A theorem on products of matrices

A new result on products of matrices is proved in the following theorem: let M_i (i=1,2,…) be a bounded sequence of square matrices, and K be the l.u.b. of the spectral radii ρ(M_i). Then for any positive number ε there is a constant A and an ordering p(j) (j = 1,2,…) of the matrices such that

$\text{∏}\text{j=1}\text{n}\text{M}{}_{\mathrm{p(j)}}\text{?A·(K+ε)}{}^{\mathrm{n}}\text{(n = 1,2,…)}$

. The ordering is well defined by p(j), a one-to-one mapping on the set of positive integers. In general the inequality does not hold for any ordering p(j) (a counterexample is provided); however, some sufficient conditions are given for the result to remain true irrespective of the order of the matrices.     

The crystal and molecular structure of tris[(triphenylstannyl)methyl]methane

The title compound, C₅₈H₅₂Sn₃, belongs to the triclinic space group P$\text{1}$, with a 10.165, b 13.365, c 18.670 Å, α 96.28, β 93.88, γ 103.15°, V = 2443.8 ų, f_w = 1105.1, Z = 2, D_calc 1.501 g cm^?3, m.p. 206.5–208°C, λ(Mo-K_$\text{α}$) 0.71069 Å. The structure was refined on 2684 nonzero reflections to an R factor of 0.044. The crystal contains molecules in which the (SnCH₂)₃CH core possesses an approximate C₃ symmetry. The three SnC(H₂) bonds are gauche to the C(4)-H bond. Repulsive interactions involving the bulky Ph₃Sn substituents lead to large SnC(H₂)C(H) angles (av. 117.3°), whereas the C(H₂)C(H)C(H₂) angles at the tertiary carbon average 111.3°. Little distortion of the Ph₃Sn groups themselves is present, since the PhSnPh angles (av. 109.8°) are almost equal to the C(H₂)SnPh angles (av. 109.9°). The molecule as a whole has no symmetry because the aromatic rings in the three Ph₃Sn groups have different orientations. The phenyl groups create a pocket in the middle of the molecule which encloses and shields the tertiary hydrogen atom. The resulting inaccessibility of this hydrogen accounts in part for the low reactivity of the title compound in redox reactions.     

An improved synthesis of cyclopropanes from stabilized phosphonates and 1,2-dioxines

Addition of stabilized Horner-Wadsworth-Emmons (HWE) phosphonates to substituted 1,2-dioxines leads to diastereomerically pure di- and trisubstituted cyclopropanes in high yields and represents a viable alternative to ylides in the cyclopropanation reaction involving 1,2-dioxines. While yields are comparable, reaction times with these stabilized phosphonates were accelerated and the diastereoselectivity for this cyclopropanation reaction was significantly greater than for the previously reported examples employing ylides.     

Interactions magnétiques dans des groupements binucléaires du Ruthénium +V

The cationic ordering in the $\text{Ba}\text{M}{}_{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{Ru}{}_{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{O}{}_3\text{(M =}\text{Mg, Ca, Cd, Sr}\text{)}$ oxides which crystallize with the 6H structure gives rise to Ru⁵⁺Ru⁵⁺ pairs in (Ru₂O₉) clusters. The magnetic properties have been analyzed on the basis of the Heisenberg-Dirac-Van Vleck model: The d electrons are localized on each Ru⁵⁺ ion and interact strongly through antiferromagnetic exchange. The susceptibility curves agree with the H.D.V.V. model. The values of the exchange integrals have been determined by fitting the experimental values.     

Fourier transform Raman spectral measurements of powdered quaternary mixtures of organic compounds: Exceptional pure component spectral reconstruction using band-target entropy minimization (BTEM)

Fourier transform Raman spectra of eight mixtures of four organic solids, namely dicyandiamide, melamine, acetamide and urea were measured. Matrices formed from these spectra were first subjected to singular value decomposition to obtain the right singular vectors. The right singular vectors were then subjected to blind source separation using band-target entropy minimization (BTEM), thus no a priori information (i.e. involving the nature of the components present, their spectra, nor their concentrations) was included in the analysis. The recovered pure component spectra are of exceptionally high quality and are consistent with pure reference spectra. Various empirical and statistical tests, such as the Euclidean norm and target transform factor analysis, were employed to assess the quality of the recovered spectra. The present results indicate the applicability of combined Raman and BTEM analysis for solid mixtures.     

Amine-terminated dendrimers as biomimetic templates for silica nanosphere formation

In diatoms, silica synthesis occurs by use of complex posttranslationally modified peptides, termed silaffins, and highly complex biological polyamine structures. Silaffin peptides have lysine residues that are modified to long-chain polyamine moieties of N-methyl derivatives of polypropylenimine to drive silica synthesis at slightly acidic pH conditions. Using polypropylenimine (PPI) and PAMAM amine-terminated dendrimers as a biomimetic analogue of the polyamine modifications of silaffins, we have demonstrated the condensation of silica nanospheres. We have shown that the dendrimers react in an amine concentration-dependent fashion yielding silica nanospheres with a distinct size distribution reminiscent of the structures produced from both the modified and nonmodified peptides extracted from diatoms. Additionally, the templates were encapsulated by the growing nanospheres and precipitated from solution in a manner similar to that previously described for the bioactive peptides and polyamines.     

Single crystal EPR studies of the reduced active site of [NiFe] hydrogenase from Desulfovibrio vulgaris Miyazaki F

In the catalytic cycle of [NiFe] hydrogenase the paramagnetic Ni-C intermediate is of key importance, since it is believed to carry the substrate hydrogen, albeit in a yet unknown geometry. Upon illumination at low temperatures, Ni-C is converted to the so-called Ni-L state with markedly different spectroscopic parameters. It is suspected that Ni-L has lost the "substrate hydrogen". In this work, both paramagnetic states have been generated in single crystals obtained from the [NiFe] hydrogenase from Desulfovibrio vulgaris Miyazaki F. Evaluation of the orientation dependent spectra yielded the magnitudes of the g tensors and their orientations in the crystal axes system for both Ni-C and Ni-L. The g tensors could further be related to the atomic structure by comparison with the X-ray crystallographic structure of the reduced enzyme. Although the g tensor magnitudes of Ni-C and Ni-L are quite different, the orientations of the resulting g tensors are very similar but differ from those obtained earlier for Ni-A and Ni-B (Trofanchuk et al. J. Biol. Inorg. Chem. 2000, 5, 36-44). The g tensors were also calculated by density functional theory (DFT) methods using various structural models of the active site. The calculated g tensor of Ni-C is, concerning magnitudes and orientation, in good agreement with the experimental one for a formal Ni(III) oxidation state with a hydride (H(-)) bridge between the Ni and the Fe atom. Satisfying agreement is obtained for the Ni-L state when a formal Ni(I) oxidation state is assumed for this species with a proton (H(+)) removed from the bridge between the nickel and the iron atom.