A categorical invariant for cubic threefolds

We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.     

The string of variable density: Perturbative and non-perturbative results

We obtain systematic approximations for the modes of vibration of a string of variable density, which is held fixed at its ends. These approximations are obtained iteratively applying three theorems which are proved in the paper and which hold regardless of the inhomogeneity of the string. Working on specific examples we obtain very accurate approximations which are compared both with the results of WKB method and with the numerical results obtained with a collocation approach. Finally, we show that the asymptotic behaviour of the energies of the string obtained with perturbation theory, worked to second order in the inhomogeneities, agrees with that obtained with the WKB method and implies a different functional dependence on the density that in two and higher dimensions.     

On the symmetry of three identical interacting particles in a one-dimensional box

We study a quantum-mechanical system of three particles in a one-dimensional box with two-particle harmonic interactions. The symmetry of the system is described by the point group D_3d$D_{3d}$. Group theory greatly facilitates the application of perturbation theory and the Rayleigh–Ritz variational method. A great advantage is that every irreducible representation can be treated separately. Group theory enables us to predict the connection between the states for the small box length and large box length regimes of the system. We discuss the crossings and avoided crossings of the energy levels as well as other interesting features of the spectrum of the system.     

THE STAUDINGER REACTION BETWEEN 2-H-1,2,3-DIAZAPHOSPHOLENES AND AROMATIC AZIDES

Abstract

The title reaction between substituted phenyl azides 2 and diastereomeric diazaphospholenes 1 gives the corresponding cyclic phosphazenes 3 with different stereochemical results. Hydrolysis of some phosphazenes 3 yield the corresponding ring-opened compounds Z-4I together with small amounts of diazaphospholene-oxide 5 and anilines 6. The configuration of the compounds obtained are established by ¹H n.m.r. spectroscopy. The results are explained invoking the formation of pentacoordinate phosphorus intermediates.     

BASE-CATALYZED HYDROGEN-DEUTERIUM EXCHANGE IN BENZO DERIVATIVES OF FIVE-MEMBERED AROMATIC HETEROCYCLES. PART VII. PRIMARY HYDROGEN ISOTOPE EFFECTS IN SOME 5- AND 6-SUBSTITUTED BENZOTHIAZOLES

Abstract

The rates of the base-catalyzed hydrogen-deuterium exchange at position 2, and the reverse, in some 5-and 6-substituted benzothiazoles are reported. The plots of log k _sH, and log k _sD against the [sgrave]_m + [sgrave]_p values of the substituents, according to the Hammett-Jaffé equations, are slightly curved. The primary hydrogen isotope effect k _sH/k _sD varies between 0.7 and 2.3 and the plot of log k _sH/k _sD against the [sgrave]_m + [sgrave]_p values of the substituents is a more pronounced curve, showing a maximum near to the [sgrave]_m + [sgrave]_p value of ?0.3. By these investigations the simple utilization of the primary hydrogen isotope effect in the prediction of reaction mechanisms seems extremely hazardous.     

SYNTHESIS AND THERMAL REARRANGEMENT OF 2-ALKOXYCARBONYL-3-ALKYL-4H-BENZO[b][1,4]THIAZINES

Abstract

The title compounds (4b-d) together with the benzothiazolines (5b-d) have been obtained by a reaction between 2,2′-dithiodianiline (1) and β-keto esters (2b-d). The reaction between 1 and the β-keto ester 2a gives 1,4-benzothiazine 3a in addition to the benzothiazoline 5a.

It has been established that the 1,4-benzothaizines 4b-d undergo an acid-catalysed thermal rearrangement involving a [1,3] shift of the sulfur atom, giving rise to the isomeric 1,4-benzothiazines 3b-d.     

Molecular Chirality Recognized by Achiral Compounds

Abstract

Three cases are described where chirality is recognized by achiral molecules, where chirality is induced into achiral compounds through interactions with chiral compounds, and lastly where induced chirality in the solid-state is utilized for an enantio-selective photoreaction. In the first instance, the thermodynamically and kinetically preferred diastereoisomer of an optically labile chromium complex depended on the nature of the achiral solvent. In the second case, for the first time 1,2-chloroethane was trapped and observed in a chiral near-eclipsed form and 1-chloropropane in the truly eclipsed form at room temperature in a 1:1 inclusion complex with an optically active host molecule. Finally, induced chirality in a prochiral compound in the solid-state was successfully employed in an enantio-selective photoreaction. In the two cases, solid-state CD provided valuable information.