This paper is concerned with mod $p$ Morita-Mumford classes $e_{n}^{(p)} \in H^{2n}(\Gamma_{g}, \mathbb{F}_{p})$ of the mapping class group $\Gamma_{g}$ of a closed oriented surface of genus $g \geq 2$, especially triviality and nontriviality of them. It is proved that $e_{n}^{(p)}$ is nilpotent if $n \equiv -1 \pmod{p-1}$, while the stable mod $p$ Morita-Mumford class $e_{n}^{(p)} \in H^{2n}(\Gamma_{\bullet}, \mathbb{F}_{p})$ is proved to be nontrivial and not nilpotent if $n \not\equiv -1 \pmod{p-1}$. With these results in mind, we conjecture that $e_{n}^{(p)}$ vanishes whenever $n \equiv -1 \pmod{p-1}$, and obtain a few pieces of supporting evidence.

We establish certain deLeeuw type theorems for Littlewood-Paley functions. By these theorems, we know that the boundedness of a Littlewood-Paley function on $\mathbb{R}^n$ is equivalent to the boundedness of its corresponding Littlewood-Paley function on the torus $\mathbb{T}^n$.

Let $X$ be a smooth rational surface. In this paper, we prove the rationality of the moduli space ${M}_{X, L}(2; c_{1}, c_{2})$ of rank two $L$-stable vector bundles $E$ on $X$ with det $(E) = c_{1} \in $ Pic$(X)$ and $c_{2}(E) = c_{2} \gg 0$.

We give a discussion on the properties of $\Delta_a(x)$ ($-1<a<0$), which is a generalization of the error term $\Delta(x)$ in the Dirichlet divisor problem. In particular, we study its oscillatory nature and investigate the gaps between its sign-changes for $-1/2\le a<0$.

Let $\beta>0$ and $W_\beta (x) = \exp (-|x|^\beta )$, $x \in \mathbb{R}$. For $c>0$, define $W_{\beta, cn} (x) = W_\beta (x)$ if $|x| \leq c^{1/\beta}\,a_{2n}$ and $W_{\beta, cn} (x) = 0$ if $|x| > c^{1/\beta}\,a_{2n}$, where $a_{2n}$ denotes Mhaskar-Rahmanov-Saff number for $W_\beta$. Let $\gamma_n (W_{\beta, cn})$ be the leading coefficient of the $n$th orthonormal polynomial corresponding to $W_{\beta, cn}$ and write $\alpha_n(W_{\beta, cn}) = \gamma_{n-1}(W_{\beta, cn})/\gamma_{n}(W_{\beta, cn})$. It is shown that if $c>1$ and $\beta$ is a positive even integer then $\alpha_n(W_{\beta, cn})/n^{1/\beta}$ has an asymptotic expansion. Also when $0<c<1$, asymptotic expansions of recursion coefficients of the truncated Hermite weights are given.

We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.

We introduce the notion of abelian system on a finite group $G$, as a particular case of the recently defined notion of kernel system (see this Journal, September 2001). Using a famous result of Suzuki on CN-groups, we determine all finite groups with abelian systems. Except for some degenerate cases, they turn out to be special linear group of rank $2$ over fields of characteristic $2$ or Suzuki groups. Our ideas were heavily influenced by [1] and [8].

Let $M$ be a manifold and let ${\cal L}$ be a sufficiently smooth second order elliptic operator in $M$ such that $(M, {\cal L})$ is a transient pair. It is first shown that if ${\cal L}$ is symmetric with respect to some density in $M$, there exists a positive ${\cal L}$-harmonic function in $M$ which dominates ${\cal L}$-Green's function at infinity. Other classes of elliptic operators are investigated and examples are constructed showing that this property may fail if the symmetry assumption is removed. Another part of the paper deals with the existence of critical points for certain ${\cal L}$-harmonic functions with periodicity properties. A class of small perturbations of second order elliptic operators is also described.

Let $K$ be a finite extension of $\mathbf{Q}_{p}^{\mathit{ur}}$ (= the maximal unramified extension of $\mathbf{Q}_{p}$) of degree $e_{K}$, $\mathcal{O}$ its integer ring, $p$ a rational prime and $r$ a positive integer. If there exists a one parameter formal group defined over $\mathcal{O}$ whose reduction is of height $2$ with a cyclic subgroup $V$ of order $p^{r}$ defined over $\mathcal{O}$, then $e_{K} \geq 2p^{l}$ (resp.~$p^{l}+p^{l-1}$) if $r = 2l+1$ (resp.~$r = 2l$).

We apply this result to a criterion for non-existence of $\mathbf{Q}$-rational point of $X_{0}^{+}(p^{r})$. (This criterion is Momose's theorem in [14] except for the cases $p = 5$ and $p = 13$, but our new proof does not require defining equations of modular curves except for the case $p = 2$.)

Let $SU(r,1)$ be the moduli space of stable vector bundles, on a smooth curve $C$ of genus $g \geq 2$, with rank $r \geq 3$ and determinant $O_C(p)$, $p \in C$; let ${\cal L}$ be the generalized theta divisor on $SU(r,1)$. In this paper we prove that the map ${\phi}_{\cal L}$, defined by ${\cal L}$, is a morphism and has degree $1$.