We consider the analytic smoothing effect for the KdV equation. That is to say, if the initial data given at $t = 0$ decays very rapidly, the solution to the Cauchy problem becomes analytic with respect to the space variable for $t > 0$. In this paper we show this effect by using the inverse scattering method which transforms the KdV equation to a linear dispersive equation whose analytic smoothing effect is shown through the properties of the Airy function.

For an odd prime $p$, we show that the $p$-primary homotopy exponent of Harper’s rank 2 finite mod-$p$ $H$-space $K_{p}$ is $p^{p^{2}+p}$. We then use this to show that the 3-primary homotopy exponent of each of the exceptional Lie groups $F_{4}$ and $E_{6}$ is $3^{12}$.

Let $E_{6}$ be the compact 1-connected exceptional Lie group of rank 6. In [9] we determined the Hopf algebra structure of $H_{*}(\Omega E_{6}; \mathbb{Z})$ by the generating variety approach of R. Bott [1]. In this case, as a generating variety we can take $EIII$, the irreducible Hermitian symmetric space of exceptional type. Then as Bott pointed out in [1], §6, we can determine the action of the mod $p$ Steenrod algebra $\mathcal{A}_{p}$ on $H^{*}(\Omega E_{6}; \mathbb{Z}_{p})$ from that on $H^{*}(EIII; \mathbb{Z}_{p})$ for all primes $p$.

In this paper, for ease of algebraic description, we compute the action of $\mathcal{A}_{p_{*}}$, the dual of $\mathcal{A}_{p}$, on $H^{*}(\Omega E_{6}; \mathbb{Z}_{p})$ for $p = 2, 3$ (For larger primes the description is easy). In the course of computation we also determine the action of $\mathcal{A}_{3}$ on $H^{*}(E_{6}/T; \mathbb{Z}_{3})$, where T is a maximal torus of $E_{6}$.

The paper is constructed as follows: In Section 2 we recall some results concerning the cohomology of some homogeneous spaces of $E_{6}$. In Section 3 by considering the action of the Weyl group on $E_{6}/T$, we determine the cohomology operations in $EIII$. Using the results obtained, in Section 4 we shall determine the cohomology operations in $\Omega E_{6}$.

Throughout this paper $\sigma _{i}(x_{1},\ldots , x_{n})$ denotes the $i$-th elementary symmetric function in the variables $x_{1},\ldots , x_{n}$.

We study Calabi-Yau 3-folds with infinitely many divisorial contractions. We also suggest a method to describe Calabi-Yau 3-folds with the infinite automorphism group.

In this paper, we introduce the powerful, new method of Wiles into establishing that a Calabi-Yau threefold defined over the field $\mathbb{Q}$ of rational numbers is modular, answering a question of Saito & Yui [SY].

We study the Euler equations of compressible isentropic gas dynamics with spherical symmetry. Due to the presence of the singularity at the origin, little is known in the case including the origin. In this paper, we prove the existence of local solutions for the case including the origin. A modified Godunov scheme is introduced to construct approximate solutions and obtain $L^{\infty}$ estimates. The convergence and consistency of the approximate solutions are proved with the aid of a compensated compactness framework.

This paper investigates the global behavior of subharmonic functions on a complete noncompact simply-connected Riemannian manifold. The authors obtain some Liouville-type theorems, a comparison theorem for the strong parabolicity of a manifold and their applications to geometry.

The image of the cycle map $CH^{*}(BF_{4})_{(3)}\to H^{*}(BF_{4})_{(3)}$ is studied by using $BP^{*}$-theory and the motivic cohomology.

We study the Euler equations of compressible isentropic gas dynamics with damping and spherical symmetry. For spherically symmetric flow, the global existence of the weak entropy $L^{\infty}$ solutions with damping isn’t still obtained. In this paper, we prove the existence of global solutions with small $L^{\infty}$ data. We construct the approximate solutions by using modified Godunov scheme. A $L^{\infty}$ bound for the approximate solutions can be obtained with the aid of the presence of the damping term.

We consider the symmetric pair $(G,K) = (U(p, q), U(p) \times U(q))$. For this pair, we classify spherical nilpotent $K_{\mathbb{C}}$-orbits which are theta lift in the stable range. For the pair $(G,K) = (U(p, p), U(p) \times U(p))$ where $p = q$, we prove that a spherical nilpotent $K_{\mathbb{C}}$-orbit must be a theta lift. As a consequence, we get a complete classification of the spherical nilpotent $K_{\mathbb{C}}$-orbits for the symmetric pair $(U(p, p), U(p) \times U(p))$.