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How do they know it is a parallelogram? Analysing geometric discourse at van Hiele Level 3

In this article, we introduce Sfard's discursive framework and use it to investigate prospective teachers' geometric discourse in the context of quadrilaterals. In particular, we focus on describing and analysing two participants' use of mathematical words and substantiation routines related to parallelograms and their properties at van Hiele level 3 thinking. Our findings suggest that a single van Hiele level of thinking encompasses a range of complexity of reasoning and differences in discourse and thus a deeper investigation of students' mathematical thinking within assigned van Hiele levels is warranted.     

Control of phase behavior and properties of vesicle gels by admixing ionic surfactants to the nonionic surfactant Brij 30

The effect of the addition of two cationic surfactants of different chain length (decyl and dodecyl trimethylammonium bromide, DeTMABr and DTMABr, respectively) and one anionic surfactant of identical chain length (sodium dodecyl sulfate, SDS) on phase behavior, structure, and macroscopic properties of a bilayer forming nonionic surfactant (Brij 30) has been investigated by means of phase studies, rheology, turbidity measurements, dynamic light scattering, and freeze-fracture transmission electron microscopy. We concentrated on DTMABr because of the generically similar behavior for the other ionic surfactants. It is found that already very small amounts of added ionic surfactant have a very pronounced effect on the phase behavior of these systems. The pure nonionic surfactant forms bilayers and has a tendency for the formation of vesicles which becomes enhanced by charging the bilayer through the incorporation of the ionic surfactant. The presence of the ionic surfactant leads to much more viscous systems, which already at a total surfactant concentration of 150 mM become gel-like. For a given surfactant concentration, the elastic properties of the gels increase largely upon the addition of ionic surfactant. This effect is strongly synergistic, requiring only very small amounts of added ionic surfactant, and the elastic properties pass through a maximum for a content of ionic surfactant of about 3-5 mol %. This behavior can be explained in a self-consistent way by a simple rheological model and by combining it with light scattering data. For the addition of larger amounts, the elastic properties decrease again and the formed vesicles become structurally less defined as one is leaving the range of conditions for forming well-defined vesicles, which are required for forming elastic vesicle gels.     

Understanding the enhanced synchronization of delay-coupled networks with fluctuating topology

We study the dynamics of networks with coupling delay, from which the connectivity changes over time. The synchronization properties are shown to depend on the interplay of three time scales: the internal time scale of the dynamics, the coupling delay along the network links and time scale at which the topology changes. Concentrating on a linearized model, we develop an analytical theory for the stability of a synchronized solution. In two limit cases, the system can be reduced to an “effective” topology: in the fast switching approximation, when the network fluctuations are much faster than the internal time scale and the coupling delay, the effective network topology is the arithmetic mean over the different topologies. In the slow network limit, when the network fluctuation time scale is equal to the coupling delay, the effective adjacency matrix is the geometric mean over the adjacency matrices of the different topologies. In the intermediate regime, the system shows a sensitive dependence on the ratio of time scales, and on the specific topologies, reproduced as well by numerical simulations. Our results are shown to describe the synchronization properties of fluctuating networks of delay-coupled chaotic maps.     

Characterizing the lower log region of the atmospheric surface layer via large-scale particle tracking velocimetry

We investigate the dynamics of a small number of droplets (N = 1, 2, 3) in microfluidic Hele–Shaw cells. We study the cases N = 1, 2, and 3 droplets and analyze the influence of the side walls. In the course of the study, we observe spontaneous alignment of droplet pairs, pair exchanges, droplet escape, multiple reflections between walls, i.e., a number of phenomena that have not been reported yet. As a whole, using pairwise far-field dipolar interactions between droplets, along with treating the walls as mirrors, allows to reproduce the observations, even though limitations in the predictability of the model are pointed out in a few cases. From a more practical prospective, the work shows that the behavior of elementary droplet assemblies can be put under acceptable experimental control in a wide variety of situations, a feature potentially interesting for self-assembly, mixing, or transport of particles in microfluidic environments.     

H-abstraction is more efficient than cis-trans isomerization in (4-methylcyclohexylidene) fluoromethane. An ab initio molecular dynamics study

Non-adiabatic molecular dynamics simulations have been performed in the fluoro-olefin (4-methylcyclohexylidene) fluoromethane (4MCF) using multiconfigurational CASSCF (complete active space self-consistent field) on-the-fly calculations. As an olefin containing a C[double bond, length as m-dash]C double bond, 4MCF is expected to undergo cis-trans isomerization after light irradiation. However, ab initio molecular dynamics shows that a preferential dissociation of atomic hydrogen is taking place after population transfer to the bright ππ* state. This state is strongly mixed with πσ* states allowing dissociation in the electronic excited state before deactivation to the ground state occurs. A minor amount of trajectories experiences F-dissociation, followed by pyramidalization at the sp(2) carbons and CHF dissociation. In contrast, the amount of trajectories undergoing torsion around the double bond, and therefore cis-trans isomerization, is marginal. The H-abstraction reaction is ultrafast, taking place in less than 60 fs.     

Phase resolved near-field imaging of propagating waves in infrared tapered slot antennas

Tapered slot antennas (TSAs) consist of a planar non-resonant structure which couples incident radiation to a propagating waveguide mode. They are commonly used at microwave and radio frequencies because they are fundamentally broadband and have small profiles. Because of their planar layout and broadband response they have recently been scaled to infrared frequencies where they have advantages for sensing and energy harvesting. We use scattering-type scanning near-field optical microscopy (s-SNOM) to study the mode transformation of two types of TSA operating in the thermal infrared (λ₀ = 10.6 μm) with respect to electric field amplitude and phase. The results agree well with simulation showing both the phase reversal across the tapered slot and the traveling of wave fronts along the tapered slot, yet they also reveal high sensitivity of device performance to inhomogeneities in the geometry or illumination. This study will aid future design and analysis of practical non-resonant antennas operating at optical and infrared frequencies.     

Stark control of a chiral fluoroethylene derivative

Hydrogen dissociation is an unwanted competing pathway if a torsional motion around the C═C double bond in a chiral fluoroethylene derivative, namely (4-methylcyclohexylidene) fluoromethane (4MCF), is to be achieved. We show that the excited state H-dissociation can be drastically diminished on time scales long enough to initiate a torsion around the C═C double bond using the nonresonant dynamic Stark effect. Potential energy curves, dipoles, and polarizabilities for the regarded one-dimensional reaction coordinate are calculated within the CASSCF method. The influence of the excitation and the laser control field is then simulated using wave packet dynamics.