We consider the initial value problem for the reduced fifth order KdV type equation: $\partial_{t}u-\partial_{x}^{5}u-10\partial_{x}(u^{3})+5\partial_{x}(\partial_{x}u)^{2}=0$ which is obtained by removing the nonlinear term $10\partial_{x}(u\partial_{x}^{2} u)$ from the fifth order KdV equation. We show the existence of the local solution which is real analytic in both time and space variables, if the initial data $\phi\in H^{s}(\mathbf{R})$ $(s>1/8)$ satisfies the condition \begin{equation*} ∑_{k=0}^{∞}\frac{A_{0}^{k}}{k!}{\|}(x\partial_{x})^{k}φ{\|}_{H^{s}}<{∞}, \end{equation*} for some constant $A_{0}(0<A_{0}<1)$. Moreover, the smoothing effect for this equation is obtained. The proof of our main result is based on the argument used in [5].

We announce the construction of toroidal partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. They are moduli spaces of log mixed Hodge structures with polarized graded quotients. We include an application to the analyticity of zero loci of normal functions.

Consider the scheme $B_{2,L}^{k}$ of stable vector bundles of rank two and fixed determinant $L$ which have at least $k$ sections. Under suitable numerical conditions and for generic $L$, we show the existence of a component of the expected dimension of $B_{2,L}^{k}$.

Let $A$ be a finite-dimensional associative algebra and $\phi$ a symmetric linear function on $A$. In this note, we will show that the pseudotrace maps defined in [6] are obtained as special cases of well-known symmetric linear functions on the endomorphism rings of projective modules. As an application of our approach, we will give proofs of several propositions and theorems in [6] for an arbitrary finite-dimensional associative algebra.