Duality and Hidden Symmetries in Interacting Particle Systems

In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the “hidden” symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP) model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1,1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing.     

A Pieri-type formula for isotropic flag manifolds

We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from the Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type (respectively, type ) Schubert polynomial by the Schur -polynomial (respectively, the Schur -polynomial ). Geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the infinite Coxeter group , identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show that most of these identities follow from the Pieri-type formula, and our analysis leads to a new partial order on the Coxeter group and formulas for many of these structure constants.

     

Asymptotic Expansions for the Distribution of the Maximum of Gaussian Random Fields

Some asymptotic results are proved for the distribution of the maximum of a centered Gaussian random field with unit variance on a compact subset S of ^N . They are obtained by a Rice method and the evaluation of some moments of the number of local maxima of the Gaussian field above an high level inside S and on the border S. Depending on the geometry of the border we give up to N+1 terms of the expansion sometimes with exponentially small remainder. Application to waves maximum is shown.     

Perturbations in the Spectra of the Rydberg Series of a Silver Atom

The main series of a silver atom with n = 5–70 are investigated. Perturbations of the np ² P _1/2,3/2 states with n = 10 and 17 caused by interaction with the selfionization 4d5s5p ⁴ P _1/2,3/2 states are detected. Energies of the selfionization states are determined.     

Polymer rings and chains consisting of doubly silyl-bridged metallocenes

With the formation of novel organometallic macromolecules in mind, the polycondensation of transition metal ions and bridged cyclopentadienyl ligands was studied. To this end solvated salts MX2 (M = Fe, Ni, and Cr) were treated with a ligand that consisted of two doubly silyl-bridged cyclopentadienyl anions. For M = Fe and diluted solutions a series of rings Oi was obtained that consisted of a minimum of six (O6) and up to 17 (O17) ferrocene moieties in the backbone. They were separated partly by medium pressure liquid chromatography. The macrocycles were established by high-resolution MALDI-TOF mass spectroscopy which also yielded the molecular weight, the polydispersion, and the mean ring size, chi n, of the mixture of reaction products. When the reaction temperature was decreased from 25 degrees C to -20 degrees C, chi n increased from 8.1 to 10.8. Ferrocene-containing chains, lambda j, with 2 < or = j < or = 12 were obtained in addition to rings in the presence of water; the terminal groups were cyclopentadiene moieties. The reaction of two ferrocene-fused cyclopentadienyl anions with [FeCl2(thf)1.5] gave chains consisting of exclusively uneven numbers of ferrocenes. For M = Ni and Cr the formation of doubly silyl-bridged nickelocenes and chromocenes was proven by NMR spectroscopy. MALDI-TOF mass spectroscopy showed nickelocene-containing chains accompanied by some rings. For M = Fe the H,H-DQF COSY spectra established the structure of O7, O8, and O9. The oxidation of the ferrocene-containing ring O7 with I2, NOPF6, and AgPF6 gave ionic species [O7]n+ which suffered from low stability. The ring-closing reaction is discussed, and the relative abundance of the various rings is related to MNDO calculations.     

Regiochemistry of S-alkyl-substituted alkynes in Pauson-Khand reactions. Is a correlation with X-ray data and charge distribution calculations of the Co(2)(CO)(6)-alkyne complexes possible?

[reaction: see text] Pauson-Khand reactions (PKR) of RSCtbd1;CR' (6-10) yielded in all cases as the main product the regioisomer with the alkyl-S group disposed alpha to the CO group (16a-22a). Correlation of these results with X-ray data and charge distribution calculations of the corresponding dicobalthexacarbonyl complexes proves that a recently postulated "trans effect" in these complexes is not suitable for predicting the regiochemical outcome of the PKR unambiguously.     

Enzymatic synthesis of chiral organophosphothioates from prochiral precursors

The phosphotriesterase from Pseudomonas diminuta has been shown to selectively cleave the pro-R p-nitrophenolate substituent from bis-p-nitrophenyl alkyl phosphothioate esters. When the alkyl substituent is methyl, ethyl, or isopropyl the enantiomeric excess of the product is >/=99%. Manipulation of the active site through mutagenesis has enabled the preparation of protein variants that preferentially hydrolyze the pro-S substituent of the target substrates. This methodology thus permits the preparation of chiral products from prochiral precursors.