Algebraic properties of Hamel functions

We say that f: ℝ → ℝ is LIF if it is linearly independent over ℚ as a subset of ℝ² and that it is a Hamel function (HF) if it is a Hamel basis of ℝ². We construct an example of HF bijection and use a similar method to prove that any function can be represented as the composition of three HF’s as well as the limit of uniformly convergent sequence of HF’s. Finally we consider products of HF’s, maximal invariant classes (with respect to several algebraic operations) and pose some open problems concerning sets of continuity points of HF’s.     

Determination of fatty acid methyl esters: Elimination of tissue-derived contamination and artefacts and preparation of a support-coated acid-modified polyester liquid phase

Fatty acid methyl esters made by transesterification of extracted plasma lipids with methanolic hydrochloric acid were found to contain substances which interfered with quantitation by gas-liquid chromatography. A method to eliminate most interfering materials and a procedure that allows reproducible wetting of an inert support with acid-modified ethylene glycol succinate are reported. Hydrocarbon contamination and artefacts produced during transesterification from substances native to tissue matrices were identified by gas chromatography/mass spectrometry and removed by chromatography over 6% hydrated alumina. Ethylene glycol succinate was treated with orthophosphoric acid to improve wetting of silanized supports and extend the thermal stability of the polymer. Quantitation was achieved by using synthetic triheptadecanoin as an internal standard.     

Preparative separation of isomeric 2-(2-quinolinyl)-1H-indene-1,3(2H)-dione monosulfonic acids of the color additive D&C Yellow No. 10 (quinoline yellow) by pH-zone-refining counter-current chromatography

The main components of the color additive D&C Yellow No. 10 (Quinoline Yellow, Color Index No. 47005), 2-(2-quinolinyl)-1H-indene-1,3(2H)-dione-6'-sulfonic acid (6SA) and 2-(2-quinolinyl)-1H-indene-1,3(2H)-dione-8'-sulfonic acid (8SA), were isolated from the dye mixture by pH-zone-refining counter-current chromatography (CCC) in the ion-exchange mode. These positional isomers were separated from a portion of dye using sulfuric acid as the retainer acid and dodecylamine as the ligand (ion exchanger). The added ligand enhanced the partitioning of the hydrophilic components in the organic stationary phase of the two-phase solvent system that consisted of isoamyl alcohol-methyl tert.-butyl ether-acetonitrile-water (3:1:1:5). Thus, separation of 1.8 g of D&C Yellow No. 10 using the above method resulted in 0.6 g of 6SA and 0.18 g of 8SA of over 99% purity. The isolated compounds were characterized by mass spectrometry and proton nuclear magnetic resonance with correlated spectroscopy assignments. The study exemplifies a new field of applications for pH-zone-refining CCC, to the separation of positional isomers of strongly hydrophylic compounds containing sulfonic acid groups.