On tt-extensions of the Hankel determinants of certain automatic sequences

In 1998, Allouche, Peyri\`ere, Wen and Wen considered the Thue--Morse sequence, and proved that all the Hankel determinants of the period-doubling sequence are odd integral numbers. We speak of $t$-extension when the entries along the diagonal in the Hankel determinant are all multiplied by~$t$. Then we prove that the $t$-extension of each Hankel determinant of the period-doubling sequence is a polynomial in $t$, whose leading coefficient is the {\it only one} to be an odd integral number. Our proof makes use of the combinatorial set-up developed by Bugeaud and Han, which appears to be very suitable for this study, as the parameter $t$ counts the number of fixed points of a permutation. Finally, we prove that all the $t$-extensions of the Hankel determinants of the regular paperfolding sequence are polynomials in $t$ of degree less than or equal to $3$

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

Fundamental Plane of Black Hole Activity in Quiescent Regime

A correlation among the radio luminosity ($L_{\rm R}$), X-ray luminosity ($L_{\rm X}$), and black hole mass ($M_{\rm BH}$) in active galactic nuclei (AGNs) and black hole binaries is known to exist and is called the "Fundamental Plane" of black hole activity. Yuan & Cui (2005) predicts that the radio/X-ray correlation index, $\xi_{\rm X}$, changes from $\xi_{\rm X}\approx 0.6$ to $\xi_{\rm X}\approx 1.2-1.3$ when $L_{\rm X}/L_{\rm Edd}$ decreases below a critical value $\sim 10^{-6}$. While many works favor such a change, there are also several works claiming the opposite. In this paper, we gather from literature a largest quiescent AGN (defined as $L_{\rm X}/L_{\rm Edd} < 10^{-6}$) sample to date, consisting of $75$ sources. We find that these quiescent AGNs follow a $\xi_{\rm X}\approx 1.23$ radio/X-ray relationship, in excellent agreement with the Yuan \& Cui prediction. The reason for the discrepancy between the present result and some previous works is that their samples contain not only quiescent sources but also "normal" ones (i.e., $L_{\rm X}/L_{\rm Edd} > 10^{-6}$). In this case, the quiescent sources will mix up with those normal ones in $L_{\rm R}$ and $L_{\rm X}$. The value of $\xi_{\rm X}$ will then be between $0.6$ and $\sim1.3$, with the exact value being determined by the sample composition, i.e., the fraction of the quiescent and normal sources. Based on this result, we propose that a more physical way to study the Fundamental Plane is to replace $L_{\rm R}$ and $L_{\rm X}$ with $L_{\rm R}/L_{\rm Edd}$ and $L_{\rm X}/L_{\rm Edd}$, respectively.Comment: 11 pages, 7 figures, accepted for publication in The Astrophysical Journa