A split-step finite difference method for nonparaxial nonlinear Schrödinger equation at critical dimension

The critical nonlinear Schrödinger equation (NLS) is the model equation for propagation of laser beam in bulk Kerr medium. One of the final stages in the derivation of NLS from the nonlinear Helmholtz equation (NLH) is to apply paraxial approximation. However, there is numerical evidence suggesting nonparaxiality prevents singularity formation in the solutions of NLS. Therefore, it is important to develop numerical methods for solving nonparaxial NLS. Split-step methods are widely used for finding numerical solutions of NLS equation. Nevertheless, these methods cannot be applied to nonparaxial NLS directly. In this study, we extend the applicability of split-step methods to nonparaxial NLS by using Padé approximant operators. In particular, split-step Crank-Nicolson (SSCN) method is used in conjunction with Padé approximants to provide examples of numerical solutions of nonparaxial NLS.     

Radial growth of Bloch functions in the complex ball

We show that N.G. Makarov’s law of the iterated logarithm governs the radial growth of weighted Bloch functions in the complex unit ball. The argument is based on an analog of M. Weiss’ theorem for special lacunary series in the ball.     

Multipliers of Fractional Cauchy Transforms

Let B_n denote the unit ball of , n ≥ 2. Given an α > 0, let denote the class of functions defined for by integrating the kernel against a complex-valued measure on the sphere . Let denote the space of holomorphic functions in the ball. A function is called a multiplier of provided that for every . In the present paper, we obtain explicit analytic conditions on which imply that g is a multiplier of . Also, we discuss the sharpness of the results obtained. This research was supported by RFBR (grant no. 08-01-00358-a), by the Russian Science Support Foundation and by the programme “Key scientific schools NS 2409.2008.1”.     

An inner function which is not weak outerUne fonction intérieure qui n'est pas faiblement extérieure

We construct a nonvanishing inner function I in the unit ball $\text{B?}\text{C}{}^{\mathrm{n}}$ such that the subspace IH^p(B) is not weakly dense in the Hardy space H^p(B), with 0<p<1. To cite this article: E. Doubtsov, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 957–960.     

The role of cloud-scale resolution on radiative properties of oceanic cumulus clouds

Both individual and combined effects of the horizontal and vertical variability of cumulus clouds on solar radiative transfer are investigated using a two-dimensional (x- and z-directions) cloud radar dataset. This high-resolution dataset of typical fair-weather marine cumulus is derived from ground-based cloud radar observations. The domain-averaged (along x-direction) radiative properties are computed by a Monte Carlo method. It is shown that (i) different cloud-scale resolutions can be used for accurate calculations of the mean absorption, upward and downward fluxes; (ii) the resolution effects can depend strongly on the solar zenith angle; and (iii) a few cloud statistics can be successfully applied for calculating the averaged radiative properties.     

In situ XPS investigation of Pt(Sn)/Mg(Al)O catalysts during ethane dehydrogenation experiments

Calcined hydrotalcite with or without added metal (Mg(Al)O, Pt/Mg(Al)O and Pt,Sn/Mg(Al)O) have been investigated with in situ X-ray photoelectron spectroscopy (XPS) during ethane dehydrogenation experiments. The temperature in the analysis chamber was 450 °C and the gas pressure was in the range 0.3-1 mbar. Depth profiling of calcined hydrotalcite and platinum catalysts under reaction, oxidation and in hydrogen-water mixture was performed by varying the photon energy, covering an analysis depth of 10-21 Å. It was observed that the Mg/Al ratio in the Mg(Al)O crystallites does not vary significantly in the analysis depth range studied. This result indicates that Mg and Al are homogeneously distributed in the Mg(Al)O crystallites. Catalytic tests have shown that the initial activity of a Pt,Sn/Mg(Al)O catalyst increases during an activation period consisting of several cycles of reduction-dehydrogenation-oxidation. The Sn/Mg ratio in a Pt,Sn/Mg(Al)O catalyst was followed during several such cycles, and was found to increase during the activation period, probably due to a process where tin spreads over the carrier material and covers an increasing fraction of the Mg(Al)O surface. The results further indicate that spreading of tin occurs under reduction conditions.A PtSn₂ alloy was studied separately. The surface of the alloy was enriched in Sn during reduction and reaction conditions at 450 °C. Binding energies were determined and indicated that Sn on the particle surface is predominantly in an oxidised state under reaction conditions, while Pt and a fraction of Sn is present as a reduced Pt-Sn alloy.     

Corrected outer functions

Given and a strictly positive continuous function on the unit circle, we construct a bounded analytic function such that a.e., and is in the Besov space on the unit disc.

     

An experimental and theoretical study of the type C enone rearrangement: mechanistic and exploratory organic photochemistry

We recently described a new photochemical rearrangement which we termed a Type C process. The reaction involves a delta to alpha aryl migration in 5-disubstituted cyclohexenones also having bulky C-3 substituents. In contrast to most cyclohexenone rearrangements, the reaction occurs via a twisted pi-pi excited triplet rather than the usual n-pi state. The electronic nature of the rearrangement was assessed using migration selectivity with p-anisyl and p-cyanophenyl groups. A synthesis of the reactants was elaborated, and the product structures were established by X-ray and NMR analysis. The reaction mechanism was established further with DFT and CASSCF computations. In the latter, localized NBO basis orbitals permitted proper selection of the active space. The nature of the diradical intermediates as well as the transition states was established computationally. Sensitization experiments with regioselectivities the same as those in direct irradiation confirmed the triplet multiplicity of the process.     

Influence of pore dimension and sorption configuration on the heat of sorption of hexane on monodimensional siliceous zeolites

Sorption of n-hexane on monodimensional pure silica SSZ-35, CIT-5, ZSM-12, and ZSM-22 zeolites with different pore dimension and on recently synthesized ITQ-29 was studied by IR spectroscopic and computational chemistry methods. Heats of sorption of n-hexane on these zeolites was determined experimentally from the temperature dependence of the intensity of IR bands of sorbed hexane as well as from theoretical calculations. Calculations have shown the different orientations of sorbed hexane molecules inside zeolite channels, which depend on the type of zeolite and loading. At high loadings, ordering of hexane inside the channels is observed due to optimization of sorbate-sorbate and sorbate-zeolite interaction energies. Such ordering is responsible for the increase of the sorption energy. A decrease of the sorption energy upon increasing the pore dimension of zeolite was observed, in agreement with results previously published in the literature. Effects of pore diameter of zeolites and ordering of molecules inside zeolite channels on the sorption energy of hexane are discussed.