We investigate the asymptotic behavior of sample functions of stable processes when t→∞. We compare our results with the iterated logarithm law, results for the first hitting time and most visited sites problems. 1 Main Notions 1. Let X1, X2... be mutually independent random variables with the same law of distribution F (x). The distribution F (x) is called strictly stable if the random variabl...

Abstract For any Liouville number $\alpha $ , all of the following are transcendental numbers: ${e}^\alpha $\log _{e}\alpha $\sin \alpha $\cos $\tan $\sinh $\cosh $\tanh $\arcsin and inverse functions evaluated at listed trigonometric hyperbolic functions, noting that wherever multiple values involved, every such value is transcendental. This remains true if ‘Liouville number’ replaced by ‘ U -...

<p style='text-indent:20px;'>In this paper we show the existence of a nonnegative solution for singular problem with logarithmic and exponential nonlinearity, namely <inline-formula><tex-math id="M1">\begin{document}$ -\Delta u = \log(u)\chi_{\{u>0\}} + \lambda f(u) $\end{document}</tex-math></inline-formula> in id="M2">\begin{document}$ \Omega id="M3"&gt...

Deep neural networks have recently achieved state-of-the-art results in many machine learning problems, e.g., speech recognition or object recognition. Hitherto, work on rectified linear units (ReLU) provides empirical and theoretical evidence on performance increase of neural networks comparing to typically used sigmoid activation function. In this paper, we investigate a new manner of improvi...