Calculs de complexité relatifs à une méthode de dissection emboîtée

Résumé Nous présentons une numérotation de type Nested Dissection des inconnues d'un système linéaireAX=B pour des ensembles de matricescreuses symétriques définies positives correspondant à des famille de graphes non orientés,à degré borné, et admettant un -thérème de séparation. Comparativement aux méthodes et résultats de Rose [9], l'algorithme présenté est plus simple, mais les théorèmes de complexité moins généraux, en raison de l'hypothèse restrictive de degré borné. En outre, les démonstrations font appel en permanence à la structure d'arbre sur la famille des séparateurs qui constitue, par ailleurs, une partition de l'ensemble des sommets du graphe initial. Nous présentons ensuite le schéma général d'implémentation dans le cadre du code d'éléments finis MODULEF pourdes problèmes plans d'éléments finis, et nous donnons quelques mesures comparatives avec la numérotation plus classique qui tend à minimiser le profil de la matrice.

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Complexity bounds for a nested dissection method

Summary A nested dissection ordering is given for solving any system of linear equationsAX=B, for the family ofsparse symmetric positive definite matrices corresponding to the class of undirected graphs withbounded degree, and satisfyinga -separator theorem. If we compare the methods and results presented by Rose [9], our algorithm is more simple, but the complexity results are less general because of the restriction of bounded degree. Besides, our proofs use continually the arborescent structure on the family of separators, which is, by another way, a partition of the set of vertices for the initial graph. Then, the general implementation scheme in the finite element package MODULEF, fortwo-dimensional finite element problems, is presented, and numerical comparisons between our ordering and the standard envelope method are given.

     

Arylselenenation of conjugated dienes by arylselenenamides in the presence of phosphorus(V) oxyhalides

Arylselenenation of conjugated and non-conjugated dienes by arylselenenamides in the presence of phosphorus(V) oxyhalides has been studied. Reactions with conjugated dienes lead to 1,4-adducts whilst only addition to double bonds takes place in reactions with non-conjugated dienes.     

Neue Macrolide und einige Sesquiterpen-Derivate aus dem Galbanum-Harz

New Macrolides and Some Sesquiterpenoid Derivatives Occurring in Galbanum Absolute Analytical investigations of a Galbanum absolute have revealed for the first time the presence of four macrolides 1 – 4 derived from the corresponding hydroxylated C₁₃–C₁₆ fatty acids by lactonization of a secondary alcoholic group with the carboxylic group. The musky odor strengthens with increasing number of members in the macrocycle, whereas the woody note is predominant in the smaller rings. Isolation, identification and synthesis of the new naturally occurring substances are described. Shyobunol ( 24 ) epishyobunol ( 25 ) as well as their acetates 26 and 27 were identified - together with 10-epijunenyl acetate ( 28 ) - in the same substrate, thus indicating 25 and 26 to be natural products. The corresponding known and already naturally occurring ketones 23 and 24 are trace components in the investigated material derived from the so-called Galbanum oleo-gum-resin.     

Hierarchical structure of the energy landscape of proteins revisited by time series analysis. I. Mimicking protein dynamics in different time scales

Time series models, which are constructed from the projections of the molecular-dynamics (MD) runs on principal components (modes), are used to mimic the dynamics of two proteins: tendamistat and immunity protein of colicin E7 (ImmE7). Four independent MD runs of tendamistat and three independent runs of ImmE7 protein in vacuum are used to investigate the energy landscapes of these proteins. It is found that mean-square displacements of residues along the modes in different time scales can be mimicked by time series models, which are utilized in dividing protein dynamics into different regimes with respect to the dominating motion type. The first two regimes constitute the dominance of intraminimum motions during the first 5 ps and the random walk motion in a hierarchically higher-level energy minimum, which comprise the initial time period of the trajectories up to 20-40 ps for tendamistat and 80-120 ps for ImmE7. These are also the time ranges within which the linear nonstationary time series are completely satisfactory in explaining protein dynamics. Encountering energy barriers enclosing higher-level energy minima constrains the random walk motion of the proteins, and pseudorelaxation processes at different levels of minima are detected in tendamistat, depending on the sampling window size. Correlation (relaxation) times of 30-40 ps and 150-200 ps are detected for two energy envelopes of successive levels for tendamistat, which gives an overall idea about the hierarchical structure of the energy landscape. However, it should be stressed that correlation times of the modes are highly variable with respect to conformational subspaces and sampling window sizes, indicating the absence of an actual relaxation. The random-walk step sizes and the time length of the second regime are used to illuminate an important difference between the dynamics of the two proteins, which cannot be clarified by the investigation of relaxation times alone: ImmE7 has lower-energy barriers enclosing the higher-level energy minimum, preventing the protein to relax and letting it move in a random-walk fashion for a longer period of time.     

