Linear Systems of Cubics Singular at General Points of Projective Space

We present an elementary proof that given a general collection of d points in Pⁿ the linear system of cubics singular on each point has the expected codimension except when n=4 and d=7. In that case the cubic is unique. This, together with previous work of the author, gives a proof of the Alexander–Hirschowitz interpolation theorem.     

-Body Quantum Scattering Theory in Two Hilbert Spaces: -Body Integral Equations

Scattering for a nonrelativistic system of distinguishable and spinless particles interacting via short-range pair potentials is considered. Half-on-shell integral equations (the CG equations) are proposed, the solutions of which determine approximate scattering amplitudes that converge to the exact scattering amplitude. It is proved, under mild H?lder integrability assumptions, that these apparently singular equations actually have a compact kernel for real energies and, consequently, a unique solution. The CG equations have a structure that is much simpler than the Yakubovskii equations and similar to that of coupled-reaction-channel equations. The driving terms look like distorted-wave Born integrals and nonorthogonality integrals. However, there is no restriction to channels with only two asymptotic bound clusters and for all channels, no matter how many bound clusters, appropriate boundary conditions are exactly satisfied. This work completes the establishment of a rigorous mathematical link between the solutions of the half-on-shell CG equations and the on-shell transition operators defined in time-dependent multichannel scattering theory, and it provides for the first time a rigorous theoretical basis for practical calculations of scattering amplitudes for certain problems with . Received October 27, 1997; accepted for publication December 29, 1997     

The interstellar gas-phase formation of CO2 – Assisted or not by water molecules?

Using state of the art methods of quantum chemistry, potential energy surfaces for the formation of and CO₂ (³B₂) from CO + O (¹D) and CO + O (³P), respectively, have been studied. At the MRSDCI level, we show that the formation of from O (³P) is strongly connected with the height of the barrier localized on the CO + O (³P) entrance channel. At the CCSD(T) level with a large basis set we calculate this barrier to be 5.9 kcal/mol. Consequently, we confirm that the gas-phase formation of CO₂ in interstellar molecular clouds is inefficient. To mimic the formation of CO₂, through the Eley–Rideal mechanism, on the water ice surfaces of interstellar grains, we have extended our study to consider the formation of CO₂ in the presence of water molecules. We show, using density functional and CCSD(T) methods, that the barrier located on the CO + O (³P) reaction entrance channel is hardly affected by the presence of water molecules. We therefore suggest that CO₂ formation, through the Eley–Rideal mechanism, on the water ice surfaces of interstellar grains, should be inefficient too.     

Cyclic Relative Difference Sets and their p-Ranks

By modifying the constructions in Helleseth et al. [10] and No [15], we construct a family of cyclic ((q ^3k –1)/(q–1), q–1, q ^3k–1, q ^3k–2) relative difference sets, where q=3 ^e . These relative difference sets are liftings of the difference sets constructed in Helleseth et al. [10] and No [15]. In order to demonstrate that these relative difference sets are in general new, we compute p-ranks of the classical relative difference sets and 3-ranks of the newly constructed relative difference sets when q=3. By rank comparison, we show that the newly constructed relative difference sets are never equivalent to the classical relative difference sets, and are in general inequivalent to the affine GMW difference sets.     

Synthesis and bioactivity of the gibberellin, 18-hydroxy-GA1 (GA132)

As part of a study to confirm putative structural assignments to new gibberellins and to furnish sufficient quantities for biological investigations, a twenty step synthesis of 18-hydroxy GA1 from gibberellic acid (GA3) is described, allowing the confirmation of structure for a new gibberellin, GA132, that occurs in developing grains of barley (Hordeum vulgare). The early part of the sequence involved cleavage of the C(3)-C(4) bond in the A-ring of a 3-oxo intermediate. The ring was then reformed as part of a "domino" process involving the conjugate addition of alkoxide to an alpha-methylene lactone moiety followed by an intramolecular aldol reaction. The bioactivities of the new GA, and its 18-hydroxy-GA4 relative, have been confirmed in dwarf barley growth and alpha-amylase induction assays.     

Comparison of Scintillation Detection Efficiencies of Depleted Uranium (DU) in Wounds

Three tissue-equivalent cylindrical wound phantoms with varying activities of DU metal imbedded at varying depths were used to compare the efficiencies of a bismuth germinate (BGO) detector, a sodium iodide (NaI), and two identical wound probes with smaller sodium iodide crystals. Our results show that the BGO detector had the highest efficiency (1.0·10^-3) and the lowest minimum detectable activity (MDA = 5.8 kBq) for the shallow depth DU phantom, relative to the other detectors. The BGO detector also had the highest peak efficiencies (1.7·10^-3 and 5.8·10^-4) and the lowest MDAs (3.5 and 10.0 kBq) for the medium and deep phantoms, respectively. Other detectors' performance data are presented.     

Modification of the velocity distribution of H(2) molecules in a supersonic beam by intense pulsed optical gradients

We report the acceleration and deceleration of H(2) molecules in a supersonic molecular beam by means of its interaction with an intense optical gradient from a nanosecond far-off-resonant optical pulse. The strong optical gradients are formed in the interference pattern of two intense optical pulses at 532 nm. The velocity distribution of the molecular beam, before and after the applied optical pulse, is measured by a velocity-mapped ion imaging technique. Changes in velocity up to 202 m s(-1)+/- 61 m s(-1) are observed in a molecular beam initially travelling at a mean speed of 563 m s(-1). We report the dependence of this change in velocity with the strength of the optical gradient applied.