Observations on the numerical stability of the Galerkin method

A simple factorization of the finite-dimensional Galerkin operators motivates a study of the numerical stability of a Galerkin procedure on the basis of its “potential stability” and the “conditioning” of its coordinate functions. Conditions sufficient for stability and conditions leading to instability are thereby identified. Numerical examples of stability and instability occurring in the application of the Galerkin method to boundary-integral equations arising in simple scattering problems are provided and discussed within this framework. Numerical instabilities reported by other authors are examined and explained from the same point of view. This revised version was published online in June 2006 with corrections to the Cover Date.     

The enumeration of homeomorphically irreducible labelled graphs

The counting series for homeomorphically irreducible labelled graphs is given. A linear recurrence equation is obtained to facilitate the tabulation of the number of such graphs on a specified number of vertices.     

Conformational analysis of trehalose disaccharides and analogues using MM3

Energy surfaces for the relative orientations of the pyranosyl rings of α,α-, α,β-, and β,β-trehalose and analogues were generated with MM3. Sixteen starting conformations of the rotatable side groups of α,β-trehalose were considered, while only 10 conformations were needed for α,α- and β,β-trehalose because of molecular symmetry. Energies were calculated at 20° increments of the two torsional angles of the glycosidic linkage, but otherwise the molecules were fully relaxed. The structure at the overall minimum for α,α-trehalose agrees well with that found in crystal structures, and also agrees with interpretations of NMR and optical rotation data. The energy surfaces for the three trehaloses differ greatly from each other, but are each similar to those for the corresponding three 2-(6-methyltetrahydropyran-2-yloxy)6-methyltetrahydropyrans. This suggests that linkage type (axial or equatorial) is more important than exocyclic substituents in determining trehalose conformations. A comparison with surfaces from the corresponding 5a-carba trehalose analogues illustrates that the exo-anomeric effect is important in determining disaccharide conformation.     

The effect of carbon sources on the single cell proteins and extracellular enzymes production byChrysonilia sitophila (TFB 27441 strain)

The single cell protein and extracellular enzyme production from a lignocellulolytic fungus,Chrysonilia sitophila, using different carbon sources were evaluated. The mycelial dry mass composition showed a high protein (39.2%) and low nucleic acid content (3.3%), as well as carbohydrate, fatty acid, fiber, and ash levels comparable with single cell proteins currently studied. Mycelial protein showed amino acid content similar to or higher than FAO standard requirements. The amino acid, fatty acid, and carotenoid composition, as well as mycelial mass yield and enzyme production, were dependent on the carbon source used. Glucose, saccharose, cellobiose, cellulose, microcrystal-line cellulose, lactose, and rice hull as carbon sources were studied.