Linear differential equations and functional analysis,V

Mathematische Annalen -     

Linear differential equations and functional analysis. IX. Almost periodic equations

During the writing of the author's doctoral dissertation on almost periodic differential equations, he received valuable suggestions from ProfessorB. L. van der Waerden concerning the theory of stability of linear equations; these suggestions became an important germ of the author's contribution to the series of papers to which this is the latest addition. It is but fitting that, the wheel having come full circle, this chapter on almost periodic equations should be dedicatedto Professorvan der Waerden on his 60th birthday.     

Characterization of CuO/Al2O3 catalysts used in the oxidation of phenol solutions

Two CuO/Al₂O₃ catalysts were synthesized at different calcination temperatures and used for the oxidation of phenol solutions. Deactivation is due to leaching of the active phase and deposition of carbonaceous material.This revised version was published online in December 2005 with corrections to the Cover Date.     

Quantitative NIR-Raman analysis of methyl-parathion pesticide microdroplets on aluminum substrates

The potential of Raman spectroscopy in the quantitative analysis of dilute organic contaminants on aluminum substrates is evidenced in this work. Methyl-parathion microdroplets, an organophosphorus pesticide, has been used as a probe for this purpose. The samples were analyzed on an aluminum foil, which is very easy to acquire and to adapt. Moreover, aluminum foil does not need a previous treatment. Linear and no-linear curves as a function of the concentration of methyl-parathion versus the Raman intensity of the 1345 and 1110 cm(-1) peaks were established by means of a simple mathematical expression. A comparison with calibration curves fits very well, allowing quantification at concentration levels as low as parts per million.     

Bifurcations of Umbilic Points and Related Principal Cycles

The simplest patterns of qualitative changes on the configurations of lines of principal curvature around umbilic points on surfaces whose immersions into ³ depend smoothly on a real parameter (codimension one umbilic bifurcations) are described in this paper.Global effects, due to umbilic bifurcations, on these configurations such as the appearance and annihilation of periodic principal lines, called also principal cycles, are also studied here.     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.