Establishing vicinal proton-proton connectivities using two-dimensional proton zero quantum coherence NMR

Phenanthro[3,4:3′,4′]phenanthro[2,1-b]thiophene ( 1 ) served as the model system to evaluate two-dimensional proton zero quantum coherence nmr in order to establish the vicinal proton-proton connectivities. The utility of the two-dimensional proton zero quantum nmr experiment has been compared with the utility of the traditional autocorrelated proton-proton (COSY) experiment. In the case of a molecule such as 1 , where the proton chemical shifts are so highly congested, the zero quantum coherence experiment provides data not obtainable from the COSY experiment.     

New reals: Can live with them,can live without them

We give a self‐contained proof of the preservation theorem for proper countable support iterations known as “tools‐preservation”, “Case A” or “first preservation theorem” in the literature. We do not assume that the forcings add reals. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Preservation of logarithmic concavity by the Mellin transform and applications to the Schrödinger equation for certain classes of potentials

We prove that the Mellin transform of a function log-concave (convex) is, after division by (+1), where is the argument of the transform, itself log-concave (convex) in . This theorem is first applied to the moments of the ground state wave function of the Schrödinger equation where the Laplacian of the central potential has a given sign, and generalized to other situations. This is used to derive inequalities linking thel ^th derivative of the ground state wave function at the origin for angular momentuml and the expectation value of the kinetic energy, and applied to quarkonium physics. A generalization to higher radial excitations is shown to be plausible by using the WKB approximation. Finally, new bounds on ground-state energies in power potentials are obtained.     

Generalized Lax pairs,the modified classical Yang-Baxter equation,and affine geometry of Lie groups

We generalize the usual Lax equationd/dt L=[M, L] byd/dt L=–(M)L, where is an arbitrary representation of a Lie algebra g (the values ofM) in a representation spaceV (the values ofL). The usual classicalr-matrix programme for Hamiltonian integrable systems is generalized tor-matrices taking values in gV. Ther-matrices are then considered as left invariant torsion-free covariant derivatives on a Lie groupK (with Lie algebraV ^*). The Classical Yang-Baxter Equation (CYBE) is equivalent to the flatness ofK whereas the Modified CYBE implies thatK is an affine locally symmetric space. An example is discussed.