Benchmark results in vector atmospheric radiative transfer

In this paper seven vector radiative transfer codes are inter-compared for the case of underlying black surface. They include three techniques based on the discrete ordinate method (DOM), two Monte-Carlo methods, the successive orders scattering method, and a modified doubling-adding technique. It was found that all codes give very similar results. Therefore, we were able to produce benchmark results for the Stokes parameters both for reflected and transmitted light in the cases of molecular, aerosol and cloudy multiply scattering media. It was assumed that the single scattering albedo is equal to one. Benchmark results have been provided by several studies before, including Coulson et al. [22], Garcia and Siewert [7], [8], Wauben and Hovenier [10], and Natraj et al. [11] among others. However, the case of the elongated phase functions such as for a cloud and with a high angular resolution is presented here for the first time. Also in difference with other studies, we make inter-comparisons using several codes for the same input dataset, which enables us to quantify the corresponding errors more accurately.     

On Nonlinear Operator Approximation with Preassigned Accuracy

In this paper, we provide a state-of-the-art survey of some recent methods in nonlinear operator approximation theory and its applications. We give existence theorems for approximating operators. and discuss corresponding numerical schemes. The new results are natural but very specific extensions of known techniques to the so-called realistic operator approximation.     

Peculiarities of pπ?pπ conjugation in aminosubstituted phosphaalkenes

Effets of p_π p_π conjugation in phosphaalkenes have been considered by taking into account the results of X-ray structural studies and ab initio quantum-chemical calculations. The structural consequences of the conjugation depend to a certain extent on the character of the substituent and the place of its attachment to the double PC bond. Depending on the location of the substituent, bond polarization may strengthen or weaken (compensate for) the conjugation effects. A high contribution of the s-character of the lone electron pair of the P atom is an essential feature of the electron structure of a phosphaalkene.     

On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H_−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well‐known Sommerfeld integral, belongs to the Sobolev space H₁(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two‐dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright © 2000 John Wiley & Sons, Ltd.     

Phase behaviour of blends with mesophase components

Temperature transitions and structural transformations were studied for blends of two thermotropic mesophase cyclolinear polymethylsiloxanes with linear PDMS of various molecular weights by means of DSC, optical polarizing microscopy, and optical interferometry. Compatibility of the components which depends on the chemical structure of cyclolinear polymethylsiloxanes, molecular weight of PDMS, composition, and temperature affect formation of mesophase in cyclolinear polymethylsiloxanes. The most interesting aspect of the phase behaviour consists in the fact that it is possible to reach compatibility of the components in the mesomorphic state for the blends of two cyclolinear polymethylsiloxanens due to various annealing regimes in one‐phase molten state.     

Certain characteristic features of phospholane and phosphorinane systems and electrophilicity of antibonding orbitals of phosphorus atom

Within the already proposed qualitative model for evaluating the influence of the structure of a molecule on the value of hyperconjugation reactions, the problem is discussed of the reasons for the increased electron-acceptor capacity of diheterophospholane and diheterophosphorinane fragments compared with that of the analogous acyclic systems, and also the dependence of the corresponding ^ph constants on the solvent and the type of the reaction series. The change in the acceptor capacity is attributed to a change in the valence angles at the phosphorus atom, and replacement of the hetero atoms in the ring bound to the phosphorus atom. A similar approach is used to explain the dimerization and oligomerization of cyclic phosphorus, silicon, and germanium compounds.Translated from Teoreticheskaya i Eksperimental'naya Khimiya, Vol. 21, No. 2, pp. 159–165, March–April, 1985.     

A study of Galerkin method for the heat convection equations

The paper investigates the Galerkin method for an initial boundary value problem for heat convection equations. New error estimates for the approximate solutions and their derivatives in strong norm are obtained.     

The bound of supports of solutions for some classes of evolutionary systems and equations

We study the property of finiteness of a support for a system of evolutionary equations and for equations with nonstationary p-Laplacian and with convective terms. New bounds of the size of the support with regard for the convective terms are obtained.