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We introduce two entire functions $f_{{\mathrm{A}}_{\frac{1}{2}\infty }}$ and $f_{{\mathrm{D}}_{\frac{1}{2}\infty }}$ in two variables. Both of them have only two critical values $0$ and $1$, and the associated maps ${\mathbf{C}}^2\to \mathbf{C}$ define topologically locally trivial fibrations over $\mathbf{C}\setminus \{0,1\}$. All critical points in the singular fibers over $0$ and $1$ are ordinary double points, and the associated vanishing cycles span the middle homology group of the general fiber, whose intersection diagram forms bi-partitely decomposed infinite quivers of type ${\mathrm{A}}_{\frac{1}{2}\infty }$ and ${\mathrm{D}}_{\frac{1}{2}\infty }$, respectively. Coxeter elements of type ${\mathrm{A}}_{\frac{1}{2}\infty }$ and...

Let $M$ be a free module over a noetherian ring. For ${\omega }_1,...,{\omega }_k\in M$, let $𝒜$ be the ideal generated by coefficients of ${\omega }_1\wedge ...\wedge {\omega }_k$. For an element $\omega \in {\bigwedge }^{p}M$ with $p\<\mathrm{prof}.𝒜$, if $\omega \wedge {\omega }_1\wedge ...\wedge {\omega }_k=0$, there exists ${\eta }_1,...,{\eta }_k\in {\bigwedge }^{p-1}M$ such that $\omega ={\sum }_{i=1}^k{\eta }_i\wedge {\omega }_i$. This is a generalization of a lemma on the division of forms due to de Rham (Comment. Math. Helv., 28 (1954)) and has some applications to the study of singularities.