We prove that semi-Riemannian manifolds satisfying some curvature condition of pseudosymmetry type are semisymmetric. We give also some inverse theorems.

The analytic representation of the generalized tempered distributions of $e^{M(kx)}$-growth with restricted order, $\mathscr{K}_{M}^{r'}(R)$, is given in terms of series of analytic wavelets. These series converge uniformly on compact subsets of the upper and lower half planes.

This paper gives a new treatment of the Clifford algebras. We represent the pinor and spinor spaces as subspaces of the Clifford algebras and use these pinors to construct isomorphisms between the Clifford algebras and the matrix algebras. In doing these we develop some spinor calculus.

In this paper we use Clifford algebra and spinor calculus to study the calibrations on Riemannian manifolds and the Grassmann manifolds. Show that for every Grassmannian, there is a map $\pi: G(k, \bm{R}^{m}) \to M$ such that every $\xi \in M$ is a calibration on $\bm{R}^{m}$ and $\pi^{-1}(\xi)$ is the contact set of $\xi$. In low dimensional cases, the calibration sets $M$ are manifolds or manifolds with singularities. We also use Clifford algebra to study the isotropy groups of calibrations.

We study some links between autoparallel distributions and the factorization of a riemannian manifold. Finally, we prove a splitting theorem for Lie groups with biinvariant metrics.

We will show that every $3$ dimensional cutting sequence is a three-distance sequence, and there are uncountable many periodic or aperiodic three-distance sequences (with $3$-symbols) which are not $3$ dimensional cutting sequences.

For an infinite set $\Gamma$, let $\ell_{2}^{f}(\Gamma)$ be the linear span of the canonical orthonormal basis of the Hilbert space $\ell_{2}(\Gamma)$, that is, \[ \ell_{2}^{f}(\Gamma) = \{ x\in\ell_{2}(\Gamma) \mid x(\gamma) = 0 \text{ except for finitely many } \gamma\in\Gamma \}.\] We denote $\ell_{2}^{f}=\ell_{2}^{f}(N)$. Let $Q=[-1,1]^{\omega}$ be the Hilbert cube. In this paper, we give characterizations of manifold modeled on the following spaces: $\ell_{2}(\Gamma) \times \ell_{2}^{f}$, $\ell_2^f(\Gamma)$ and $\ell_{2}^{f}(\Gamma)\times Q$, where $\ell_{2}(\Gamma)\times \ell_{2}$ and $\ell_{2}(\Gamma)\times Q$ are homeomorphic to $\ell_{2}(\Gamma)$. Our results are obtained by suitable alteration and modification of the separable case due to Bestvina and Mogilski.

The hypergeometric series reduce to many elementary functions. Many of them are known to have the continued fraction expansions. Few of them for certain values, however, are known to have the simple continued fraction expansions which keep regularities. In this paper we show more simple continued fraction expansions holding regularities.

We consider the non-characteristic Cauchy problem for the degenerate nonlinear parabolic equation $|u|^{\alpha}u_{t}=\triangle u$, where we assume that $1/2 \lt \alpha \lt 1$. This equation is based on the fast diffusion model. And we prove the unique continuation property for the above problem.

We prove some $C^{\infty}$ and Gevrey well-posedness results for hyperbolic equations with singular coefficients.