Filtrations of symmetric powers via crystalline cohomology

Consider a finite dimensional vector space over a field of prime characteristic. We show that two differently constructed filtrations of the symmetric powers of this space coincide. One of them arises from the representation theory of the general linear group. In order to get the other one we interpret the symmetric powers as crystalline cohomology groups and use the action of the Frobenius endomorphism on this cohomology. Dedicated to Jacques Tits on his sixtieth birthday Supported in part by the NSF.     

Multinuclear enantiopure titanium self‐assembly complexes—synthesis,characterization and application to organic synthesis

Hexa‐ and nonanuclear titanium complexes were obtained by self‐assembly of titanium(IV)‐tert‐butoxide and D ‐mandelic acid. Suitable single crystals of these complexes were characterized by X‐ray structure analysis. When used with these complexes, aldol adducts were isolated with a high degree of regioselectivity in direct aldol additions of aromatic and aliphatic aldehydes to functionalized unsymmetrical ketones. High syn‐diastereoselectivities were obtained in aldol additions of enolizable aldehydes with hydroxyacetone and methoxyacetone. Copyright © 2007 John Wiley & Sons, Ltd.     

Scalar dynamic nuclear polarization of protons in trifluoroacetic acid solutions

Strongly scalar ¹H intermolecular dynamic nuclear polarization (DNP) of solvent ¹H nuclei in solutions of organic free radicals has now been observed at room temperature. Transient protonation of the CF₃COOH/pyridyl imidazoline-1-oxyl radical system accounts for the observed effects.     

Martinaxiom und die Beschreibung gewisser Homomorphismen in der Theorie der ℵ1-freien abelschen Gruppen

P(I) denotes the abelian group of integer-valuated functions defined on the set I, P_κ (I) its subgroup consisting of all elements, which support has size less than the infinite cardinal κ and Q_κ (I) the quotient P(I)/P_κ(I). We formulate a statement I(κ) in the following way: For an arbitrary set I you can lift every homomorphism from P (?) to Q_κ(I) to a homomorphism from P(?) to P(I). Using Martin's Axiom we show, that L(?) is independent of ZFC.     

Microwave spectrum and structure of cyanamide: Semirigid bender treatment

An extensive study of the microwave spectrum of cyanamide has been undertaken, the analysis being based in part on semirigidbender calculations by the methods of Bunker and Szalay. Inversion lines of NH₂CN, $\text{K}{}_{\mathrm{?1}}\text{= 2}{}^{\mathrm{a}}\text{Q}$ branches and a number of vibrational satellites of the J = 2?1 transition were observed. A two-vibrational-state Hamiltonian was used to fit simultaneously the 0⁺ and 0^? microwave data and yielded rotational constants X, Y, Z, D_J, D_JK, d₁, H_JK as well as the inversion splitting and the μ_yz-connecting matrix element. Vibrational satellite data of seven isotopic species and infrared frequencies of NH₂CN were included in the semirigid bender calculations: The NCN spine is nonlinear by ca. 5° in the equilibrium structure of the molecule. Also, $\text{r}{}_{\mathrm{<mtext>NH</mtext>}}\text{A}\text{?}\text{= 0.9994 + 0.0144?}{}^2$; <HNH/2 = 60.39° ? 0.1134?²; $\text{r}{}_{\mathrm{<mtext>NC</mtext>}}\text{A}\text{?}\text{= 1.3301 + 0.0327?}{}^2$ (? is the inversion angle in rad); $\text{r}{}_{\mathrm{<mtext>CN</mtext>}}\text{= 1.1645}\text{A}\text{?}$ fixed. The inclusion of the NC bond flexing was necessary in order to reproduce the observed vibrational satellite patterns of NH₂CN, NHDCN, and ND₂CN. The barrier to inversion of the amino group is 510 ± 6 cm^?1 with minima at ±45.0 ±0.2°. The inversion dipole moment is 0.91 ± 0.02 Debye.