We report simple rigidity theorems for Euler products under deformations of Euler factors. Certain products of the Riemann zeta function are rigid in the sense that there exist no deformations which preserve the meromorphy on $$.

In this paper, we give a necessary and sufficient condition under which a woven frame can be preserved when applying an operator to it, which provides a correction to one existing corresponding result due to Rahimi et al. We also present some new results on the stability of woven frames under perturbation of two different operators.

Let $$ be a discontinuous group for the 3-dimensional anti-de Sitter space $$. In this article, we discuss a growth rate of the counting of $$-orbits at infinity and the discrete spectrum of the hyperbolic Laplacian of the complete anti-de Sitter manifold $$.