On a q-difference Painlev\'e III equation: II. Rational solutions

Rational solutions for a $q$-difference analogue of the Painlev\'e III equation are considered. A Determinant formula of Jacobi-Trudi type for the solutions is constructed.Comment: Archive version is already official. Published by JNMP at http://www.sm.luth.se/math/JNMP

Hypergeometric solutions to the q-Painlev\'e equation of type A4(1)A_4^{(1)}

We consider the q-Painlev\'e equation of type $A_4^{(1)}$ (a version of q-Painlev\'e V equation) and construct a family of solutions expressible in terms of certain basic hypergeometric series. We also present the determinant formula for the solutions.Comment: 16 pages, IOP styl

Invariant varieties of periodic points for some higher dimensional integrable maps

By studying various rational integrable maps on $\mathbf{\hat C}^d$ with $p$ invariants, we show that periodic points form an invariant variety of dimension $\ge p$ for each period, in contrast to the case of nonintegrable maps in which they are isolated. We prove the theorem: {\it `If there is an invariant variety of periodic points of some period, there is no set of isolated periodic points of other period in the map.'}Comment: 24 page

Multi-indexed Wilson and Askey-Wilson Polynomials

As the third stage of the project multi-indexed orthogonal polynomials, we present, in the framework of 'discrete quantum mechanics' with pure imaginary shifts in one dimension, the multi-indexed Wilson and Askey-Wilson polynomials. They are obtained from the original Wilson and Askey-Wilson polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of 'virtual state solutions' of type I and II, in a similar way to the multi-indexed Laguerre, Jacobi and (q-)Racah polynomials reported earlier.Comment: 30 pages. Three references added. To appear in J.Phys.A. arXiv admin note: text overlap with arXiv:1203.586

Study of the GeV to TeV morphology of the γ Cygni SNR (G 78.2+2.1) with MAGIC and Fermi-LAT: Evidence for cosmic ray escape

Context. Diffusive shock acceleration (DSA) is the most promising mechanism that accelerates Galactic cosmic rays (CRs) in the shocks of supernova remnants (SNRs). It is based on particles scattering caused by turbulence ahead and behind the shock. The turbulence upstream is supposedly generated by the CRs, but this process is not well understood. The dominant mechanism may depend on the evolutionary state of the shock and can be studied via the CRs escaping upstream into the interstellar medium (ISM). Aims. Previous observations of the γ Cygni SNR showed a difference in morphology between GeV and TeV energies. Since this SNR has the right age and is at the evolutionary stage for a significant fraction of CRs to escape, our aim is to understand γ-ray emission in the vicinity of the γ Cygni SNR. Methods. We observed the region of the γ Cygni SNR with the MAGIC Imaging Atmospheric Cherenkov telescopes between 2015 May and 2017 September recording 87 h of good-quality data. Additionally, we analysed Fermi-LAT data to study the energy dependence of the morphology as well as the energy spectrum in the GeV to TeV range. The energy spectra and morphology were compared against theoretical predictions, which include a detailed derivation of the CR escape process and their γ-ray generation. Results. The MAGIC and Fermi-LAT data allowed us to identify three emission regions that can be associated with the SNR and that dominate at different energies. Our hadronic emission model accounts well for the morphology and energy spectrum of all source components. It constrains the time-dependence of the maximum energy of the CRs at the shock, the time-dependence of the level of turbulence, and the diffusion coefficient immediately outside the SNR shock. While in agreement with the standard picture of DSA, the time-dependence of the maximum energy was found to be steeper than predicted, and the level of turbulence was found to change over the lifetime of the SNR. © 2023 EDP Sciences. All rights reserved