We aim to generalize Eichler’s strong approximation theorem, which is known for a division algebra $D$ over a global field $K$, to the case that $K$ is a general PF field. First we show by an example that the generalized theorem is false for $SL_{1}(D)$. But if we replace $SL_{1}(D)$ by the commutator group $[D^{\times},D^{\times}]$, the generalizaion may be possible. Though its validity is not yet known, in this paper we decompose the generalized theorem into two parts, one of which can be formulated in a more general case that $K$ is the quotient field of a Dedekind domain. Further, we prove the equivalence of four approximaion properties (a)~(a'''), which was open in our previous paper.

Let $P_{n}^{d}(\mathbb{R})$ denote the space consisiting of all monic polynomials $f(z) \in \mathbb{R}[z]$ of degree $d$ which have no real roots of multplicity $\geq n$. In this paper we study the homotopy types of the spaces $P_{n}^{d}(\mathbb{R})$ for the case $n = 3$.

We consider the steady compressible Navier-Stokes equations in the isentropic regime in a bounded domain of $\mathbb{R}^{3}$. We show that the renormalized continuity equation holds even if the density is not square integrable. We use this result to prove existence of weak solutions under the sole hypothesis $\gamma > 3/2$ for the adiabatic constant.

In this paper, we study the existence and uniqueness of solutions to stochastic equations in infinite dimension with an integral-Lipschitz condition for the coefficients.

We construct a family of examples of non-zero relative homology classes on some infinite dimensional spaces of symplectic forms.