For a positive integer $k$ and a certain arithmetic progression $A$, there exist infinitely many quadratic fields $\mathbf{Q}(\sqrt{-d})$ whose class numbers are divisible by $k$ and $d\in A$. From this, we have a linear congruence of the representation numbers of integers as sums of three squares.

The equivariant cobordism class of a small cover over a simple convex polytope is determined by its tangential representation set. Since the tangential representation can be identified with the characteristic function of the simple convex polytope, by using characteristic functions we determine the number of small covers over cubes and the product of at most three simplices up to equivariant cobordism.

The aim in this paper is to develop the time-weighted energy method for quasi-linear hyperbolic systems of viscoelasticity. As a consequence, we prove the global existence and decay estimate of solutions for the space dimension $n\geq 2$, provided that the initial data are small in the $L^{2}$-Sobolev space.

We propose the viewpoint that the $r$-ple zeta-function of Euler-Zagier type can be regarded as a specialization of the zeta-function associated with the root system of $C_{r}$ type. From this viewpoint, we can see that Zagier’s well-known formula for multiple zeta values (MZVs) coincides with Witten’s volume formula associated with a sub-root system of $C_{r}$ type. Based on this observation, we generalize Zagier’s formula and also give analogous results which correspond to a sub-root system of $B_{r}$ type. We announce those results as well as some relevant results for partial multiple zeta values.