Bijection between Conjugacy Classes and Irreducible Representations of Finite Inverse Semigroups

In this paper we show that the irreducible representations of a finite inverse semigroup $S$ over an algebraically closed field $F$ are in bijection with the conjugacy classes of $S$ if the characteristic of $F$ is zero or a prime number that does not divide the order of any maximal subgroup of $S$

Hard X-ray emission and 44^{44}Ti line features of Tycho Supernova Remnant

A deep hard X-ray survey of the INTEGRAL satellite first detected the non-thermal emission up to 90 keV in the Tycho supernova (SN) remnant. Its 3 -- 100 keV spectrum is fitted with a thermal bremsstrahlung of $kT\sim 0.81\pm 0.45$ keV plus a power-law model of $\Gamma \sim 3.01\pm 0.16$. Based on the diffusive shock acceleration theory, this non-thermal emission, together with radio measurements, implies that Tycho remnant may not accelerate protons up to $>$PeV but hundreds TeV. Only heavier nuclei may be accelerated to the cosmic ray spectral "knee". In addition, we search for soft gamma-ray lines at 67.9 and 78.4 keV coming from the decay of radioactive $^{44}$Ti in Tycho remnant by INTEGRAL. A bump feature in the 60-90 keV energy band, potentially associated with the $^{44}$Ti line emission, is found with a marginal significance level of $\sim$ 2.6 $\sigma$. The corresponding 3 $\sigma$ upper limit on the $^{44}$Ti line flux amounts to 1.5 $\times$ 10$^{-5}$ ph cm$^{-2}$ s$^{-1}$. Implications on the progenitor of Tycho SN, considered to be the prototype of type Ia SN, are discussed.Comment: 15 pages, 4 figures, accepted for publication in Ap