Exploring the long-term variability and evolutionary stage of the interacting binary DQ Velorum

To progress in the comprehension of the double periodic variable (DPV) phenomenon, we analyse a series of optical spectra of the DPV system DQ Velorum during much of its long-term cycle. In addition, we investigate the evolutionary history of DQ Vel using theoretical evolutionary models to obtain the best representation for the current observed stellar and orbital parameters of the binary. We investigate the evolution of DQ Vel through theoretical evolutionary models to estimate the age and the mass transfer rate which are compared with those of its twin V393 Scorpii. Donor subtracted spectra covering around 60% of the long-term cycle, allow us to investigate time-modulated spectral variations of the gainer star plus the disc. We compare the observed stellar parameters of the system with a grid of theoretical evolutionary tracks computed under a conservative and a non-conservative evolution regime. We have found that the EW of Balmer and helium lines in the donor subtracted spectra are modulated with the long-term cycle. We observe a strenghtening in the EWs in all analysed spectral features at the minimum of the long-term cycle which might be related to an extra line emission during the maximum of the long-term variability. Difference spectra obtained at the secondary eclipse support this scenario. We have found that a non-conservative evolutionary model is a better representation for the current observed properties of the system. The best evolutionary model suggests that DQ Vel has an age of 7.40 x 10^{7} yr and is currently in a low mass transfer rate (-9.8x10^{-9} Msun/yr) stage, after a mass transfer burst episode. Comparing the evolutionary stages of DQ Vel and V393 Sco we observed that the former is an older system with a lower mass transfer rate. This might explain the differences observed in the physical parameters of their accretion discs.Comment: 10 pages, 13 figure

Moments of a single entry of circular orthogonal ensembles and Weingarten calculus

Consider a symmetric unitary random matrix $V=(v_{ij})_{1 \le i,j \le N}$ from a circular orthogonal ensemble. In this paper, we study moments of a single entry $v_{ij}$. For a diagonal entry $v_{ii}$ we give the explicit values of the moments, and for an off-diagonal entry $v_{ij}$ we give leading and subleading terms in the asymptotic expansion with respect to a large matrix size $N$. Our technique is to apply the Weingarten calculus for a Haar-distributed unitary matrix.Comment: 17 page

General moments of the inverse real Wishart distribution and orthogonal Weingarten functions

Let $W$ be a random positive definite symmetric matrix distributed according to a real Wishart distribution and let $W^{-1}=(W^{ij})_{i,j}$ be its inverse matrix. We compute general moments $\mathbb{E} [W^{k_1 k_2} W^{k_3 k_4} ... W^{k_{2n-1}k_{2n}}]$ explicitly. To do so, we employ the orthogonal Weingarten function, which was recently introduced in the study for Haar-distributed orthogonal matrices. As applications, we give formulas for moments of traces of a Wishart matrix and its inverse.Comment: 29 pages. The last version differs from the published version, but it includes Appendi