Two-layer Locally Repairable Codes for Distributed Storage Systems

In this paper, we propose locally repairable codes (LRCs) with optimal minimum distance for distributed storage systems (DSS). A two-layer encoding structure is employed to ensure data reconstruction and the designated repair locality. The data is first encoded in the first layer by any existing maximum distance separable (MDS) codes, and then the encoded symbols are divided into non-overlapping groups and encoded by an MDS array code in the second layer. The encoding in the second layer provides enough redundancy for local repair, while the overall code performs recovery of the data based on redundancy from both layers. Our codes can be constructed over a finite field with size growing linearly with the total number of nodes in the DSS, and facilitate efficient degraded reads.Comment: This paper has been withdrawn by the author due to inaccuracy of Claim

A decades-long fast-rise-exponential-decay flare in low-luminosity AGN NGC 7213

We analysed the four-decades-long X-ray light curve of the low-luminosity active galactic nucleus (LLAGN) NGC 7213 and discovered a fast-rise-exponential-decay (FRED) pattern, i.e. the X-ray luminosity increased by a factor of $\approx 4$ within 200d, and then decreased exponentially with an $e$-folding time $\approx 8116$d ($\approx 22.2$ yr). For the theoretical understanding of the observations, we examined three variability models proposed in the literature: the thermal-viscous disc instability model, the radiation pressure instability model, and the tidal disruption event (TDE) model. We find that a delayed tidal disruption of a main-sequence star is most favourable; either the thermal-viscous disk instability model or radiation pressure instability model fails to explain some key properties observed, thus we argue them unlikely.Comment: Accepted for publication in MNRAS, updated version after proof correction

Vertex operator algebras of Argyres-Douglas theories from M5-branes

We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d N=(2, 0) theory of type $J$ on a punctured sphere. We denote the AD theories as $(J^b[k],Y)$, where $J^b[k]$ and $Y$ represent an irregular and a regular singularity respectively. We restrict to the `minimal' case where $J^b[k]$ has no associated mass parameters, and the theory does not admit any exactly marginal deformations. The VOA corresponding to the AD theory is conjectured to be the W-algebra $\mathcal{W}^{k_{2d}}(J,Y)$, where $k_{2d}=-h+ \frac{b}{b+k}$ with $h$ being the dual Coxeter number of $J$. We verify this conjecture by showing that the Schur index of the AD theory is identical to the vacuum character of the corresponding VOA, and the Hall-Littlewood index computes the Hilbert series of the Higgs branch. We also find that the Schur and Hall-Littlewood index for the AD theory can be written in a simple closed form for $b=h$. We also test the conjecture that the associated variety of such VOA is identical to the Higgs branch. The M5-brane construction of these theories and the corresponding TQFT structure of the index play a crucial role in our computations.Comment: 35 pages, 1 figure, v2: minor corrections, referenced adde