Decompositions of spaces of measures

Let M be the Banach space of sigma-additive complex-valued measures on an abstract measurable space. We prove that any closed, with respect to absolute continuity norm-closed, linear subspace L of M is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon-Nikodym theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen-Wintner purity theorem for our decompositions.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

Steric effects in the selective hydrogenation of cinnamaldehyde to cinnamyl alcohol using an Ir/C catalyst

The liquid phase selective hydrogenation of cinnamaldehyde to cinnamyl alcohol has been carried out over a graphite-supported iridium catalyst. The effect of reaction parameters such as temperature, pressure, concentration of reactant, the effect of addition of product to the feed and pre-reduction of the catalyst were studied. In situ pre-reduction of the catalyst with hydrogen had a very significant enhancing effect on the conversion of cinnamaldehyde and selectivity of the catalyst to cinnamyl alcohol. Kinetic analysis of the pre-reduced catalyst showed that the reaction is zero order with respect to cinnamaldehyde and first order with respect to hydrogen. The reaction follows an Arrhenius behaviour with an activation energy of 37 kJ mol(-1). Detailed analysis of the reaction showed that hydrogenation of the C=C double bond to give hydrocinnamaldehyde predominantly occurred at low conversions of cinnamaldehyde (

Multiple Evidence for Gold(I)center dot center dot center dot Silver(I) Interactions in Solution

[AuAg3(C6F5)(CF3CO2)(3)(CH2PPh3)](n) (2) was prepared by reaction of [Au(C6F5)(CH2PPh3)] (1) and [Ag(CF3CO2)] (1:3). The crystal structures of complexes I and 2 were determined by X-ray diffraction, and the latter shows a polymeric 2D arrangement built by Au - Ag, Ag - Ag, and Ag - O contacts. The metallophilic interactions observed in 2 in the solid state seem to be preserved in concentrated THF solutions, as suggested by EXAFS, pulsed-gradient spin-echo NMR, and photophysical studies, which showed that the structural motif [AuAg3(C6F5)(CF3CO2)(3)(CH2PPh3)] is maintained under such conditions. Time-dependent DFT calculations agree with the experimental photophysical energies and suggest a metal-to-ligand charge-transfer phosphorescence process. Ab initio calculations give an estimated interaction energy of around 60 kJ mol(-1) for each Au - Ag interaction.

Desulfurisation of oils using ionic liquids: selection of cationic and anionic components to enhance extraction efficiency

Extraction of dibenzothiophene from dodecane using ionic liquids as the extracting phase has been investigated for a range of ionic liquids with varying cation classes (imidazolium, pyridinium, and pyrrolidinium) and a range of anion types using liquid-liquid partition studies and QSPR (quantitative structure-activity relationship) analysis. The partition ratio of dibenzothiophene to the ionic liquids showed a clear variation with cation class (dimethylpyridinium > methylpyridinium > pyridinium approximate to imidazolium approximate to pyrrolidinium), with much less significant variation with anion type. Polyaromatic quinolinium-based ionic liquids showed even greater extraction potential, but were compromised by higher melting points. For example, 1-butyl-6-methylquinolinium bis{(trifluoromethyl)sulfonyl} amide (mp 47 degrees C) extracted 90% of the available dibenzothiophene from dodecane at 60 degrees C.

Individual differences in trajectories of arithmetical development in typically achieving 5-to 7-year-olds

The arithmetical performance of typically achieving 5- to 7-year-olds (N = 29) was measured at four 6-month intervals. The same seven tasks were used at each time point: exact calculation, story problems, approximate arithmetic, place value, calculation principles, forced retrieval, and written problems. Although group analysis showed mostly linear growth over the 18-month period, analysis of individual differences revealed a much more complex picture. Some children exhibited marked variation in performance across the seven tasks, including evidence of difficulty in some cases. Individual growth patterns also showed differences in developmental trajectories between children on each task and within children across tasks. The findings support the idea of the componential nature of arithmetical ability and underscore the need for further longitudinal research on typically achieving children and of careful consideration of individual differences. (C) 2009 Elsevier Inc. All rights reserved.