Centers and Azumaya loci of finite WW-algebras

In this paper, we study the center $Z$ of the finite $W$-algebra $\mathcal{T}(\mathfrak{g},e)$ associated with a semi-simple Lie algebra $\mathfrak{g}$ over an algebraically closed field \mathds{k} of characteristic $p\gg0$, and an arbitrarily given nilpotent element $e\in\mathfrak{g}$. We obtain an analogue of Veldkamp's theorem on the center. For the maximal spectrum $\text{Specm}(Z)$, we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $\mathcal{T}(\mathfrak{g},e)$.Comment: 31 page

Matroidal structure of generalized rough sets based on symmetric and transitive relations

Rough sets are efficient for data pre-process in data mining. Lower and upper approximations are two core concepts of rough sets. This paper studies generalized rough sets based on symmetric and transitive relations from the operator-oriented view by matroidal approaches. We firstly construct a matroidal structure of generalized rough sets based on symmetric and transitive relations, and provide an approach to study the matroid induced by a symmetric and transitive relation. Secondly, this paper establishes a close relationship between matroids and generalized rough sets. Approximation quality and roughness of generalized rough sets can be computed by the circuit of matroid theory. At last, a symmetric and transitive relation can be constructed by a matroid with some special properties.Comment: 5 page

Second Repeating FRB 180814.J0422+73: Ten-year Fermi-LAT Upper Limits and Implications

The second repeating fast radio burst source, FRB 180814.J0422+73, was detected recently by the CHIME collaboration. We use the ten-year Fermi Large Area Telescope archival data to place a flux upper limit in the energy range of 100 MeV−10 GeV at the position of the source, which is ~1.1 × 10−11 erg cm−2 s−1 for a six-month time bin on average, and ~2.4 × 10−12 erg cm−2 s−1 for the entire ten-year time span. For the maximum redshift of z = 0.11, the ten-year upper limit of luminosity is ~7.3 × 1043 erg s−1. We utilize these upper limits to constrain the fast radio burst (FRB) progenitor and central engine. For the rotation-powered young magnetar model, the upper limits can pose constraints on the allowed parameter space for the initial rotational period and surface magnetic field of the magnetar. We also place significant constraints on the kinetic energy of a relativistic external shock wave, ruling out the possibility that there existed a gamma-ray burst (GRB) beaming toward Earth during the past ten years as the progenitor of the repeater. The case of an off-beam GRB is also constrained if the viewing angle is not much greater than the jet opening angle. All of these constraints are more stringent if FRB 180814.J0422+73 is at a closer distance