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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

While we would like to predict exact values, available incomplete information is rarely sufficient - usually allowing only to predict conditional probability distributions. This article discusses hierarchical correlation reconstruction (HCR) methodology for such prediction on example of usually unavailable bid-ask spreads, predicted from more accessible data like closing price, volume, high/low price, returns. In HCR methodology we first normalize marginal distributions to nearly uniform like in copula theory. Then we model (joint) densities as linear combinations of orthonormal polynomials, getting its decomposition into (mixed) moments. Then here we model each moment (separately) of predicted variable as a linear combination of mixed moments of known variables using least squares linear regression - getting accurate description with interpretable coefficients describing linear relations between moments. Combining such predicted moments we get predicted density as a polynomial, for which we can e.g. calculate expected value, but also variance to evaluate uncertainty of such prediction, or we can use the entire distribution e.g. for more accurate further calculations or generating random values. There were performed 10-fold cross-validation log-likelihood tests for 22 DAX companies, leading to very accurate predictions, especially when using individual models for each company as there were found large differences between their behaviors. Additional advantage of the discussed methodology is being computationally inexpensive, finding and evaluation a model with hundreds of parameters and thousands of data points takes a second on a laptop., Comment: 10 pages, 7 figures

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

The rapidly improving performance of modern hardware renders convolutional codes obsolete, and allows for the practical implementation of more sophisticated correction codes such as low density parity check (LDPC) and turbo codes (TC). Both are decoded by iterative algorithms, which require a disproportional computational effort for low channel noise. They are also unable to correct higher noise levels, still below the Shannon theoretical limit. In this paper, we discuss an enhanced version of a convolutional-like decoding paradigm which adopts very large spaces of possible system states, of the order of $2^{64}$. Under such conditions, the traditional convolution operation is rendered useless and needs to be replaced by a carefully designed state transition procedure. The size of the system state space completely changes the correction philosophy, as state collisions are virtually impossible and the decoding procedure becomes a correction tree. The proposed decoding algorithm is practically cost-free for low channel noise. As the channel noise approaches the Shannon limit, it is still possible to perform correction, although its cost increases to infinity. In many applications, the implemented decoder can essentially outperform both LDPC and TC. This paper describes the proposed correction paradigm and theoretically analyzes the asymptotic correction performance. The considered encoder and decoder were verified experimentally for the binary symmetric channel. The correction process remains practically cost-free for channel error rates below 0.05 and 0.13 for the 1/2 and 1/4 rate codes, respectively. For the considered resource limit, the output bit error rates reach the order of $10^{-3}$ for channel error rates 0.08 and 0.18. The proposed correction paradigm can be easily extended to other communication channels; the appropriate generalizations are also discussed in this study., Comment: 14 pages, 7 figures, submitted to IEEE Transactions on Information Theory