Using Mehler kernel, we give an example of nontrivial solution of the homogeneous Cauchy problem of the Hermite heat equation, which is, for each $t$, bounded in the space variables.

Let $p$ be an odd prime, $\mathbf Z_p$ the ring of $p$-adic integers, and $l$ a prime number different from $p$. We have shown in [1] that, if $l$ is a sufficiently large primitive root modulo $p^2$, then the $l$-class group of the $\mathbf Z_p$-extension over the rational field is trivial. We shall modify part of the proof of the above result and see, in the case $p\leq 7$, that the result holds without assuming $l$ to be sufficiently large.

The gradient map associated to a regular open convex cone gives a diffeomorphism from the cone onto its dual cone. If the cone is homogeneous, the inverse of the map is known to be equal to the gradient map associated to the dual cone. However, we show that this is no longer true for a general case by presenting a simple counterexample.

We consider a parameterized family of continuous functions, which contains as its members Bourbaki's and Perkins's nowhere differentiable functions as well as the Cantor-Lebesgue singular functions.

$N$-extended affine Weyl groups are Weyl groups associated to $n$-extended affine root systems introduced by K. Saito [1]. We calculate the growth series of the n-extended affine Weyl group of type $A_1$ with a generator system of an $n$-toroidal sense.

The best constant of Sobolev inequality in an $n$ dimensional Euclidean space is found by means of the theory of reproducing kernel and Green function. The concrete form of the best constant is also found in the case of Sobolev space $W^2(\mathbf{R}^n)$ ($n=2,3$).

Characterization of conformally flat bundle metric on $S^1$-principal bundle is studied. It is shown further that there are infinitely many compact conformally flat $S^1$-principal bundles which are new examples, besides the Hopf fibration.