Braided differential operators on quantum algebras

We propose a general scheme of constructing braided differential algebras via algebras of “quantum exponentiated vector fields” and those of “quantum functions”. We treat a reflection equation algebra as a quantum analog of the algebra of vector fields. The role of a quantum function algebra is played by a general quantum matrix algebra. As an example we mention the so-called RTT algebra of quantized functions on the linear matrix group GL(m)$GL\left(m\right)$. In this case our construction essentially coincides with the quantum differential algebra introduced by S. Woronowicz. If the role of a quantum function algebra is played by another copy of the reflection equation algebra we get two different braided differential algebras. One of them is defined via a quantum analog of (co)adjoint vector fields, the other algebra is defined via a quantum analog of right-invariant vector fields. We show that the former algebra can be identified with a subalgebra of the latter one. Also, we show that “quantum adjoint vector fields” can be restricted to the so-called “braided orbits” which are counterparts of generic GL(m)$GL\left(m\right)$-orbits in gl^∗(m)$g{l}^{\ast }\left(m\right)$. Such braided orbits endowed with these restricted vector fields constitute a new class of braided differential algebras.     

On q-Deformed $${{\mathfrak{gl}}_{\ell+1}}$$-Whittaker Function III

In this paper, we identify q-deformed \mathfrakgl_l+1{\mathfrak{gl}_{\ell+1}}-Whittaker functions with a specialization of the Macdonald polynomials. This provides a representation of q-deformed \mathfrakgl_l+1{\mathfrak{gl}_{\ell+1}}-Whittaker functions in terms of the Demazure characters of affine Lie algebra [^(\mathfrakgl)]_l+1{\widehat{\mathfrak{gl}}_{\ell+1}}. We also define a system of dual Hamiltonians for q-deformed \mathfrakgl_l+1{\mathfrak{gl}_{\ell+1}}-Toda chains and give a new integral representation for the q-deformed \mathfrakgl_l+1{\mathfrak{gl}_{\ell+1}}-Whittaker functions. Finally, we represent the q-deformed \mathfrakgl_l+1{\mathfrak{gl}_{\ell+1}}-Whittaker function as a matrix element of a quantum torus algebra.     

Chiodo formulas for the <Emphasis Type="Italic">r</Emphasis>-th roots and topological recursion

We analyze Chiodo’s formulas for the Chern classes related to the r-th roots of the suitably twisted integer powers of the canonical class on the moduli space of curves. The intersection numbers of these classes with \(\psi \)-classes are reproduced via the Chekhov–Eynard–Orantin topological recursion. As an application, we prove that the Johnson-Pandharipande-Tseng formula for the orbifold Hurwitz numbers is equivalent to the topological recursion for the orbifold Hurwitz numbers. In particular, this gives a new proof of the topological recursion for the orbifold Hurwitz numbers.     

The Residually Weakly Primitive Geometries of M 22

We announce the classification of all firm and residually connected geometries satisfying the conditions (I P)₂ and (2T)₁ and on which the Mathieu group M ₂₂ acts flag-transitively and residually weakly primitively. The complete list of geometries is available as a supplement to this note [6].     

Absence of localization in a class of Schrödinger operators with quasiperiodic potential

We prove that a class of discrete Schrödinger operators with a quasiperiodic potential taking only a finite number of values, exhibits purely continuous spectrum; in particular they cannot have localized eigenvectors.Groupe de Recherche 048 of C.N.R.S.     

Thermodynamics for the zero-level set of the Brownian bridge

The random set of instants where the Brownian bridge vanishes is constructed in terms of a random branching process. The Hausdorff measure supported by this set is shown to be equivalent to the partition function of a special class of disordered systems. This similarity is used to show rigorously the existence of a phase transition for this particular class of disordered systems. Moreover, it is shown that at high temperature the specific free energy has the strong self-averaging property and that at low temperature it has no self-averaging property. The unicity at high-temperature and the existence of many limits at low temperature are established almost surely in the disorder.Work supported by the Swiss National Science Foundation     

Robust prediction and interpolation for vector stationary processes

Summary Robust multivariate prediction and interpolation problems for statistically contaminated vector valued second order stationary processes are considered. The statistical contamination is modeled by requiring that the spectral density matrices of the processes lie within certain nonparametric classes. Both prediction and interpolation are then formalized as games whose saddle point solutions are sought. Finally, such solutions are found and analyzed, for two specific multivariate spectral classes.Research supported by the Air Force Office of Scientific Research under Grants AFOSR-83-0229 and AFOSR-82-0030     

On the Distribution of Activity Time in PERT

In PERT analysis the activity-time distribution is assumed to be a beta distribution, and the mean and variance of the activity time are estimated on the basis of the ‘pessimistic’, ‘most likely’ and ‘optimistic’ completion times, which are subjectively determined by an analyst. In this paper, on the basis of the study of the PERT assumptions, we present an improvement of these estimates. It is also shown that, by means of additional reasonable assumptions, the activity-time distribution in PERT analysis may be essentially simplified.     

