Direct determination of rate constants for coupling between aromatic radical anions and alkyl and benzyl radicals by laser-flash photolysis

Coupling rates between the radicals methyl, n-, sec-, tert-butyl and benzyl (R.) and the aromatic radical anions of 1,4-dicyanonaphthalene, 9,10-dicyanoanthracene and fluorenone (A-.) have been obtained using a new laser-flash photolysis method. The radicals R. and the radical anions A-. were generated by a photoinduced electron transfer reaction between the aromatic compound A and the alkyl or benzyl triphenylborate anion RB(Ph)3-. For the first time the rate constants of the coupling reaction between methyl and benzyl radicals with aromatic radical anions have been obtained. For all the measured coupling rate constants an average value of k1 = 1.9 x 10(9) M-1 s-1 was found with a relatively small variation in the coupling rates (0.8-2.9 x 10(9) M-1 s-1). The results demonstrate that the coupling rate k1 is insensitive to changes in the steric and electronic properties of the radicals and the structure and standard potentials of the aromatic radical anions.     

Polymer-induced turbulence modifications in an impinging jet

This effort explores the impact of dilute polymer solutions on the turbulence characteristics in a submerged liquid impinging-jet configuration. Turbulent impinging jets are commonly used in technological applications such as drying, scouring, cooling, or heating due to an enhancement in transport characteristics in the impingement region under certain nozzle-to-wall configurations. Previous efforts have identified significant turbulence modifications in the presence of dilute concentrations of polymer in both bounded and unbounded flows, though the former has received considerably more attention. To this end, particle-image velocimetry measurements were taken for an axisymmetric turbulent impinging jet with a nozzle-to-wall distance H/D = 6.8 and nominal Reynolds number of 26,000. Measurements were performed for both plain water and dilute polymer solutions of polyethylene oxide at concentrations of 50 and 100 ppm. The mean and turbulence characteristics of these three flows are contrasted and it is observed that the two polymer solutions modify both the mean and turbulent characteristics of the jet in all three regions of interest (the free-jet, impingement, and wall-jet regions). Of interest, the 50 ppm case yielded a slight suppression of the turbulence in the free-jet region accompanied by a longer axial length of the potential core compared to the case of plain water. In contrast, the 100 ppm case exhibits clear enhancement of the turbulence in the free-jet region and a shortening of the potential core length. The effect of polymer was opposite in the impingement and wall-jet regions wherein the turbulence was slightly suppressed in the 100 ppm case in a manner consistent with the onset of the Toms effect in this wall-bounded region of the flow.     

Microsoft Excel as a supplement in a high school statistics course

This study was conducted in the spring of 2002 at Arlington High School in a Probability and Statistics course for junior and senior level students. Two sections of the class were compared with respect to the use/non-use of EXCEL in the course. The experimental group (the one using EXCEL) outperformed the control group on five of the six teacher-created unit tests and on both of the criterion referenced assessments used by the district; however, the differences were not statistically significant at the 0.05 level. Furthermore, the experimental group expressed very positive feelings towards the use of the software in a survey administered at the conclusion of the course.     

Gabor Frames in $${\ell }^2({\mathbb {Z}})$$ ℓ 2 ( Z ) and Linear Dependence

We prove that an overcomplete Gabor frame in \({\ell }^2({\mathbb {Z}})\) generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in \({\ell }^2({\mathbb {Z}})\) with modulation parameter 1 / M and translation parameter N for some \(M,N\in {\mathbb {N}},\) and generated by a finite sequence g in \({\ell }^2({\mathbb {Z}})\) with K nonzero entries.

     

Frame properties of systems arising via iterated actions of operators

Motivated by recent progress in dynamical sampling we prove that every frame which is norm-bounded below can be represented as a finite union of sequences ${\{{(T_j)}^n{\phi }_j\}}_{n=0}^{\infty },j=1,\dots ,J$ for some bounded operators $T_j$ and elements ${\phi }_j$ in the underlying Hilbert space. The result is optimal, in the sense that it turns out to be problematic to replace the collection of generators ${\phi }_1,\dots ,{\phi }_J$ by a singleton: indeed, for linearly independent frames we prove that we can represent the frame in terms of just one system ${\{T^n\phi \}}_{n=0}^{\infty }$, but unfortunately this representation often forces the operator T to be unbounded. Several examples illustrate the connection of the results to typical frames like Gabor frames and wavelet frames, as well as generic constructions in arbitrary separable Hilbert spaces.     

DFT Study of the Molybdenum‐Catalyzed Deoxydehydration of Vicinal Diols

The mechanism of the molybdenum‐catalyzed deoxydehydration (DODH) of vicinal diols has been investigated using density functional theory. The proposed catalytic cycle involves condensation of the diol with an Mo^VI oxo complex, oxidative cleavage of the diol resulting in an Mo^IV complex, and extrusion of the alkene. We have compared the proposed pathway with several alternatives, and the results have been corroborated by comparison with the molybdenum‐catalyzed sulfoxide reduction recently published by Sanz et al. and with experimental observations for the DODH itself. Improved understanding of the mechanism should expedite future optimization of molybdenum‐catalyzed biomass transformations.