Osculating spaces to secant varieties

We generalize the classical Terracini’s Lemma to higher order osculating spaces to secant varieties. As an application, we address with the so-called Horace method the case of thed-Veronese embedding of the projective 3-space. This research is part of the T.A.S.C.A. project of I.N.d.A.M., supported by P.A.T. (Trento) and M.I.U.R. (Italy).     

Critical exponents of discrete groups and -spectrum

Let be a noncompact semisimple Lie group and an arbitrary discrete, torsion-free subgroup of . Let be the bottom of the spectrum of the Laplace-Beltrami operator on the locally symmetric space , and let be the exponent of growth of . If has rank , then these quantities are related by a well-known formula due to Elstrodt, Patterson, Sullivan and Corlette. In this note we generalize that relation to the higher rank case by estimating from above and below by quadratic polynomials in . As an application we prove a rigiditiy property of lattices.

     

Lipschitz and path isometric embeddings of metric spaces

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C ¹ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite Hausdorff dimension can be embedded in some Euclidean space via a Lipschitz map.     

Quantitative analysis of two-dimensional gel-separated proteins using isotopically marked alkylating agents and matrix-assisted laser desorption/ionization mass spectrometry

We describe a simple approach for the relative quantification of individual proteins within a mixture. The method is based on the differential labelling of the mixtures by use of a commercially available acrylamide and deuterium-labelled [2,3,3'-d(3)]-acrylamide to alkylate proteins prior to two-dimensional (2-D) gel electrophoresis. The tryptic digests of the separated proteins were subjected to reflector matrix-assisted laser desorption/ionization time-of-flight (MALDI-TOF) analysis and the relative peak heights of cysteine-containing peptides were used to quantify their precursor proteins. This approach was tested for the relative quantification of proteins within an artificial mixture of standard proteins and for proteins observed in a 2-D map of rat serum. A good correlation was found between the measured ratios derived from MALDI-TOF data and those theoretically calculated prior to 2-D analysis via known mixing ratios of the two alkylating reagents. The described procedure has proved to be effective for comparative measurements of protein abundances within the investigated mixtures.     

Spectral properties and absolute rate constants for beta-scission of ring-substituted cumyloxyl radicals. A laser flash photolysis study

A laser flash photolysis study of the spectral properties and beta-scission reactions of a series of ring-substituted cumyloxyl radicals has been carried out. All cumyloxyl radicals display a broad absorption band in the visible region of the spectrum, which decays on the microsecond time scale, leading to a strong increase in absorption in the UV region of the spectrum, which is attributed to the corresponding acetophenone formed after beta-scission of the cumyloxyl radicals. The position of the visible absorption band is red-shifted by the presence of electron-donating ring substituents, while a blue-shift is observed in the presence of electron-withdrawing ring substituents, suggesting that + R ring substituents promote charge separation in the excited cumyloxyl radical through stabilization of the partial positive charge on the aromatic ring of an incipient radical zwitterion. Along this line, an excellent Hammett-type correlation between the experimentally measured energies at the visible absorption maxima of the cumyloxyl radicals and sigma(+) substituent constants is obtained. A red-shift is also observed on going from MeCN to MeCN/H(2)O for all cumyloxyl radicals, pointing toward a specific effect of water. The ring substitution does not influence to a significant extent the rate constants for beta-scission of the cumyloxyl radicals, which varies between 7.1 x 10(5) and 1.1 x 10(6) s(-1), a result that suggests that cumyloxyl radical beta-scission is not governed by the stability of the resulting acetophenone. Finally, k(beta) increases on going from MeCN to the more polar MeCN/H(2)O 1:1 for all cumyloxyl radicals, an observation that reflects the increased stabilization of the transition state for beta-scission through increased solvation of the incipient acetophenone product.     

Allylation of exocyclic N-Acyliminium ions generated from chiral N-[1-(phenylsulfonyl)alkyl]oxazolidin-2-ones

N-[1-(Phenylsulfonyl)alkyl]oxazolidin-2-ones are successfully prepared by condensation of the corresponding optically active oxazolidin-2-ones with aldehydes and benzenesulfinic acid. At low temperature, in the presence of titanium tetrachloride, these sulfones are converted into N-acyliminium ions, which react with allyltrimethylsilane with a variable degree of stereoselectivity. The best results are obtained with (R)-5,5-dimethyl-4-phenyloxazolidin-2-one as a chiral auxiliary. Cleavage of the oxazolidin-2-one ring with lithium/ammonia affords the corresponding homoallylamines, which reveal an absolute configuration opposite that expected on the basis of the usual steric effects. A complexation of the Lewis acid with the N-acyliminium ion may be responsible of this rather unusual stereochemical outcome.