We first interpret circles in Riemannian Symmetric space by Lie algebro-theoretic formalism. In particular, it is a solution of the system of ordinary differential equation of first order. We divide circles into 3-types. We investigate closedness and simpleness for such circles in compact Hermitian symmetric spaces. Consequently, we find many open holomorphic circles and non-simple circles. Note that there exist no non-simple circles and no open holomorphic circles in compact Riemannian symmetric space of rank one.

Let $R$ be a ring, and let $\tau$ be a torsion theory for R-mod. We give a necessary condition for every R-module to have a $\tau-$torsionfree cover; this necessary condition is close to the known sufficient condition. Then we present a method for computing, $\tau-$torsionfree covers of modules that can be embedded in $Q_{\tau}$-modules where $Q_{\tau}$ is the quotient ring for $\tau$.

Let $X=(X, d)$ be a metric space. By USCC(X, I), we denote the space of upper semi-continuous multi-valued functions $\varphi:X\rightarrow I=[0,1]$ such that each $\varphi(x)$ is a closed interval. Each $\varphi\in USCC(X, I)$ can be identified with its graph, which is a closed subset of $X\times I$. The space USCC(X, I) admits the Hausdorff metric induced by the product metric on $X\times I$. In this paper, by proving the converse of Fedorchuk’s result, we show that USCC(X, I) is homeomorphic to the Hilbert cube $Q=[-1,1]^{\omega}$ if and only if $X$ is infinite, locally connected and compact. In case $X$ is a dense subset of a locally connected metric space $Y$ such that $Y\backslash X$ is locally nonseparating in $Y$, USCC $(X, I)$ can be regarded as a subspace of USCC $(Y, I)$. It is also proved that the pair (USCC $(Y, I)$ , USCC $(X, I))$ is homeomorphic to $(Q, s)$ if and only if $X\neq Y,$ $X$ is $G_{\delta}$ in $Y$, and $Y$ is compact, where $s=(-1,1)^{\omega}\subset Q$.

We study the Cauchy problem for $u_{tt}-\Delta u+V(x)|u|^{p-1}u=0$ with $x\in R^{n}$. The function $V(x)$ is positive and regular. The exponent $p$ is subcritical or critical. By the aid of Shatah-Struwe technique (cf.[7]), we prove the existence of the global classical solution with suitable hypotheses on $V(x):V(x)>0,3\leq n\leq 7$ or $V(x)=|x|^{2}$, $n=3$. To approach this second case we cannot follow directly the argument used in [7]: we need and prove weighted nonlinear estimates in Besov spaces.

In this note, we shall aim, using explicit theta constants,to describe the Chow-form of an elliptic normal curve of degree 4 embedded in $P^{3}$ by using theta functions, at any point of the upper half plane. In this case, the embedded elliptic curve is a complete intersection which is defined by two quadratic forms. In order to calculate its Chow-form, we use the elimination theory. Our main result is Th. 1' (§2). In our case, Chow-forms are divisors of the Grassmann variety of lines in $P^{3}$. From the theorem, we see that the Chow-forms of elliptic normal curves of degree 4 lie on a linear subspace $M$ (cf. 2.3) of dimension 23. §2 is the main part of this note.

In §3, using the theory of elliptic modular forms of level 4, we consider the geometric meaning of Th. 1'. Then our theorem shows that the Chow point is given by modular forms of weight 2 (of weight 1 in the traditional sense) and of level 4. The compactification of the moduli space of level 4 is isomorphic to a plane conic. Thus, we see that the Chow points of the projective elliptic normal curves of degree 4 determined by the points of the upper half plane form a rational curve of degree 4 in $M$ which is essentially the image of the 2-uple embedding of the above plane conic (Th. 2, §3. cf. Comments after the proof of Th. 2). Furthermore, we see that the correspondence which assigns the Chow point to a point of the upper half plane can be extended to the inequivalent cusps of the principal congruence subgroup of level 4 (Cor. 1, §3). Finally, we give a remark about the structure of the corresponding component of the Chow variety parametrizing l-cycles of degree 4 in $P^{3}$ (Cor. 2, §3).

We consider the scalar curvature problem with nonzero Dirichlet boundary data and with prescribed scalar function changing sign as a nonlinear eigenvalue problem. We use the super-sub solution method and our early result [MW] to obtain a solution of the problem and give a lower bound on the eigenvalue.

We study continua on which each nonconstant open mapping is light. W. Makuchowski asked if this property is preserved under atomic mappings. It is known that this is true under an additional assumption of arcwise connectedness of the domain continuum. We show that in general this is not true.

The purpose of this paper is to calculate the $b-$function of the regular simple prehomogeneous vector space $(GL(1)^{2}\times SL(7)$, $\Lambda_{3}+\Lambda_{1}^{*})$ by the aid of microlocal method using the holonomy diagram of relative invariants.