``Quasi-elliptic'' functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated. A related structure has appeared recently in the computation of Feynman integrals. The two approaches are related by a sequence of polynomials closely tied to the Eulerian polynomials.

We present, theoretical predictions and Monte Carlo simulations, for a simple three matrix model that exhibits an exotic phase transition. The nature of the transition is very different if approached from the high or low temperature side. The high temperature phase is described by three self interacting random matrices with no background spacetime geometry. As the system cools there is a phase transition in which a classical two-sphere condenses to form the background geometry. The transition has an entropy jump or latent heat, yet the specific heat diverges as the transition is approached from low temperatures. We find no divergence or evidence of critical fluctuations when the transition is approached from the high temperature phase. At sufficiently low temperatures the system is described by small fluctuations, on a background classical two-sphere, of a U(1) gauge field coupled to a massive scalar field. The critical temperature is pushed upwards as the scalar field mass is increased. Once the geometrical phase is well established the specific heat takes the value 1 with the gauge and scalar fields each contributing 1/2.

The first prototype of a photo-detection unit of the future KM3NeT neutrino telescope has been deployed in the deep waters of the Mediterranean Sea. This digital optical module has a novel design with a very large photocathode area segmented by the use of 31 three inch photomultiplier tubes. It has been integrated in the ANTARES detector for in-situ testing and validation. This paper reports on the first months of data taking and rate measurements. The analysis results highlight the capabilities of the new module design in terms of background suppression and signal recognition. The directionality of the optical module enables the recognition of multiple Cherenkov photons from the same $^{40}$K decay and the localization bioluminescent activity in the neighbourhood. The single unit can cleanly identify atmospheric muons and provide sensitivity to the muon arrival directions.

We discuss the prospects for a progress to be brought by ASTRO-H in the understanding of the physics of particle acceleration in astrophysical environments. Particular emphasis will be put on the synergy with gamma-ray astronomy, in the context of the rapid developments of recent years. Selected topics include: shock acceleration in supernova remnants (SNRs) and in clusters of galaxies, and the extreme particle acceleration seen in gamma-ray binaries. Since the hydrodynamics and thermal properties of shocks in these objects are covered in other white papers, we focus on the aspects related to the process of particle acceleration. In the case of SNRs, we emphasize the importance of SXS and HXI observations of the X-ray emission of young SNRs dominated by synchrotron radiation, particularly SNR RX J1713.7-3946. We argue that the HXI observations of young SNRs, as a byproduct of SXS observations dedicated for studies of the shock dynamics and nucleosynthesis, will provide powerful constraints on shock acceleration theories. Also, we discuss gamma-ray binary systems, where extreme particle acceleration is inferred regardless of the nature (a neutron star or a black hole) of the compact object. Finally, for galaxy clusters, we propose searches for hard X-ray emission of secondary electrons from interactions of ultra-high energy cosmic rays accelerated at accretion shocks. This should allow us to understand the contribution of galaxy clusters to the flux of cosmic rays above 10^18 eV.

We construct the fuzzy spaces based on the three non-trivial co-adjoint orbits of the exceptional simple Lie group, $G_2$.

We study the $N$-dependent behaviour of $\mathrm{2d}$ causal set quantum gravity. This theory is known to exhibit a phase transition as the analytic continuation parameter $\beta$, akin to an inverse temperature, is varied. Using a scaling analysis we find that the asymptotic regime is reached at relatively small values of $N$. Focussing on the $\mathrm{2d}$ causal set action $S$, we find that $\beta \langle S\rangle$ scales like $N^\nu$ where the scaling exponent $\nu$ takes different values on either side of the phase transition. For \beta > \beta_c we find that $\nu=2$ which is consistent with our analytic predictions for a non-continuum phase in the large $\beta$ regime. For \beta<\beta_c we find that $\nu=0$, consistent with a continuum phase of constant negative curvature thus suggesting a dynamically generated cosmological constant. Moreover, we find strong evidence that the phase transition is first order. Our results strongly suggest that the asymptotic regime is reached in $\mathrm{2d}$ causal set quantum gravity for $N \gtrsim 65$.