We show that for any natural number $n$ and any knot $K$, there are infinitely many unknotting number one knots, all of whose Vassiliev invariants of order less than or equal to $n$ coincide with those of $K$.

Let $\varSigma$ be an open Riemann surface and Hol($\varSigma$) be the Lie algebra of holomorphic vector fields on $\varSigma$. We fix a projective structure (i.e. a local $\text{SL}_2(\mathbf{C})$-structure) on $\varSigma$. We calculate the first group of cohomology of Hol($\varSigma$) with coefficients in the space of linear holomorphic operators acting on tensor densities, vanishing on the Lie algebra $\text{sl}_2(\mathbf{C})$. The result is independent on the choice of the projective structure. We give explicit formulas of 1-cocycles generating this cohomology group.

To solve non-symmetric linear equations, we have proposed a generalized SOR method, named the improved SOR method with orderings, and for an $n\times n$ tridiagonal matrix, we have given $n$ selections of the multiple relaxation parameters which satisfy $\rho(\mathcal{L}_{\varPhi})=0$ and correspond to the reciprocal numbers of the pivots of Gaussian elimination, where $\mathcal{L}_{\varPhi}$ is the $n\times n$ iterative matrix of this method. In this paper, using the ``essential dimensions-reductions for error vectors'', we investigate the numbers of all conditions for the multiple relaxation parameters which satisfy $\rho(\mathcal{L}_{\varPhi})=0$. As a result, adding to $n$ known selections of the multiple relaxation parameters, we find another type of selections of the multiple relaxation parameters and we conclude that such numbers of conditions are totally $2^{n-1}$ cases for an $n\times n$ tridiagonal matrix. Examples of such selections of multiple relaxation parameters are also contained. For an $n\times n$ Hessenberg matrix, we also obtain the similar results.

Let $(M,g)$ be a Riemannian manifold of constant sectional curvature $\kappa$ and $(TM, \tilde{g})$ be the tangent bundle of $M$ equipped with the Cheeger-Gromoll metric induced by $g$. We give necessary and sufficient conditions for $TM$ having positive scalar curvature. This gives counterexamples to a stated theorem of Sekizawa.

We will give an explicit description of extensions of the group scheme (resp.\ the formal group scheme) which gives a deformation of the additive group scheme to the multiplicative group scheme (resp.\ the additive formal group scheme to the multiplicative formal group scheme) by the additive group scheme (resp.\ the additive formal group scheme) over an algebra for which all prime numbers except a given prime $p$ is invertible.

Suppose that $p_{\theta}$ is a probability density of sample $X$, $T$ is a mapping, $\textit{g}\kern0.5pt_{\theta}(t)$ is an induced probability density by $T$ and $k_{\theta}(x)$ is a conditional density given $T=t$. Then, the following results are proved under some conditions. (a) $L^2$-differentiability of the family $(\sqrt{p_{\theta}})$ is equivalent to that of $(\sqrt{\textit{g}\kern0.5pt_{\theta}})$ and $(\sqrt{k_{\theta}})$. (b) Regularity of the family $(p_{\theta})$ is equivalent to that of $(\textit{g}\kern0.5pt_{\theta})$ and $(k_{\theta})$.

We shall give a simple proof of the weak type $L^1$ inequality for the $K$-bi-invariant Hardy-Littlewood maximal functions on non-compact real rank one semisimple Lie groups. For higher rank groups we do under an assumption which holds for the most parts. And on $SU(n,n+k)$ we introduce a maximal operator defined by the characteristic function supported on a cube, and show that the operator also satisfies the weak $L^1$ property.

We find conditions under which a Ricci positive metric can be deformed in a tubular neighbourhood of some submanifold to agree with another given Ricci positive metric. We insist that this final metric has everywhere positive Ricci curvature.

Let $p$ be an odd prime and $r$ a divisor of $p-1$. We present a characterization of metacyclic extensions of degree $pr$ containing a given cyclic extension of degree $r$ over a field of characteristic other than $p$. Furthermore, we give a method of constructing polynomials with Galois groups which are Frobenius groups of degree $p$.