Generalizations of a result of Christensen on renewal sequences and linear recurrences

This note extends some recent work of Christensen on simple closed form solutions for linear recurrences, employing connections with renewal sequences. We obtain results for linear recursive sequences defined via monotone coefficients, and provide a broader result in the third-order case. A related open question is posed.     

Slope and Line of Best Fit: A Transfer of Knowledge Case Study

This paper brings together research on slope from mathematics education and research on line of best fit from statistics education by considering what knowledge of slope students transfer to a novel task involving determining the placement of an informal line of best fit. This study focuses on two students who transitioned from placing inaccurate to accurate lines of best fit during a task‐based interview. The analysis focuses on describing shifts in slope reasoning that accompanied the change to accurately placed lines, and investigates factors that may have influenced the shift in reasoning. The results have implications for the teaching of both slope and the line of best fit.     

“Instant Methylide” Modification of the Corey–Chaykovsky Cyclopropanation Reaction

Treatment of various electron‐deficient alkenes in DMSO with stable, dry, equimolar mixtures of either Me₃S(O)I/KOt‐Bu or Me₃S(O)I/NaH cleanly afforded the corresponding substituted cyclopropanes in good yields and short reaction times (<20 min for reactions at 50–60°C using 0.4–4.0 mmol alkene).     

On the chemical processing of hydrocarbon surfaces by fast oxygen ions

Solid methane (CH(4)), ethane (C(2)H(6)), and ethylene (C(2)H(4)) ices (thickness: 120 ± 40 nm; 10 K), as well as high-density polyethylene (HDPE: [C(2)H(4)](n)) films (thickness: 130 ± 20 nm; 10, 100, and 300 K), were irradiated with mono-energetic oxygen ions (Φ ~ 6 × 10(15) cm(-2)) of a kinetic energy of 5 keV to simulate the exposure of Solar System hydrocarbon ices and aerospace polymers to oxygen ions sourced from the solar wind and planetary magnetospheres. On-line Fourier-transform infrared spectroscopy (FTIR) was used to identify the following O(+) induced reaction pathways in the solid-state: (i) ethane formation from methane ice via recombination of methyl (CH(3)) radicals, (ii) ethane conversion back to methane via methylene (CH(2)) retro-insertion, (iii) ethane decomposing to acetylene via ethylene through successive hydrogen elimination steps, and (iv) ethylene conversion to acetylene via hydrogen elimination. No changes were observed in the irradiated PE samples via infrared spectroscopy. In addition, mass spectrometry detected small abundances of methanol (CH(3)OH) sublimed from the irradiated methane and ethane condensates during controlled heating. The detection of methanol suggests an implantation and neutralization of the oxygen ions within the surface where atomic oxygen (O) then undergoes insertion into a C-H bond of methane. Atomic hydrogen (H) recombination in forming molecular hydrogen and recombination of implanted oxygen atoms to molecular oxygen (O(2)) are also inferred to proceed at high cross-sections. A comparison of the reaction rates and product yields to those obtained from experiments involving 5 keV electrons, suggests that the chemical alteration of the hydrocarbon ice samples is driven primarily by electronic stopping interactions and to a lesser extent by nuclear interactions.     

Hybrid quantum mechanics/molecular mechanics-based molecular dynamics simulation of acid-catalyzed dehydration of polyols in liquid water

We use the conversion of protonated glycerol to acrolein for a case study of the mechanism of acid-catalyzed dehydration of polyols in aqueous environments. We employ hybrid Quamtum Mechanics/Molecular Mechanics Molecular Dynamics (QM/MM MD) simulations with biased sampling and perform free energy calculations for the elementary steps of the reaction. We investigate the effects of solvent dynamics and in particular the role of quantum mechanical water in the dehydration mechanism. We present results supporting a mechanism that proceeds via water-mediated proton transfers and thus through an enol intermediate. We find that the first dehydration may take place by two, low-energy pathways requiring, respectively, 20.9 and 18.8 kcal/mol of activation free energy. The second dehydration requires 19.9 kcal/mol of activation free energy while for the overall reaction we compute a free energy change of -8 kcal/mol.     

Injection of atoms and molecules in a superfluid helium fountain: Cu and Cu2He(n) (n = 1, ..., ∞)

We introduce an experimental platform designed around a thermomechanical helium fountain, which is aimed at investigating spectroscopy and dynamics of atoms and molecules in the superfluid and at its vapor interface. Laser ablation of copper, efficient cooling and transport of Cu and Cu(2) through helium vapor (1.5 K < T < 20 K), formation of linear and T-shaped Cu(2)-He complexes, and their continuous evolution into large Cu(2)-He(n) clusters and droplets are among the processes that are illustrated. Reflection is the dominant quantum scattering channel of translationally cold copper atoms (T = 1.7 K) at the fountain interface. Cu(2) dimers mainly travel through the fountain unimpeded. However, the conditions of fountain flow and transport of molecules can be controlled to demonstrate injection and, in particular, injection into a nondivergent columnar fountain with a plug velocity of about 1 m/s. The experimental observables are interpreted with the aid of bosonic density functional theory calculations and ab initio interaction potentials.     

Reactions of sulfur dioxide on calcium carbonate single crystal and particle surfaces at the adsorbed water carbonate interface

Sulfur dioxide reactions with calcium carbonate interfaces at 296 K in the presence and absence of adsorbed water result in the formation of adsorbed sulfite and sulfate. The extent of reaction is significantly enhanced, approximately five- to ten-fold for particulate and single crystal CaCO(3) (calcite), respectively, in the presence of adsorbed water between 30 and 85% RH. Atomic force microscopy following the reaction shows that adsorbed water facilitates surface reactivity by enhancing the mobility of surface ions, giving rise to the formation of nanometer sized product crystallites approximately 1 nm in height. Simultaneous with the formation of these crystallites is pitting and etching of the underlying substrate, which occurs preferentially in the vicinity of monoatomic surface steps. In the absence of water, there is little pitting and no evidence for the formation of crystallites. X-Ray photoelectron core and valence band spectra confirm the presence of two sulfur adsorbed species, SO and SO, with nearly equal amounts of SO and SO in the absence of adsorbed water and approximately five times more SO relative to SO in the presence of adsorbed water. From these data, it is proposed that the nanometer-sized crystallites are composed primarily of CaSO(3).