Algebras satisfying a Capelli identity

The sequence of cocharacters (c.c.s.) of a P.I. algebra is studied. We prove that an algebra satisfies a Capelli identity if, and only if, all the Young diagrams associated with its cocharacters are of a bounded height. This result is then applied to study the identities of certain P.I. algebras, includingF _k .     

Symmetry analysis of the potential energy surfaces for the photochemical decomposition of formaldehyde

Orbital Correspondence Analysis in Maximum Symmetry (OCAMS) is applied to the decomposition pathways of formaldehyde to H₂ + CO and to H + HCO. The symmetry adapted nuclear motions, which are preferentially incorporated in the energetically favoured fragmentation pathways on both the ground and excited state surfaces are singled out. The results of this analysis are in full agreement with those of published potential energy surfaces and consistent with the results of experimental investigations reported in the literature. The nuclear motions favouring the various processes thus appear to be deducible from considerations of orbital symmetry.This work was carried out during tenure of a Minerva Foundation grant to one of us. (E.A.H.)     

Generalized Thrackles and Geometric Graphs in $${\mathbb{R}}^3$$ with No Pair of Strongly Avoiding Edges

We define the notion of a geometric graph in . This is a graph drawn in with its vertices drawn as points and its edges as straight line segments connecting corresponding points. We call two disjoint edges of G strongly avoiding if there exists an orthogonal projection of to a two dimensional plane H such that the projections of the two edges on H are contained in two different rays, respectively, with a common apex that create a non-acute angle. We show that a geometric graph on n vertices in with no pair of strongly avoiding edges has at most 2n − 2 edges. As a consequence we get a new proof to Vázsonyi’s conjecture about the maximum number of diameters in a set of n points in . This research was supported by THE ISRELI SCIENCE FOUNDATION (grant No. 938/06).     

Coiled to diffuse: brownian motion of a helical bacterium

We employ real-time three-dimensional confocal microscopy to follow the Brownian motion of a fixed helically shaped Leptospira interrogans (LI) bacterium. We extract from our measurements the translational and the rotational diffusion coefficients of this bacterium. A simple theoretical model is suggested, perfectly reproducing the experimental diffusion coefficients, with no tunable parameters. An older theoretical model, where edge effects are neglected, dramatically underestimates the observed rates of translation. Interestingly, the coiling of LI increases its rotational diffusion coefficient by a factor of 5, compared to a (hypothetical) rectified bacterium of the same contour length. Moreover, the translational diffusion coefficients would have decreased by a factor of ～1.5, if LI were rectified. This suggests that the spiral shape of the spirochaete bacteria, in addition to being employed for their active twisting motion, may also increase the ability of these bacteria to explore the surrounding fluid by passive Brownian diffusion.     

Graded tensor products

We study the polynomial identities of regular algebras, introduced in [A. Regev, T. Seeman, Z₂-graded tensor products of P.I. algebras, J. Algebra 291 (2005) 274-296]. For example, a finite-dimensional algebra is regular if it has a basis whose multiplication table satisfies some commutation relations. The matrix algebra M_n(F) over the field F is regular, which is closely related to M_n(F) being Z_n-graded. We study the polynomial identities of various types of tensor products of such algebras. In particular, using the theory of Hopf algebras, we prove a far reaching extension of the A⊗B theorem for Z₂-graded PI algebras.     

Dimerization of C60: Symmetry Considerations

Orbital‐symmetry analysis (OCAMS) of the dimerization of C₆₀ via [2+2] cycloaddition indicates that the reactant monomers should approach one another along a pathway in which C_2h symmetry is conserved. Point‐by‐point computations (AM1/UHF) confirm this prediction: a low‐energy pathway leads to a single‐bonded dimer 2 with C_2h symmetry. Closure to the stable D_2h dimer 1 is effected by relatively facile rotation about the single bond. A similar symmetry analysis was performed on a second isomer 3 with D_2h symmetry, the moieties of which are linked by two two‐atom chains. It raises the possibility that 3 , the so‐called `window' isomer, may be interconvertible with 1 along a pathway that retains C_i (S₂) symmetry. Although the computational results indicate that C₆₀ is in thermal equilibrium with its stable dimer 1 at moderate temperatures, the latter is not observed in the gas phase for thermodynamic reasons. According to THERMO computations (AM1/RHF), the equilibrium is shifted strongly towards the monomer pair at temperatures where vaporization of the solid C₆₀ is observed (>400°).     

From polynomial to formal identities

Polynomial identities, when multiplied in the group algebra of the symmetric group by a two sided ideal of non identities, imply various formal (combinatorial) identities.     

A simple perturbational approach for the determination of favourable reaction pathways

A simple perturbational approach has been used to establish the symmetry conditions for energetically favourable nuclear motions on a potential energy surface. In particular, the operational rules of Orbital Correspondence Analysis in Maximum Symmetry (OCAMS) for specifying the symmetry species of the nuclear displacements, which make a symmetry forbidden pathway symmetry-allowed, have been derived. A general, symmetry-independent, procedure is then proposed for finding the energetically most favourable pathway by referring to the form of the overlap density function of non-correlating orbitals. The method is demonstrated by selecting from among the several symmetry-allowed nuclear motions on the potential energy surface for the H₂ + D₂ exchange reaction, that which is energetically most favourable.     

Identification of a mixed-valence triclinic product formed by the partial bromination of K2Pt(CN)4

X-ray diffraction measurements as well as the results of chemical analysis are presented as evidence that the partial bromination of K₂Pt(CN)₄ results in at least two products which differ in their structural features and stoichiometry. Triclinic crystals grow from acidic aqueous solution whereas the known tetragonal phase grows from basic solutions. The Pt-Pt spacing is 2.95 Å in the triclinic phase as compared to a Pt-Pt spacing of 2.88 Å in the tetragonal phase.