Development and validation of a liquid chromatographic/tandem mass spectrometric assay for the quantitation of nucleoside HIV reverse transcriptase inhibitors in biological matrices

Besides liquid chromatographic (LC)/UV methods adapted to therapeutic drug monitoring, there is still a need for more powerful techniques that can be used for pharmacological research and clinical purposes. We developed an LC method coupled with tandem mass spectrometry (MS/MS) to separate, detect and quantify with high sensitivity the nucleoside analogues used in multitherapies (zidovudine, stavudine, zalcitabine, didanosine, lamivudine and abacavir) in plasma and in the intracellular medium. We worked on two essential issues: (i) the need to use two ionization modes in order to achieve the best sensitivity, which leads to the optimization of the chromatographic separation of drugs detected in the positive ionization mode and drugs detected in the negative ionization mode, and (ii) the need to optimize the extraction step in order to enhance sample recovery. The peripheral blood mononuclear cells were lysed in Tris buffer-MeOH. A clean-up procedure was performed by solid-phase extraction only for plasma samples. The LC separation was carried out on a Zorbax Stable Bond C(18) column followed by MS/MS analysis after electrospray ionization in either the negative or positive mode. The positive ionization mode was applied at the beginning of the run to detect zalcitabine and lamivudine, then the ionization mode was changed to negative for the detection of didanosine, stavudine, internal standard and zidovudine. The calibration range for all the analytes was 0.5-200 ng ml(-1). The recoveries were between 64 and 90%, with coefficients of variation (CVs) lower than 15%. The inaccuracy (bias) was +/-15% with CVs always lower than 12%. The analytes were stable at room temperature and in the extraction solvent for at least 24 h, after storage at -80 degrees C for 3 months, after three freeze-thaw cycles and in the injection solvent after 48 h at 4 degrees C. Together with the measurement of intracellular triphosphorylated metabolites thanks to the powerful plasma and intracellular assay method for intact drugs, it is possible to describe the behaviour of nucleoside analogues against HIV through plasma pharmacokinetics, cell membrane diffusion including drug transport involvement, and also the intracellular metabolism.     

Quinobis(imidazolylidene): Synthesis and Study of an Electron‐Configurable Bis(N‐Heterocyclic Carbene) and Its Bimetallic Complexes

Reaction of bromanil with N,N′‐dimesitylformamidine followed by deprotonation with NaN(SiMe₃)₂ afforded 1,1′,3,3′‐tetramesitylquinobis(imidazolylidene) ( 1 ), a bis(N‐heterocyclic carbene) (NHC) with two NHC moieties connected by a redox active p‐quinone residue, in 72 % yield of isolated compound. Bimetallic complexes of 1 were prepared by coupling to FcN₃ ( 2 ) or FcNCS ( 3 ; Fc=ferrocenyl) or coordination to [M(cod)Cl] ( 4 a or 4 b , where M=Rh or Ir, respectively; cod=1,5‐cyclooctadiene). Treatment of 4 a and 4 b with excess CO(g) afforded the corresponding [M(CO)₂Cl] complexes 5 a and 5 b , respectively. Analysis of 2 – 5 by NMR spectroscopy and X‐ray diffraction indicated that the electron‐deficient quinone did not significantly affect the inherent spectral properties or coordination chemistry of the flanking imidazolylidene units, as compared to analogous NHCs. Infrared spectroscopy and cyclic voltammetry revealed that decreasing the electron density at ML_n afforded an increase in the stretching energy and a decrease in the reduction potential of the quinone, indicative of metal–quinone electronic interaction. Differential pulse voltammetry and chronoamperometry of the metal‐centered oxidations in 2 – 4 revealed two single, one‐electron peaks. Thus, the metal atoms bound to 1 are oxidized at indistinguishable potentials and do not appear electronically coupled. However, the metal–quinone interaction was used to increase the electron density at coordinated metal atoms. Infrared spectroelectrochemistry revealed that the average ν_CO values for 5 a and 5 b decreased by 14 and 15 cm^?1, respectively, upon reduction of the quinone embedded within 1 . These shifts correspond to 10 and 12 cm^?1 decreases in the Tolman electronic parameter of this ditopic ligand.