On Normed Lattices and Their Banach Completions

It is known that the Banach completion Y = bX of a normed lattice X need not preserve the properties to be Dedekind complete or σ-Dedekind complete. In this paper it is proved that the countable interpolation property and the property to be sequentially order complete are preserved under the Banach completion. To prove this results we found some sufficient conditions (which are close to necessary ones) on X which secure for Y to have the countable interpolation property and (respectively) to be sequentially order complete. These conditions are obtained with the help of the newly developed techniques based on representations of normed lattices. It is well known that order continuity, and σ-order continuity of a norm are preserved under the Banach completion. Here necessary and sufficient conditions on X to secure these properties in Y are discussed. Mathematics Subject Classification 2000: 46B42, 46E15     

On the Structure of the Completion of a Normed Lattice

Let X be a normed lattice and Y be the norm completion of X with a natural embedding π : X → Y . By the Kawai- Luxemburg theorem, X is embedded as an order dense set and π preserves all suprema and infima iff X satisfies the condition (A_o ) (i.e., the norm has pseudo σ-Lebesgue property). Let X_o be the largest ideal in X having the condition (A_o); let Y_(o) be the band in Y generated by πX_o and Y₍₁₎ be the complementary band to Y_(o). The structure of Y and, in particular, of the bands Y_(o) and Y₍₁₎ are studied. The conditions for Y_(o) to be a projection band and πX_o to be topologically dense in Y_(o) are obtained.     

Lasing efficiency of dyes incorporated into a nanoporous glass-polymer composite

We have used microsecond coherent pumping to study the lasing efficiency of active elements based on a nanoporous glass-polymer composite activated by organic dyes, as a function of the spectral composition of the pump within the main absorption band of the dye. We have shown that under the indicated excitation conditions, the lasing efficiency can exceed 40% and can be comparable with the lasing of liquid elements. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 75, No. 5, pp. 708–716, September–October, 2008.     

New Oscillatory Process in Aqueous Solutions of Compounds Containing Carbonyl and Amino Groups1

Frequency-correlated oscillations of chemiluminescence and the redox potential are observed during the amino-carbonyl reaction, which occurs in aqueous solutions of methylglyoxal and various amino-containing compounds (ethanolamine, glycine, and -alanine) at room temperature. The duration and amplitude of oscillations depend on the composition and concentration of the starting reactants, pH, stirring, and surface area through which oxygen diffuses into the system.