How Fast can a Subharmonic Function Decay Along a Ray?

Much is known about the connection between the growth and decayof subharmonic functions. The results indicate that there isa general principle: asubharmonic function cannot decay ‘toofast’ relative to its growth.Three theorems are provedwhich, together with work previously published elsewhere, givea fairly complete account of how this principle works out fora subharmonic function having extremal decay along a ray. 1991Mathematics Subject Classification: 30D20, 31A05.     

The nickel-substituted quasi-Wells-Dawson-type polyfluoroxometalate,

A series of transition metal substituted polyfluorooxometalates (PFOM) [M(L)H2F6NaW17)55]q-, M= Zn2+ , Co2+, Mn2+, Fc2+, Ru2+, Ni2+ and V5+ and L=H2O, O2-, of quasi-Wells-Dawson structure, was synthesized. In the series prepared, only the nickel-substituted polyfluorooxometalate was capable of catalytic activation of hydrogen peroxide in biphasic reaction media, the reaction leading mainly to the selective epoxidation of alkenes and alkenols. The manganese-, cobalt-, ruthenium-, iron-, vanadium-, and zinc-substituted polyfluorooxometalates were catalytically inactive, although, except for the zinc polyfluorooxometalate, very significant catalase activity was observed. Oxidation of thianthrene showed that sulfoxides were oxidized more easily than sulfides. Kinetic profiles of cyclooctene epoxidation showed that the reaction was zero order in both cyclooctene and hydrogen peroxide. Hydrogen peroxide was consumed at a rate 40% higher than the rate of epoxidation of cyclooctene. The reaction appears to proceed through an intermediate peroxo/hydroperoxo species that was observed in the IR spectrum. Atomic absorption, IR and 19F NMR spectroscopy indicated that the [Ni(H2O)H2F6NaW17O55]9- compound was stable under reaction conditions.     

Qualitative properties of modified equations

Suppose that a consistent one-step numerical method of orderr is applied to a smooth system of ordinary differential equations.Given any integer m 1, the method may be shown to be of orderr + m as an approximation to a certain modified equation. Ifthe method and the system have a particular qualitative propertythen it is important to determine whether the modified equationsinherit this property. In this article, a technique is introducedfor proving that the modified equations inherit qualitativeproperties from the method and the underlying system. The techniqueuses a straightforward contradiction argument applicable toarbitrary one-step methods and does not rely on the detailedstructure of associated power series expansions. Hence the conclusionsapply, but are not restricted, to the case of Runge-Kutte methods.The new approach unifies and extends results of this type thathave been derived by other means: results are presented forintegral preservation, reversibility, inheritance of fixed points.Hamiltonian problems and volume preservation. The techniquealso applies when the system has an integral that the methodpreserves not exactly, but to order greater than r. Finally,a negative result is obtained by considering a gradient systemand gradient numerical method possessing a global property thatis not shared by the associated modified equations.     

Nitrosonium salts, NO+ X- (X = B(3,5-diCF3Ph)4(-) or PW12O40(3-)), as electrophilic catalysts for alkene activation in arene alkylation and dimerization reactions

It has been found that in apolar reaction media the nitrosonium cation (NO+) activated alkenes under mild conditions toward electrophilic substitution of arene substrates to yield the alkylated arene with Markovnikov orientation. In the absence of arenes the alkenes react with themselves to yield a mixture of dimeric alkenes. The nitrosonium cation can be dissolved in the reaction medium by using the tetrakis-(bis-(3,5-trifluromethyl)phenyl) borate anion, where upon the reactions occur effectively at 30 degrees C. Alternatively an insoluble, heterogeneous catalyst was prepared so as to yield a NO+ cation with a polyoxometalate (PW12O403-) anion. This catalyst was generally more effective and selective toward a broader range of substrates at 70 degrees C.     

Anisotropy of optical,electrical, and photoelectrical properties of amorphous hydrogenated silicon films modified by femtosecond laser irradiation

Two types of independent anisotropic structures have been formed simultaneously in amorphous hydrogenated films by applying a femtosecond laser pulse to them, i.e., a structure with a period of several micrometers to several tens of micrometers and a structure with a period of several hundred nanometers. The formation mechanisms of these strictures are different, which allows us to orient them relative to each other in a desirable way. Both structures independently influence the optical properties of the modified films, which causes the diffraction of transmitted light and making the films polarization-sensitive. The conductivity of the modified films correlates with the mutual orientation of the anisotropic structures, whereas no interrelation between the photoconductivity and optical performance of the modified films has been observed.