Lower Bounds for the Height and Size of the Ideal Class Group in CM-Fields

 We prove that, under the assumption of the Generalized Riemann Hypothesis, the exponent of the ideal class group of a CM-field goes to infinity with its absolute discriminant. This gives a positive answer to a question raised by Louboutin and Okazaki [4]. Received September 10, 2001; in revised form April 5, 2002     

The synthesis and reactivity of the heptanuclear dianion [Os7(CO)20]2− including the synthesis and structural characterizations of the compounds [Os7(CO)20(AuPEt)3 2] and [Os8(CO)20(η6-C6H6]

Reduction of the heptaosmium cluster [Os₇(CO)₂₁] With [Et₄N][NH₄) gives the cluster dianion [Os₇(CO)₂₀]^2–,1, in high yield. The reaction of the dianion with [AuPR ₃Cl] (R=Et or Ph) in the presence of TlPF₆ forms [Os₇((CO)₂₀(AuPR ₃)₂] [R=Et (2a);R = Ph(2b)] in 80% yield, while the corresponding reaction with (Os(C₆H₆)(CH₃CN)₃]²⁺ gives [Os₈(CO)₂₀ ( ⁶-C₆H₆)] (3) in reasonable yield (ca. 30%). The dianion,1, and the clusters2 and3 have been fully characterized by bout spectroscopic and crystallographic methods. The crystal structure of the [Ph₄P]⁺ salt of1 shows that the metals in the anion adopt a capped octahedral geometry, with all twenty carbonyl ligands in terminal sites. The metal core geometry in2a is best described as a tricapped octahedron, and is based on the structure of the dianion1 with two adjacent octahedral faces capped by the Au atoms of the two AuPEt₃ groups. In a similar fashion, the geometry of3 is related to that of1 with the addition of an Os(C₆H₆) unit capped to a triangular face, to give a bicapped octahedral framework.     

Densité des Points à Coordonnées Multiplicativement Indépendantes

Nous montrons que pour toute sous-variété algébrique d'un tore multiplicatif (non contenue dans un sous-groupe algébrique propre), on peut choisir un ensemble Zariski dense de points algébriques de hauteur contrôlée, dont toutes les coordonnées sont multiplicativement indépendantes. Cet énoncé précise et généralise un théorème de S. Zhang qui lie la hauteur projective d'une varété au minimum essentiel de la hauteur des points algébriques de celle-ci. En tenant compte d'un résultat précédent des auteurs sur le problème de Lehmer généralisé à un tore, nous en déduisons une minoration pour la hauteur normalisée d'une sous-variété d'un tore. Cette dernière est optimale à un «-prés» en le degré géométrique de la variété étudiée (confer une conjecture du second auteur avec P. Philippon).In this article, we prove that on any subvariety of a multiplicative torus which is not contained in a proper algebraic subgroup, one can find a Zariski dense set of algebraic points of small height whose coordinates are multiplicatively independent. This statement generalizes an earlier result of S. Zhang which links the projective height of a variety with the essential minimum of its algebraic points. Taking into account an earlier result of the authors on the Lehmer problem generalized to a multiplicative torus, one deduces a lower bound for the normalized height of subvarieties of multiplicative groups. This lower bound is optimal up to an in the geometric degree of the variety studied (confer a conjecture by the second author and P. Philippon).     

An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy

Using the approach of Rulla (1996 SIAM J. Numer. Anal. 33, 68-87)for analysing the time discretization error and assuming moreregularity on the initial data, we improve on the error boundderived by Barrett and Blowey (1996 IMA J. Numer. Anal. 16,257-287) for a fully practical piecewise linear finite elementapproximation with a backward Euler time discretization of amodel for phase separation of a multi-component alloy.     

Evaluation of electron-withdrawing group effects on heme binding in designed proteins: implications for heme a in cytochrome c oxidase

Heme a, the metalloporphyrin cofactor unique to cytochrome c oxidases, differs from the more common heme b by two chemical modifications, a C-2 hydroxyethylfarnesyl group and a C-8 formyl group. To elucidate a role of the C-8 formyl group, we compare the heme affinity, spectroscopy, and electrochemistry of a heme a mimic, Fe(diacetyldeuterioporphyrin IX) or Fe(DADPIX), with heme b, Fe(protoporphryrin IX) or Fe(PPIX), incorporated into a designed heme protein. The [Delta7-H3m]2 protein ligand, or maquette, selected for this study contains two equivalent bis-(3-methyl-L-histidine) heme binding sites within a four-alpha-helix bundle scaffold. The spectroscopic data on Fe(PPIX) and Fe(DADPIX) bound to [Delta7-H3m]2 demonstrate that these complexes are excellent synthetic analogues for natural cytochromes b and a, respectively. Comparison of the spectroscopic, electrochemical, and equilibrium thermodynamic data measured for the Fe(PPIX)-[Delta7-H3m]2 maquette with the previously reported Fe(PPIX)-[Delta7-His]2 complex demonstrates that changing the heme axial ligands to 3-methyl-L-histidine from L-histidine does not alter the resulting heme protein properties significantly in either oxidation state. Heme binding studies demonstrate that [Delta7-H3m]2 binds two ferrous Fe(DADPIX) or Fe(PPIX) moieties with similar dissociation constant values. However, in the ferric state, the data show that [Delta7-H3m]2 only binds a single Fe(DADPIX) and that one 2500-fold weaker than oxidized Fe(PPIX). The data demonstrate that the 4.6 kcal mol(-1) weakened affinity of [Delta7-H3m]2 for oxidized Fe(DADPIX) results in the majority of the 160 mV, 3.7 kcal mol(-1), positive shift in the heme reduction potential relative to Fe(PPIX). These data indicate that a role of the formyl group on heme a is to raise the iron reduction potential, thus making it a better electron acceptor, but that it does so by destabilizing the affinity of bis-imidazole sites for the ferric state.     

Co-ordination behaviour of a novel bisthiourea tripodal ligand: structural, spectroscopic and electrochemical properties of a series of transition metal complexes

The synthesis of a thiourea substituted derivative of tris(pyridyl-2-methyl)amine (TPA) is reported. Two of the three pyridine rings are substituted in the 6-position with benzoylthiourea groups. These thiourea groups undergo intramolecular hydrogen bonding to form six-membered rings which leaves one N-H group available to form hydrogen bonds with other molecules. This reports details how the complexation of this new ligand with transition metal ions yields complexes with differing geometries. Seven co-ordinate Mn(II) and Cd(II), six co-ordinate Ni(II) and five co-ordinate Co(II), Cu(II) and Zn(II) complexes have been isolated.