Monolayer metal coverage of fullerenes: the optical response

A first-principles investigation of the photoabsorption cross section of the specially stable cluster Li₁₂C₆₀ is carried out, including the icosahedral symmetry of the cluster and treating the ions via ab initio pseudopotentials. The role of the coating metal is assessed by computing the same quantity for the fullerene “seed”. It is found that the main absorption features are determined by the carbon molecule, both at low and high excitation energies, in spite of a reduction of 60% for the ionization threshold of Li₁₂C₆₀ as compared to C₆₀. Nonetheless the lithium coating is responsible for small yet clearly observable effects throughout the spectrum, in particular for a more structured and broad strength distribution at excitation energies below the ionization threshold of C₆₀, and a double peak structure in the Mie resonance at ≈20 eV.     

Oxygen equilibrium pressure as a measure of oxygen binding energy in the V2O5−x system

Using the redox mechanism of vanadia catalyst action as a starting point, oxygen binding energy and its measurement have been discussed. Experimental results on the equilibrium oxygen pressure are presented and two types of equilibrium are distinguished: chemisorbed and redox. It has been shown that univariant redox equilibrium is appropriate for characterizing vanadia catalysts.     

Thermal Analysis of the System Na3AlF6–NaVO3

Summary. The phase diagram of the system Na₃AlF₆–NaVO₃ was determined by means of thermal analysis. The system is a simple binary eutectic one. The eutectic point was estimated at x(NaVO₃) = 0.975 and t _eut = 617°C. The XRD patterns of samples after thermal analysis revealed the presence of cryolite and NaVO₃ only supporting the above assumption of a simple eutectic binary system.     

Changes in surface composition of Fe-Si-Cr-K oxide alkylation catalyst: the marked role of potassium ferrites

The influence of gas-phase methylation of phenol on the state of the Fe-Si-Cr-K oxide catalyst surface was investigated by thermodesorption of K atoms and ions. Changes in potassium desorption energies, as determined from Arrhenius-like plots, varied from 3.07 eV to 1.21 eV for the atoms and from 2.59 eV to 2.89 eV for ion desorption from the active and deactivated samples, respectively. The results were discussed in terms of transformation of the catalyst surface and formation of β-ferrite.     

Phlogophites intercalated with Al2O3 pillars and modified with transition metals as catalysts of the DeNOx process

Natural phlogophite, pre-treated with acids and intercalated with alumina pillars, was used as catalytic support. Transition metals (Fe, Cu) were deposited on the surface of the modified clay materials by an ion-exchange method. The obtained samples were characterized with respect to structure (XRD), texture (BET), composition (EPMA) and chemical nature of the deposited transition metals species (UV-vis-DRS). The phlogophite based materials have been found to be active and selective catalysts of the DeNOx process. The Fe-containing samples were catalytically active at lower temperatures than the clays modified with copper. A competitive ammonia oxidation by oxygen decreased the effectiveness of the DeNOx process in the high temperature range.     

New Carbonyl Compounds in the High-Boiling Fraction of Lavender Oil. 3rd Communication

The occurrence of a series of new constituents which can be considered as Diels-Alder adducts of methyl vinyl ketone and ocimene (→1–4), myrcene (→ 9 , 10 ) or β-far-nesene ( → 11 , 12 ), respectively, was reported. Furthermore, the structures of four isomeric cyclohexene derivatives could be established as adducts 21–24 of (E, Z)- and (E, E)-1,3,5-undecatrience and methyl vinyl ketone. Another series of constituents having the norbornane skeleton represents adducts 25–32 , and 33–40 of methyl cyclopentadiene and 1-octen-3-one or methyl vinyl ketone, respectively. In accordance with Alder's endo-rule the endo-isomers are preponderant in the natural as well as in the synthetic mixtures. Most of these constituents could also be identified in a lavender absolute as well as in a freshly prepared hexane extract of lavender flowers (Lavandula officinalis CHAIX ).