Characterizing Interaction with Visual Mathematical Representations

This paper presents a characterization of computer-based interactions by which learners can explore and investigate visual mathematical representations (VMRs). VMRs (e.g., geometric structures, graphs, and diagrams) refer to graphical representations that visually encode properties and relationships of mathematical structures and concepts. Currently, most mathematical tools provide methods by which a learner can interact with these representations. Interaction, in such cases, mediates between the VMR and the thinking, reasoning, and intentions of the learner, and is often intended to support the cognitive tasks that the learner may want to perform on or with the representation. This paper brings together a diverse set of interaction techniques and categorizes and describes them according to their common characteristics, goals, intended benefits, and features. In this way, this paper aims to provide a preliminary framework to help designers of mathematical cognitive tools in their selection and analysis of different interaction techniques as well as to foster the design of more innovative interactive mathematical tools. An effort is made to demonstrate how the different interaction techniques developed in the context of other disciplines (e.g., information visualization) can support a diverse set of mathematical tasks and activities involving VMRs.     

Electron trajectories in molecular orbitals

The time-dependent Schrödinger equation can be rewritten so that its interpretation is no longer probabilistic. Two well-known and related reformulations are Bohmian mechanics and quantum hydrodynamics. In these formulations, quantum particles follow real, deterministic trajectories influenced by a quantum force. Generally, trajectory methods are not applied to electronic structure calculations as they predict that the electrons in a ground-state, real, molecular wavefunction are motionless. However, a spin-dependent momentum can be recovered from the nonrelativistic limit of the Dirac equation. Therefore, we developed new, spin-dependent equations of motion for the quantum hydrodynamics of electrons in molecular orbitals. The equations are based on a Lagrange multiplier, which constrains each electron to an isosurface of its molecular orbital, as required by the spin-dependent momentum. Both the momentum and the Lagrange multiplier provide a unique perspective on the properties of electrons in molecules.     

Incomplete similarity of a plane turbulent wall jet on smooth and transitionally rough surfaces

This study assesses the hypothesis of incomplete similarity for a plane turbulent wall jet on smooth and transitionally rough surfaces. Typically, a wall jet is considered to consist of two regions: an inner layer and an outer layer. The degree to which these two regions reach equilibrium with each other and interact to produce the property of self-similarity remains an open question. In this study, the analysis of the outer and inner regions indicates that each region is characterised by a half-width which exhibits its own distinct dependence on the streamwise distance x from the slot, and a single self-similar structure for both regions does not exist. More specifically, the inner and outer layers of the wall jet exhibit different scaling laws, which results in two self-similar mean velocity profiles, both of which retain a dependence on the slot height H. As such, incomplete similarity of the wall jet on smooth and transitionally rough surfaces is confirmed by this study. In addition, comparison of the experimental results for the transitionally rough surface with the smooth wall case indicates that the surface roughness modifies the development of the mean velocity profile in both the inner and outer regions, although the effect on the outer region is relatively small and close to the experimental uncertainty.     

Flow around impulsively started square prisms

The flow around square and diamond prisms and a circular cylinder impulsively set into motion was studied experimentally using the particle image velocimetry (PIV) technique. The experiments were conducted in water in an X-Y towing tank for Reynolds numbers from Re=200-1000. The temporal development of the near-wake recirculation zone, and its pair of primary eddies, was examined from the initial start until the wake became asymmetric, at a dimensionless elapsed time of t^?=4 or 5. For both bodies, the length of the recirculation zone, the streamwise location of the primary eddies, and the strength of the primary eddies increased with time following the impulsive start, while the cross-stream spacing of the eddy centres remained nearly constant. The recirculation zones of the square and diamond prisms were longer than that of the impulsively started circular cylinder. For t^?>2, the primary eddy strength, maximum vorticity, and cross-stream spacing of the primary eddies, were the same for both the square prism and circular cylinder. The diamond prism had the strongest primary eddies and highest maximum values of vorticity. A comparison of recirculation zone length data for impulsively started bluff bodies of six different cross-sections illustrated the effects of afterbody and forebody shape, with the normal flat plate (no afterbody and no forebody) having the longest recirculation zone and the circular cylinder (rounded afterbody and rounded forebody) having the shortest recirculation zone.     

Jahn–Teller Effect in Circulenes: X‐ray Diffraction Study of Coronene and Corannulene Radical Anions

Single‐crystal X‐ray diffraction studies of two polyaromatic radical anions crystallized as sodium salts, namely [Na(DME)₃]⁺[C₂₀H₁₀^?] ( 1 ) and [Na(DME)₃]⁺[C₂₄H₁₂^?] ( 2 ) are reported. This allowed the first structural evaluation of Jahn–Teller (JT) effects for monoreduced circulenes and a comparison between bowl‐shaped corannulene and planar coronene. The C_s and D_2h symmetrical distortions are found to fit the experimental data for C₂₀H₁₀^.? and C₂₄H₁₂^.?, respectively. The continuous symmetry measure (CSM) analysis was carried out to provide a quantitative measure of the JT distortions in 1 and 2 . In addition, the X‐ray crystallographic results were fully supported by DFT calculations.     

Indoprofen upregulates the survival motor neuron protein through a cyclooxygenase-independent mechanism

Most patients with the pediatric neurodegenerative disease spinal muscular atrophy have a homozygous deletion of the survival motor neuron 1 (SMN1) gene, but retain one or more copies of the closely related SMN2 gene. The SMN2 gene encodes the same protein (SMN) but produces it at a low efficiency compared with the SMN1 gene. We performed a high-throughput screen of approximately 47,000 compounds to identify those that increase production of an SMN2-luciferase reporter protein, but not an SMN1-luciferase reporter protein. Indoprofen, a nonsteroidal anti-inflammatory drug (NSAID) and cyclooxygenase (COX) inhibitor, selectively increased SMN2-luciferase reporter protein and endogenous SMN protein and caused a 5-fold increase in the number of nuclear gems in fibroblasts from SMA patients. No other NSAIDs or COX inhibitors tested exhibited this activity.