indicated in the title.

     

混合Bitsadze-Lavrentév-Tricomi边值问题

It is well known that F. G. Tricomi (1923) is the originator of the theoryof boundary value problems for mixed type equations by establishing the Thicomi equation: y·u_xx+u_yy=0 which is hyperbolic for y < 0, elliptic for y=0. and parabolic for y= 0 and then applied it in the theory of transonic flows.Then A.V.Bitsadze together with M. A . Lavrent′ev (1950) established the Bitsadze Lavre nt′ev equation: sgn( y ) ·u_xx+u_yy=0 where sgn(y) = 1 for y > 0, = -1 for y<0, 0 for y=0 with the discontinuous coefficient sgn( y ) of u_xx, while in the case of Tricomi equation the corre sponding coefficient y is continuous. In this paper we establish the mixed Bitsadze Lavrent′ev Tricomi equation. Lu=K(y)·u_xx+sgn(x) ·u_yy+r(x,y)·u=f(x,y), where the coefficient K=K(y) of u_xx is increasing continuous and coefficient M=sgn(x) of u_yy discontinuous, r=r(x,y) is once continuously differentiable, f=f(x,y) continuous. Finally we prove the uniqueness of quasi regular solutions and observe that these new results can bbe applied in fluid dynamics.     

An investigation on planar dielectric light collectors for illumination and solar energy applications

In this paper the performances of inhomogeneous dielectric slabs as solar light collectors for both illumination and solar energy applications are analysed.     

Localization and the integer-quantized Hall effect: Diffusion constant for high Landau levels and white noise potential

The diffusion constant and the diagonal conductivity for non-interacting electrons in a two-dimensional, disordered system are studied. A homogeneous magnetic field perpendicular to the electron system is assumed. For weak short-range random potentials and high fields the Landau quantum numbern can be used as expansion parameter. In the limit of high Landau levels the system shows metallic behaviour. Corrections for finiten decrease the conductivity and indicate localized states in the whole energy band. A breakdown of the expansion and stronger localization are observed only for the lowest Landau levels if the typical experimental length scale of the quantized Hall effect is used.