Let $R$ be a Noetherian local ring with the maximal ideal $\mathfrak{m}$. Assume that $R$ contains ideals $I$ and $J$ satisfying the conditions (1) $I\subseteq J$, (2) $I : \mathfrak{m}\nsubseteq J$, and (3) $J$ is $\mathfrak{m}$-full, that is $\mathfrak{m}J : x = J$ for some $x \in \mathfrak{m}$. Then the theorem says that $R$ is a regular local ring, if the projective dimension $\mathrm{pd}_{R}I$ of $I$ is finite. Let $\mathfrak{q} = (a_{1}, a_{2}, \ldots , a_{t})R$ be an ideal in a Noetherian local ring $R$ generated by a maximal $R$-regular sequence $a_{1}, a_{2}, \ldots , a_{t}$ and let $\bar{\mathfrak{q}}$ denote the integral closure of $\mathfrak{q}$. Then, thanks to the theorem applied to the ideals $I = \mathfrak{q} : \mathfrak{m}$ and $J = \mathfrak{\bar{q}}$, it follows that $I^{2} = \mathfrak{q}I$, unless $R$ is a regular local ring. Consequences are discussed.

Abhyankar-Moh and Suzuki proved that if an irreducible polynomial $f \in \mathbf{C}[x_{1}, x_{2}]$ in two complex variables $x_{1}$ and $x_{2}$ defines the affine plane curve $C = (f = 0) \subset \mathbf{A}^{2}$, which is isomorphic to the affine line: $C \cong \mathbf{A}^{1}$, then $f$ is a variable of $\mathbf{C}[x_{1}, x_{2}]$, i.e., there exists a polynomial $g \in\mathbf{C}[x_{1}, x_{2}]$ such that $\mathbf{C}[f, g] = \mathbf{C}[x_{1}, x_{2}]$ (cf. [A-M75], [Su74]). In this article, we prove under some additional assumptions that the similar result holds in the three-dimensional case, namely, if an irreducible polynomial $f \in \mathbf{C}[x_{1}, x_{2}, x_{3}]$ in three complex variables $x_{1}$, $x_{2}$ and $x_{3}$ defines the hypersurface $S = (f = 0) \subset \mathbf{A}^{3}$, which is isomorphic to the affine plane: $S \cong \mathbf{A}^{2}$, then $f$ is a variable of $\mathbf{C}[x_{1}, x_{2}, x_{3}]$, i.e., there are polynomials $g, h \in \mathbf{C}[x_{1}, x_{2}, x_{3}]$ such that $\mathbf{C}[f, g, h] = \mathbf{C}[x_{1}, x_{2}, x_{3}]$. Moreover, we shall determine the detailed form of such a polynomial $f \in\mathbf{C}[x_{1}, x_{2}, x_{3}]$ for the special case.

A generalization of the Buckdahn-Föllmer formula is obtained by considering a composite transformation $\xi (x, F(x))$ in the framework of the Ramer-Kusuoka formula where $F(x)$ takes values in a finite dimensional space. The point is to establish the chain rule for composite Wiener functionals through the continuity of the substitution. The localization argument makes it possible to deal in our framework with the transformations studied by C. Donati-Martin, H. Matsumoto and M. Yor [5]. Our formula gives a new approach to the study of quadratic Wiener functionals.

Let $f$ be an area preserving monotone twist diffeomorphism on the annulus. In this paper, we prove the equivalence of the following three conditions: (i) the ;annulus is foliated by circles invariant under $f$. (ii) any periodic point of $f$ is of Birkhoff type, and (iii) all iterations $f^{n}$ are twist diffeomorphisms.