Rough matroids based on coverings

The introduction of covering-based rough sets has made a substantial contribution to the classical rough sets. However, many vital problems in rough sets, including attribution reduction, are NP-hard and therefore the algorithms for solving them are usually greedy. Matroid, as a generalization of linear independence in vector spaces, it has a variety of applications in many fields such as algorithm design and combinatorial optimization. An excellent introduction to the topic of rough matroids is due to Zhu and Wang. On the basis of their work, we study the rough matroids based on coverings in this paper. First, we investigate some properties of the definable sets with respect to a covering. Specifically, it is interesting that the set of all definable sets with respect to a covering, equipped with the binary relation of inclusion $\subseteq$, constructs a lattice. Second, we propose the rough matroids based on coverings, which are a generalization of the rough matroids based on relations. Finally, some properties of rough matroids based on coverings are explored. Moreover, an equivalent formulation of rough matroids based on coverings is presented. These interesting and important results exhibit many potential connections between rough sets and matroids.Comment: 15page

Implications of Fermi-LAT observations on the origin of IceCube neutrinos

The IceCube (IC) collaboration recently reported the detection of TeV-PeV extraterrestrial neutrinos whose origin is yet unknown. By the photon-neutrino connection in $pp$ and $p\gamma$ interactions, we use the \fermi-LAT observations to constrain the origin of the IC detected neutrinos. We find that Galactic origins, i.e., the diffuse Galactic neutrinos due to cosmic ray (CR) propagation in the Milky Way, and the neutrinos from the Galactic point sources, may not produce the IC neutrino flux, thus these neutrinos should be of extragalactic origin. Moreover, the extragalactic gamma-ray bursts (GRBs) may not account for the IC neutrino flux, the jets of active galactic nuclei may not produce the IC neutrino spectrum, but the starburst galaxies (SBGs) may be promising sources. As suggested by the consistency between the IC detected neutrino flux and the Waxman-Bahcall bound, GRBs in SBGs may be the sources of both the ultrahigh energy, $>10^{19}$eV, CRs and the $1-100$~PeV CRs that produce the IC detected TeV-PeV neutrinos.Comment: JCAP accepted version; 8 pages, 2 figs; discussion on blazar origin added; conclusion unchange

One Fits All: A Unified Synchrotron Model Explains GRBs with FRED-Shaped Pulses

The analysis of gamma-ray burst (GRB) spectra often relies on empirical models like the Band function, which lacks a distinct physical explanation. Previous attempts to couple physical models with observed data have been confined to individual burst studies, where the model is fitted to segmented spectra with independent physical parameters. These approaches frequently fail to explain the spectral evolution, which should be governed by a consistent set of physical conditions. In this study, we propose a novel approach by incorporating the synchrotron radiation model to provide a self-consistent explanation for a selection of single-pulse GRBs. Our sample is carefully chosen to minimize contamination from overlapping pulses, allowing for a comprehensive test of the synchrotron model under a unified physical condition, such as a single injection event of electrons. By tracing the evolution of cooling electrons in a decaying magnetic field, our model predicts a series of time-dependent observed spectra that align well with the observed data. Remarkably, using a single set of physical parameters, our model successfully fits all time-resolved spectra within each burst. Additionally, our model accurately predicts the evolution of some key features of GRBs such as the spectral peak $E_{\rm p}$ and light curve shapes, all of which are consistent with observations. Our findings strongly support the notion that the spectral and temporal evolution in GRB pulses originates from the expansion of the GRB emission region with an initial radius of approximately $10^{15}$ cm, with synchrotron radiation being the underlying emission mechanism.Comment: 25 pages, 18 figures, 4 table

Poly[diaqua­(μ-oxalato)(μ-2-oxido­pyridinium-3-carboxyl­ato)praseo­dymium(III)]

In the title complex, [Pr(C6H4NO3)(C2O4)(H2O)2]n, each PrIII ion is coordinated by eight O atoms from two 2-oxynicotinate ligands, two oxalate ligands and two water mol­ecules, displaying a distorted bicapped square-anti­prismatic geometry. The carboxyl­ate groups link adjacent praseodymium metal centres, forming layers parallel to the bc plane. The crystal packing is stabilized by inter­molecular O—H⋯O and N—H⋯O hydrogen bonds

[meso-Tetra­kis(4-heptyl­oxyphen­yl)porphyrinato]nickel(II)

In the title compound, [Ni(C72H84N4O4)], the four-coordinate NiII ion in the middle of the planar 24-membered porphyrin ring is located on a crystallograpic inversion center, with Ni—N distances of 1.946 (2)–1.951 (2) Å. The 4-heptyl­oxyphenyl groups are twisted with respect to the porphyrin mean plane, the dihedral angles being 88.5 (3) and 79.1 (2)°