We introduce multivariable q-analogues of the determinant and show a sufficient condition for an interval of the symmetric group to have a symmetric chain decomposition in the Bruhat order by using the expansion formulas.

We prove that cylinders over curves of finite type and planes are the only ruled surfaces in Euclidean spaces with finite type Gauss map and circular cylinders are the only tubes in Euclidean 3-space with finite type Gauss map.

Let $a\in\mathbf{Z}$ such that $a\neq 1$, $a\neq-2^{17}$ and $(17,a)=1$. Let $\alpha_1,\alpha_2,\ldots,\alpha_{17}$ denote the roots of $x^{17}+ax+a=0$. It is shown that every prime ideal is unramified in $\mathbf{Q}(\alpha_1,\alpha_2,\ldots,\alpha_{17})/\mathbf{Q}(\alpha_1)$ if and only if $a=2^{62}n^2+4605612312119580521n+1149886651258880054$ for some $n\in\mathbf{Z}$.

The notion of an entangled linear order was first introduced by Avraham and Shelah [1]. Subsequently, Todorcevic [5] generalized it to higher cardinals and mentioned it is useful to solve problems such as the productivity of chain conditions and the square bracket partition relations. He also showed that if $\mathbf{wCH}(\mu)$ holds there is a $2^{\mu}$-entangled linear order of size $2^{\mu}$. From this, we can immediately observe that $\mathbf{GCH}$ implies the full existence of entangled linear orders. In this paper we will show that such full existence occurs also in the Easton's models in which we can arbitrarily determine the powers of infinite regular cardinals.