The Fibonacci sequences can be identified with 1-dimensional quasiperiodic tilings by the canonical projection method. We divide the window of the canonical projection method into smaller intervals by using local configurations. Then, we show that the intervals which appears in the window are divided into the ratio at $1:1/\tau:1$ ad infinitum.

In this paper, it is pointed out that ${\mathcal M}(A, I)_{cof}$ is an Abelian full subcategory of the category ${\mathcal M}(A)$ consisting of all $A$-modules for a principal ideal $I$ over a noetherian ring $A$.

We characterize the surjective linear isometries on $C^{(n)} [0, 1]$ and Lip$[0, 1]$. Here $C^{(n)} [0, 1]$ denotes the Banach space of $n$-times continuously differentiable functions on $[0, 1]$ equipped with the norm \begin{equation*} \| f \| = \sum_{k=0}^{n-1}|f^{(k)} (0)| + \sup _{x \in [0, 1]} | f^{(n)} (x) | \quad (f \in C^{(n)} [0, 1]) , \end{equation*} and Lip$[0, 1]$ denotes the Banach space of Lipschitz continuous functions on $[0, 1]$ equipped with the norm \begin{equation*} \| f \| = |f(0)| + \mathop{\operatorname{ess \, sup}} _{x \in [0, 1]} | f' (x) | \quad (f \in {\rm Lip}[0, 1]). \end{equation*}

In this paper, we characterize some monotonicity property of absolute normalized norms on $\mathbb{C}^n$ (resp. $\mathbb{R}^n$) by means of their corresponding continuous convex functions.