Laws of the occupation times on a half line are studied for one-dimensional diffusion processes. The asymptotic behavior of the distribution function is determined in terms of the speed measure.

Let $F_{r,p}=V_{r,p}(L^p(X,m))$ be the abstract space of Bessel potentials and $\mu$ a positive smooth Radon measure on $X$. For $2\leq p\leq q < \infty$, we give necessary and sufficient criteria for the boundedness of $V_{r,p}$ from $L^p(X,m)$ into $L^p(X,\mu)$, provided $F_{r,p}$ is contractive. Among others, we shall prove that the boundedness is equivalent to a capacitary type inequality. Further we give necessary and sufficient conditions for $F_{r,p}$ to be compactly embedded in $L^q(\mu)$. Our method relies essentially on establishing a \textit{capacitary strong type inequality}.

We will give a complete characterization of all regular Dirichlet subspaces of $H^1(I)$ for a finite open interval $I$ by a certain family of scale functions. Each associated diffusion will be constructed from a reflecting Brownian motion on a closed interval by a time change and a transformation of the state space.

Anti-Wick operators are a class of pseudodifferential-type operators that are also known under the names of Time-frequency localization operators or Gabor-Toeplitz operators or wave packets. We consider a general version of Wick operators, depending on different window functions, and we investigate the related calculus, obtaining a composition formula that extends the known results [1, 3, 5, 15, 16, 18].

Consider a nonlinear system of two Klein-Gordon equations with masses $m$ and $\mu$. We construct a solution whose amplitude is modulated by the nonlinear interaction when $\mu = m$ or $\mu = 3m$, whereas, when $\mu \ne m$ and $\mu \ne 3m$, the influence of the nonlinearity is negligible and the solution behaves like a free solution as $t \to \infty$.

Let $u$ be a biharmonic Green potential on the unit ball $\mathbf{B}$ of $\mathbf{R}^{n}$. We show that \begin{equation*} \lim_{r\to 1}(1-r)^{n-2-(n-1)/p}\mathcal{M}_p(u,r)=0 \end{equation*} for $p$ such that $1\le p<(n-1)/(n-4)$ in case $n\ge 5$ and $1\le p<\infty$ in case $n\le 4$. Further, if $n\ge 5$ and $(n-1)/(n-4)\le p<(n-1)/(n-5)$, then it is shown that \begin{equation*} \liminf_{r\to 1}(1-r)^{n-2-(n-1)/p}\mathcal{M}_p(u,r)=0. \end{equation*} Finally we show that these limits characterize biharmonic Green potentials among super-biharmonic functions on $\mathbf{B}$.

The Exponential attractor, one of notions of limit set in infinite-dimensional dynamical systems, is known to have strong robustness and is known to be constructed under a simple compact smoothing condition. In this paper, we study a dynamical system determined from the Cauchy problem for a quasilinear abstract parabolic evolution equation. We give a general strategy for constructing the exponential attractor and apply the abstract result to a chemotaxis-growth system in non smooth domain.

The $\alpha$-parabolic Bergman space $\bm{b}^p_\alpha$ is the set of all $p$-th integrable solutions $u$ of the equation $(\partial/\partial t + (-\Delta)^{\alpha})u = 0$ on the upper half space, where $0 < \alpha \leq 1$ and $1 \leq p \leq \infty$. The Huygens property for the above $u$ will be obtained. After verifying that the space $\bm{b}^p_\alpha$ forms a Banach space, we discuss the fundamental properties. For example, as for the duality, $(\bm{b}^p_\alpha)^* \cong \bm{b}^q_\alpha$ for $p > 1$ and $(\bm{b}^1_\alpha)^* \cong \mathcal{B}_\alpha/ \mathbf{R}$ are shown, where $q$ is the exponent conjugate to $p$ and $\mathcal{B}_\alpha$ is the $\alpha$-parabolic Bloch space.

The object of this paper is to find the number of hyperelliptic curves in the bicanonical pencil of a Godeaux surface whose torsion group is $\mathbb{Z}_3$, or $\mathbb{Z}_4$, or $\mathbb{Z}_5$.

By using the correspondence between locally finitely presented additive categories and rings with enough idempotents, we study several properties of such rings in terms of the associated categories, and conversely. In particular, it is shown that a ring $R$ (with enough idempotents) is right perfect and the categories of finitely presented right and left $R$-modules are dual to each other if and only if the categories of projective and of injective right $R$-modules are equivalent.

The main purpose of this paper is to study the fourth power mean of the general Gauss sums, and give two exact calculating formulae.

We show that the two cuspidal unipotent characters of a finite Chevalley group $E_7(q)$ have Schur index $2$, provided that $q$ is an even power of a (sufficiently large) prime number $p$ such that $p\equiv 1 \bmod 4$. The proof uses a refinement of Kawanaka's generalized Gelfand--Graev representations and some explicit computations with the \textit{CHEVIE} computer algebra system.

Let $R_n$ be the coinvariant algebra of the symmetric group $S_n$. The algebra has a natural gradation. For a fixed $l$ ($1\leq l \leq n$), let $R_n(k;l)$ ($0\leq k\leq l-1$) be the direct sum of all the homogeneous components of $R_n$ whose degrees are congruent to $k$ modulo $l$. In this article, we will show that for each $l$ there exists a subgroup $H_{l}$ of $S_n$ and a representation $\Psi(k;l)$ of $H_{l}$ such that each $R_n(k;l)$ is induced by $\Psi(k;l)$.

We prove a divisibility theorem for elements in the cut-and-paste group, or the $SK$-group of $G$-manifolds, $G$ a finite abelian group of odd order. As an application we obtain necessary and sufficient conditions for that a closed $G$-manifold is equivariantly cobordant to the total space of $G$-fibration over the circle.

Small Cover is an $n$-dimensional manifold endowed with a $\mathbf{Z}_2^n$ action whose orbit space is a simple convex polytope $P$. It is known that a small cover over $P$ is characterized by a coloring of $P$ which satisfies a certain condition. In this paper we shall investigate the topology of small covers by the coloring theory in combinatorics. We shall first give an orientability condition for a small cover. In case $n=3$, an orientable small cover corresponds to a four colored polytope. The four color theorem implies the existence of orientable small cover over every simple convex $3$-polytope. Moreover we shall show the existence of non-orientable small cover over every simple convex $3$-polytope, except the $3$-simplex.

We present a class of knots associated with labelled generic immersions of intervals into the plane and compute their Gordian numbers and 4-dimensional invariants. At least 10\,\% of the knots in Rolfsen's table belong to this class of knots. We call them track knots. They are contained in the class of quasipositive knots. In this connection, we classify quasipositive knots and strongly quasipositive knots up to ten crossings.