We consider asymptotic stability of a small solitary wave to supercritical $1$-dimensional nonlinear Schrödinger equations \[iu_t+u_{xx}=Vu \pm |u|^{p-1} u \quad \text{for} (x, t) \in \mathbb{R} \times \mathbb{R},\] in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai \cite{18} in the $3$-dimensional case using the endpoint Strichartz estimate. To prove asymptotic stability of solitary waves, we need to show that a dispersive part $v (t, x)$ of a solution belongs to $L^2_t (0, \infty ; X)$ for some space $X$. In the $1$-dimensional case, this property does not follow from the Strichartz estimate alone. In this paper, we prove that a local smoothing estimate of Kato type holds globally in time and combine the estimate with the Strichartz estimate to show $\|(1+x^2)^{-3/4} v \|_{L^{\infty}_x L^2_t} < \infty$, which implies the asymptotic stability of a solitary wave.

Arithmetic of K3 surfaces defined over finite fields is investigated.In particular, we show that any K3 surface $X$ of finite height over a finite field $k$ of characteristic $p \geq 5$ has a quasi-canonical lifting $Z$ to characteristic $0$, and that for any such $Z$, the endormorphism algebra of the transcendental cycles $V(Z)$, as a Hodge module, is a CM field over $\mathbb{Q}$.The Tate conjecture for the product of certain two K3 surfaces is also proved. We illustrate by examples how to determine explicitly the formal Brauer group associated to a K3 surface over $k$. Examples discussed here are all of hypergeometric type.

We show that if a lineary compact $R$-module $M$ and local homology modules $H^I_i(M)$ satisfy the finite condition for co-associated primes for all $i<s$, then the set $\textrm{Coass}_R (H^I_s(M))$ is finite.

This work is a continuation of the papers [BZ1] and [BZ2]. Here we prove some estimates for the sum of codimensions of singularities of affine planar rational curves.

We investigate relative cohomology functors on subcategories of abelian categories via Auslander-Buchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms and introduce a notion of perfection for this context.Our main theorem is a balance result for relative cohomology that simultaneously recovers theorems of Holm and the current authors as special cases.

We consider a magnetic Laplacian $-\Delta_A=(id+A)^{\star} (id+A)$ on the Poincaré upper-half plane $mathbb{H}$ when the magnetic field $dA$ is infinite at the infinity such that $-\Delta_A$ has pure discret spectrum. We give the asymptotic behavior of the counting function of the eigenvalues.

In the stable homotopy group $\pi_{p^nq+(p+1)q-1}(V(1))$ of the Smith-Toda spectrum $V(1)$, the author constructed an essential element $\varpi_n$ for $n\geq 3$ at the prime greater than three. Let $\beta_s^{\ast}\in[V(1), S]_{spq+(s-1)q-2}$ denote the dual of the generator $\beta_{s}^{\prime\prime} \in \pi_{s(p+1)q}(V(1))$, which defines the $\beta$-element $\beta_s$. In this paper, the author shows that the composite $\alpha_1 \beta_1 \xi_s \in \pi_{p^nq+(s+1)pq+sq-6}(S)$ for $1<s<p-2$ is non-trivial, where $\xi_s=\beta_{s-1}^{\ast} \varpi_n \in \pi_{p^nq+spq+(s-1)q-3}(S)$ and $q=2(p-1)$. As a corollary, $\xi_s$, $\alpha_1 \xi_s$ and $\beta_1 \xi_s$ are also non-trivial for $1<s<p-2$.

In this note we show that, for a map of (weak or ordinary) $k$-fold homotopy coalgebras and under certain dimension and connectivity restrictions, the homotopy coalgebrastructure is inherited by the homotopy cofibre of the given map.

We estimate the variation of the Faltings-Moriwaki modular height under isogenies of elliptic curves. In particular, we reprove the Faltings finiteness theorem for elliptic curves over fields of finite type over $\mathbb{Q}$. Our approach is \lq\lqeffective\rq\rq in the sence that we generalize Raynaud's upper bound to higher dimensional cases.

We consider the unconditional uniqueness for the Cauchy problem of the derivative nonlinear Schrödinger equation. The proof is based on the Gauge transformation and the Fourier restriction method.