Training teachers of mathematics virtually: Opportunities and threats

We reflect on our experiences working with 35 Teach First Trainees in June and July 2020 in an entirely online environment. Trainees had had no opportunity to spend any time in a classroom or with young people before commencing on an 80% timetable in September, and we have (still) not met any of the trainees face to face. In this reflective piece we describe and reflect on the changes we made to our pedagogy, especially those which produced an unexpectedly rich and sophisticated discourse, as well as the challenges

The Vector Analyzing Power in Elastic Electron-Nucleus Scattering

The vector analyzing power A_n is calculated for elastic electron scattering from a variety of spin zero nuclei at energies from 14 MeV to 3 GeV. Time reversal symmetry insures that A_n vanish in first Born approximation. Therefore A_n depends on Coulomb distortions and can be large for scattering from heavy nuclei. The vector analyzing power is a potential source of systematic error for parity violation experiments. We find that A_n=-0.361 ppm for the kinematics of the Parity Radius Experiment (PREX) involving 850 MeV electrons scattering at six degrees from 208Pb. This is comparable to the parity violating asymmetry. However for HAPPEX He involving 3 GeV electrons scattering on 4He we find that A_n is very small.Comment: 6 pages, 6 figures, submitted to Phys. Rev.

Line-of-sight effects on observability of kink and sausage modes in coronal structures with imaging telescopes

Kink modes of solar coronal structures, perturbing the loop in the direction along the line-of-sight (LOS), can be observed as emission intensity disturbances propagating along the loop provided the angle between the LOS and the structure is not ninety degrees. The effect is based upon the change of the column depth of the loop (along the LOS) by the wave. The observed amplitude of the emission intensity variations can be larger than the actual amplitude of the wave by a factor of two and there is an optimal angle maximizing the observed amplitude. For other angles this effect can also attenuate the observed wave amplitude. The observed amplitude depends upon the ratio of the wave length of kink perturbations to the width of the structure and on the angle between the LOS and the axis of the structure. Sausage modes are always affected negatively from the observational point of view, as the observed amplitude is always less than the actual one. This effect should be taken into account in the interpretation of wave phenomena observed in the corona with space-borne and ground-based imaging telescopes

Exact and approximate dynamics of the quantum mechanical O(N) model

We study a quantum dynamical system of N, O(N) symmetric, nonlinear oscillators as a toy model to investigate the systematics of a 1/N expansion. The closed time path (CTP) formalism melded with an expansion in 1/N is used to derive time evolution equations valid to order 1/N (next-to-leading order). The effective potential is also obtained to this order and its properties areelucidated. In order to compare theoretical predictions against numerical solutions of the time-dependent Schrodinger equation, we consider two initial conditions consistent with O(N) symmetry, one of them a quantum roll, the other a wave packet initially to one side of the potential minimum, whose center has all coordinates equal. For the case of the quantum roll we map out the domain of validity of the large-N expansion. We discuss unitarity violation in the 1/N expansion; a well-known problem faced by moment truncation techniques. The 1/N results, both static and dynamic, are also compared to those given by the Hartree variational ansatz at given values of N. We conclude that late-time behavior, where nonlinear effects are significant, is not well-described by either approximation.Comment: 16 pages, 12 figrures, revte

Transverse waves in a post-flare supra-arcade

Observations of propagating transverse waves in an open magnetic field structure with the Transition Region And Coronal Explorer (TRACE) are presented. Waves associated with dark tadpole-like sunward moving structures in the post-flare supra-arcade of NOAA active region 9906 on the 21st of April 2002 are analysed. They are seen as quasi-periodic transverse displacements of the dark tadpole tails, with periods in the range of 90–220 s. Their phase speeds and displacement amplitudes decrease as they propagate sunwards. At heights of 90 and 60 Mm above the post-flare loop footpoints the phase speeds are in the ranges 200–700 km s −1 and 90–200 km s −1 respectively. Furthermore, for consecutive tadpoles the phase speeds decrease and periods increase as a function of time. The waves are interpreted as propagating fast magnetoacoustic kink waves guided by a vertical, evolving, open structure

Resumming the large-N approximation for time evolving quantum systems

In this paper we discuss two methods of resumming the leading and next to leading order in 1/N diagrams for the quartic O(N) model. These two approaches have the property that they preserve both boundedness and positivity for expectation values of operators in our numerical simulations. These approximations can be understood either in terms of a truncation to the infinitely coupled Schwinger-Dyson hierarchy of equations, or by choosing a particular two-particle irreducible vacuum energy graph in the effective action of the Cornwall-Jackiw-Tomboulis formalism. We confine our discussion to the case of quantum mechanics where the Lagrangian is $L(x,\dot{x}) = (1/2) \sum_{i=1}^{N} \dot{x}_i^2 - (g/8N) [ \sum_{i=1}^{N} x_i^2 - r_0^2 ]^{2}$. The key to these approximations is to treat both the $x$ propagator and the $x^2$ propagator on similar footing which leads to a theory whose graphs have the same topology as QED with the $x^2$ propagator playing the role of the photon. The bare vertex approximation is obtained by replacing the exact vertex function by the bare one in the exact Schwinger-Dyson equations for the one and two point functions. The second approximation, which we call the dynamic Debye screening approximation, makes the further approximation of replacing the exact $x^2$ propagator by its value at leading order in the 1/N expansion. These two approximations are compared with exact numerical simulations for the quantum roll problem. The bare vertex approximation captures the physics at large and modest $N$ better than the dynamic Debye screening approximation.Comment: 30 pages, 12 figures. The color version of a few figures are separately liste

Chaos in effective classical and quantum dynamics

We investigate the dynamics of classical and quantum N-component phi^4 oscillators in the presence of an external field. In the large N limit the effective dynamics is described by two-degree-of-freedom classical Hamiltonian systems. In the classical model we observe chaotic orbits for any value of the external field, while in the quantum case chaos is strongly suppressed. A simple explanation of this behaviour is found in the change in the structure of the orbits induced by quantum corrections. Consistently with Heisenberg's principle, quantum fluctuations are forced away from zero, removing in the effective quantum dynamics a hyperbolic fixed point that is a major source of chaos in the classical model.Comment: 6 pages, RevTeX, 5 figures, uses psfig, changed indroduction and conclusions, added reference

Time evolution of the chiral phase transition during a spherical expansion

We examine the non-equilibrium time evolution of the hadronic plasma produced in a relativistic heavy ion collision, assuming a spherical expansion into the vacuum. We study the $O(4)$ linear sigma model to leading order in a large-$N$ expansion. Starting at a temperature above the phase transition, the system expands and cools, finally settling into the broken symmetry vacuum state. We consider the proper time evolution of the effective pion mass, the order parameter $\langle \sigma \rangle$, and the particle number distribution. We examine several different initial conditions and look for instabilities (exponentially growing long wavelength modes) which can lead to the formation of disoriented chiral condensates (DCCs). We find that instabilities exist for proper times which are less than 3 fm/c. We also show that an experimental signature of domain growth is an increase in the low momentum spectrum of outgoing pions when compared to an expansion in thermal equilibrium. In comparison to particle production during a longitudinal expansion, we find that in a spherical expansion the system reaches the ``out'' regime much faster and more particles get produced. However the size of the unstable region, which is related to the domain size of DCCs, is not enhanced.Comment: REVTex, 20 pages, 8 postscript figures embedded with eps