We study the difference splitting scheme for numerical calculation of stable solutions a two-dimensional linear hyperbolic system with dissipative boundary conditions in case constant coefficients lower terms. A discrete analogue Lyapunov function is constructed and an priori estimate obtained it. The allows us to assert exponential stability solution.

In this paper, some classes of concerned multiplication operators consisting analytic and hyperbolic functions are defined considered. Furthermore, properties such as boundedness compactness the new discussed. Finally, a general class weighted Bloch is characterized by metric spaces.

Let $X$ be a compact, hyperbolic surface of genus $g\geq 2$. In this paper, we prove that the twisted Selberg and Ruelle zeta functions, associated with an arbitrary, finite-dimensional, complex representation $\chi$ $\pi_1(X)$ admit meromorphic continuation to $\mathbb{C}$. Moreover, study behaviour function at $s=0$ point, it has zero order $\dim(\chi)(2g-2)$.