Determining Modes for Continuous Data Assimilation in 2D Turbulence

We study the number of determining modes necessary for continuous data assimilation in the two-dimensional incompressible Navier–Stokes equations. Our focus is on how the spatial structure of the body forcing affects the rate of continuous data assimilation and the number of determining modes. We treat this problem analytically by proving a convergence result depending on the H ^–1 norm of f and computationally by considering a family of forcing functions with identical Grashof numbers that are supported on different annuli in Fourier space. The rate of continuous data assimilation and the number of determining modes is shown to depend strongly on the length scales present in the forcing.     

Advancing Mathematical Activity: A Practice-Oriented View of Advanced Mathematical Thinking

The purpose of this article is to contribute to the dialogue about the notion of advanced mathematical thinking by offering an alternative characterization for this idea, namely advancing mathematical activity. We use the term advancing (versus advanced) because we emphasize the progression and evolution of students' reasoning in relation to their previous activity. We also use the term activity, rather than thinking. This shift in language reflects our characterization of progression in mathematical thinking as acts of participation in a variety of different socially or culturally situated mathematical practices. For these practices, we emphasize the changing nature of students' mathematical activity and frame the process of progression in terms of multiple layers of horizontal and vertical mathematizing.     

Total surface areas of Group IVA organometallic compounds: Predictors of toxicity to algae and bacteria

There is a high correlation between molecular surface area (TSA) of triorganotin and triorganolead compounds and their toxicity towards a bacterium (Escherichia coli) and an alga (Selenastrum capricornutum). Parallel attempts to correlate other Group IVA organometals incorporating silicon or germanium were unsuccessful. It was further demonstrated, however, that a high correlation was obtainable between certain series of compounds with the same organic substituent but different metal centers involving all Group IVA elements. In both instances, the inability to obtain a quantitative structure-activity relationship (QSAR) for all systems studied appears to be a function of the solubility of the compounds. While organotin TSA values have been found to correlate well with their toxicities toward various organisms, this study clearly suggests that this type of QSAR can be readily extended to include other organometal systems, provided that there is no solubility problem and the toxicity is a function of the hydrophobicity of the organometal compounds.     

Sulfur adsorption on Fe(1 1 0): a DFT study

The adsorption of atomic S on the Fe(1 1 0) surface is examined using density functional theory (DFT). Three different adsorption sites are considered, including the atop, hollow and bridge sites and the S is adsorbed at a quarter monolayer coverage in a p(2 × 2) arrangement. The hollow site is found to be the most stable, followed by the bridge and atop sites. At all three sites, S adsorption results in relatively minor surface reconstruction, with the most significant being that for the hollow site, with lateral displacements of 0.09 Å. Comparisons between S-adsorbed and pure Fe surfaces revealed reductions in the magnetic moments of surface-layer Fe atoms in the vicinity of the S. At the hollow site, the presence of S causes an increase in the surface Fe d-orbital density of states between 4 and 5 eV. However, S adsorption has no significant effect on the structure and magnetic properties of the lower substrate layers.     

Accreting neutron stars as probes of dense matter physics

There are over 100 accreting neutron stars in our galaxy, in which matter (typically H/He) is tidally transferred from a secondary companion to the neutron star. Accretion of this matter perturbs the thermal structure of the interior away from that of an isolated cooling neutron star. In this paper. we review how this accretion induces reactions in the crust of the neutron star that keep the interior hot. If the accretion is intermittent, then the heated surface layers are directly observable when accretion stops. This heating also affects the unstable ignition of light elements in the neutron star envelope. Observations of the neutron star cooling following an accretion outburst can in principle constrain the thermal properties of the crust and core.     

Anionic hyperconjugation

Although hyperconjugation involving alkyl groups to a carbocation is a well-established concept, the analogous charge delocalization generated in anions by electronegative substitution has long been a subject of controversy. We have investigated this phenomenon for the β-hydroxyethyl and β-trifluoroethyl anions using ab initio electronic structure calculations. Split valence basis sets augmented by diffuse functions were used with fully optimized geometries. Three dimensional molecular orbital plots clearly show an in-plane HOMO-LUMO mixing of a C-X σ antibonding orbital with the carbon lone pair which yields hyperconjugative π bonding in the anti conformation. Deformation density maps further demonstrate that this delocalization leads to the development of a π component as well as a previously unrecognized sigma enhancement. These results support and extend the work of Apeloig and that of Schleyer and Kos, but are in opposition to the induction hypothesis of Streitwieser and Holtz. The recent experimental determination of the acidity of (CF₃)₃CH by Tatlow $\text{et al}$ and their interpretation in terms of fluorine hyperconjugation are also in accord with this work.