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Variable \gama-ray emission from blazars, one of the most powerful classes of astronomical sources featuring relativistic jets, is a widely discussed topic. In this work, we present the results of a variability study of a sample of 20 blazars using \gama-ray (0.1--300~GeV) observations from Fermi/LAT telescope. Using maximum likelihood estimation (MLE) methods, we find that the probability density functions that best describe the $\gamma$-ray blazar flux distributions use the stable distribution family, which generalizes the Gaussian distribution. The results suggest that the average behavior of the \gama-ray flux variability over this period can be characterized by log-stable distributions. For most of the sample sources, this estimate leads to standard log-normal distribution ($\alpha=2$). However, a few sources clearly display heavy tail distributions (MLE leads to $\alpha<2$), suggesting underlying multiplicative processes of infinite variance. Furthermore, the light curves were analyzed by employing novel non-stationarity and autocorrelation analyses. The former analysis allowed us to quantitatively evaluate non-stationarity in each source -- finding the forgetting rate (corresponding to decay time) maximizing the log-likelihood for the modeled evolution of the probability density functions. Additionally, evaluation of local variability allows us to detect local anomalies, suggesting a transient nature of some of the statistical properties of the light curves. With the autocorrelation analysis, we examined the lag dependence of the statistical behavior of all the $\{(y_t,y_{t+l})\}$ points, described by various mixed moments, allowing us to quantitatively evaluate multiple characteristic time scales and implying possible hidden periodic processes., Comment: 13 pages, 9 Figures, MNRAS accepted

Variable \gama-ray emission from blazars, one of the most powerful classes of astronomical sources featuring relativistic jets, is a widely discussed topic. In this work, we present the results of a variability study of a sample of 20 blazars using \gama-ray (0.1--300~GeV) observations from Fermi/LAT telescope. Using maximum likelihood estimation (MLE) methods, we find that the probability density functions that best describe the $\gamma$-ray blazar flux distributions use the stable distribution family, which generalizes the Gaussian distribution. The results suggest that the average behavior of the \gama-ray flux variability over this period can be characterized by log-stable distributions. For most of the sample sources, this estimate leads to standard log-normal distribution ($\alpha=2$). However, a few sources clearly display heavy tail distributions (MLE leads to $\alpha<2$), suggesting underlying multiplicative processes of infinite variance. Furthermore, the light curves were analyzed by employing novel non-stationarity and autocorrelation analyses. The former analysis allowed us to quantitatively evaluate non-stationarity in each source -- finding the forgetting rate (corresponding to decay time) maximizing the log-likelihood for the modeled evolution of the probability density functions. Additionally, evaluation of local variability allows us to detect local anomalies, suggesting a transient nature of some of the statistical properties of the light curves. With the autocorrelation analysis, we examined the lag dependence of the statistical behavior of all the $\{(y_t,y_{t+l})\}$ points, described by various mixed moments, allowing us to quantitatively evaluate multiple characteristic time scales and implying possible hidden periodic processes., Comment: 13 pages, 9 Figures, MNRAS accepted

Enforcing distributions of latent variables in neural networks is an active subject. It is vital in all kinds of generative models, where we want to be able to interpolate between points in the latent space, or sample from it. Modern generative AutoEncoders (AE) like WAE, SWAE, CWAE add a regularizer to the standard (deterministic) AE, which allows to enforce Gaussian distribution in the latent space. Enforcing different distributions, especially topologically nontrivial, might bring some new interesting possibilities, but this subject seems unexplored so far. This article proposes a new approach to enforce uniform distribution on d-dimensional torus. We introduce a circular spring loss, which enforces minibatch points to be equally spaced and satisfy cyclic boundary conditions. As example of application we propose multiple-path morphing. Minimal distance geodesic between two points in uniform distribution on latent space of angles becomes a line, however, torus topology allows us to choose such lines in alternative ways, going through different edges of $[-\pi,\pi]^d$. Further applications to explore can be for example trying to learn real-life topologically nontrivial spaces of features, like rotations to automatically recognize 2D rotation of an object in picture by training on relative angles, or even 3D rotations by additionally using spherical features - this way morphing should be close to object rotation., Comment: 5 pages, 5 figures