$\ell_{p}$ is characterized by the convergence of a series defined by an $L_{p}$-function on the real line $\mathbf{R}$.

The notion of principally tame regular sequences is introduced for systems of polynomials with a weight vector. As an application, construction of formal solutions is discussed for the systems of nonlinear differential equations which belong to the fourth Painlevé hierarchy with a large parameter.

We will investigate the dynamics of a holomorphic self-map $f$ of a compact complex manifold $M$ such that the sequence $\{ f^{n}\}_{n\ge 1}$ has at least one subsequence which converges uniformly on $M$.