Reflection positivity and invertible topological phases

We implement an extended version of reflection positivity (Wick-rotated unitarity) for invertible topological quantum field theories and compute the abelian group of deformation classes using stable homotopy theory. We apply these field theory considerations to lattice systems, assuming the existence and validity of low energy effective field theory approximations, and thereby produce a general formula for the group of Symmetry Protected Topological (SPT) phases in terms of Thom's bordism spectra; the only input is the dimension and symmetry group. We provide computations for fermionic systems in physically relevant dimensions. Other topics include symmetry in quantum field theories, a relativistic 10-fold way, the homotopy theory of relativistic free fermions, and a topological spin-statistics theorem.Comment: 136 pages, 16 figures; minor changes/corrections in version 2; v3 major revision; v4 minor revision: corrected proof of Lemma 9.55, many small changes throughout; v5 version for publication in Geometry & Topolog

A switch to reduce resistivity in smoothed particle magnetohydrodynamics

Artificial resistivity is included in Smoothed Particle Magnetohydrodynamics simulations to capture shocks and discontinuities in the magnetic field. Here we present a new method for adapting the strength of the applied resistivity so that shocks are captured but the dissipation of the magnetic field away from shocks is minimised. Our scheme utilises the gradient of the magnetic field as a shock indicator, setting {\alpha}_B = h|gradB|/|B|, such that resistivity is switched on only where strong discontinuities are present. The advantage to this approach is that the resistivity parameter does not depend on the absolute field strength. The new switch is benchmarked on a series of shock tube tests demonstrating its ability to capture shocks correctly. It is compared against a previous switch proposed by Price & Monaghan (2005), showing that it leads to lower dissipation of the field, and in particular, that it succeeds at capturing shocks in the regime where the Alfv\'en speed is much less than the sound speed (i.e., when the magnetic field is very weak). It is also simpler. We also demonstrate that our recent constrained divergence cleaning algorithm has no difficulty with shock tube tests, in contrast to other implementations.Comment: 8 pages, 7 figures, accepted for publication in MNRA

Diffuse Non-thermal X-ray Emission: Evidence for Cosmic-ray Acceleration at the Shock Front in IC1262

We report the first localization of diffuse, non-thermal, X-ray emission from a nearby galaxy cluster. Using Chandra data, we have isolated a diffuse non-thermal X-ray component with a photon index, Gamma_ X = 2.21 +0.14 -0.15 and a flux of 9.5 +1.1 -2.5 x 10^-5 photons cm^-2 s^-1 keV^-1 at 1 keV, that extends from ~1'.5 to ~2'.5 to the south of the X-ray flux peak. Comparison to simulations implies that the diffuse non-thermal emission is produced by primary electrons, accelerated at shocks to relativistic velocities. Using these results and the flux and hardness maps produced with data from the Chandra Advanced CCD Imaging Spectrometer, we conclude that a smaller subclump coming from the north merged with IC1262. The offset of the cD galaxy from the X-ray peak and large peculiar velocity indicate that the subclump's impact parameter was to the west and on the near side of IC1262.Comment: 5 pages, 4 figures, 2 tables. Accepted by ApJ

Towards Economic Models for MOOC Pricing Strategy Design

MOOCs have brought unprecedented opportunities of making high-quality courses accessible to everybody. However, from the business point of view, MOOCs are often challenged for lacking of sustainable business models, and academic research for marketing strategies of MOOCs is also a blind spot currently. In this work, we try to formulate the business models and pricing strategies in a structured and scientific way. Based on both theoretical research and real marketing data analysis from a MOOC platform, we present the insights of the pricing strategies for existing MOOC markets. We focus on the pricing strategies for verified certificates in the B2C markets, and also give ideas of modeling the course sub-licensing services in B2B markets