Operator Product Expansions give algebraic structures on subspaces of quasi-primary vectors in superconformal algebras. The structures characterize the structures of superconformal algebras if they meet a criteria, while in some cases the spaces of quasi-primary vectors are finite dimensional. As an application the complete list of simple physical conformal superalgebras is given by classifying the corresponding algebraic structures on finite dimensional vector spaces. The list contains a one-parameter family of superconformal algebras with $4$ supercharges that is simple for general values.

We pose a variational problem for surfaces whose solutions are a geometric model for thin films with gravity which is partially supported by a given contour. The energy functional contains surface tension, a gravitational energy and a wetting energy, and the Euler-Lagrange equation can be expressed in terms of the mean curvature of the surface, the curvatures of the free boundary and a few other geometric quantities. Especially, we study in detail a simple case where the solutions are vertical planar surfaces bounded by two vertical lines. We determine the stability or instability of each solution.

The cohomology $H^{*}(\mathrm{\Gamma },E)$ of an arithmetic subgroup $\mathrm{\Gamma }$ of a connected reductive algebraic group $G$ defined over $\mathbf{Q}$ can be interpreted in terms of the automorphic spectrum of $\mathrm{\Gamma }$. In this frame there is a sum decomposition of the cohomology into the cuspidal cohomology ( i.e., classes represented by cuspidal automorphic forms for $G$) and the so called Eisenstein cohomology. The present paper deals with the case of a quasi split form $G$ of $\mathbf{Q}$-rank two of a unitary group of degree four. We describe in detail the Eisenstein series which give rise to non-trivial cohomology classes and the cuspidal automorphic forms for the Levi components of parabolic $\mathbf{Q}$-subgroups to which these classes are attached. Mainly the generic case will be treated, i.e., we essentially suppose that the coefficient system $E$ is regular.

We give a method to determine Martin boundaries of product domains for elliptic equations in skew product form via Widder type uniqueness theorems for parabolic equations. It is shown that the fiber of the Martin boundary at infinity of the base space degenerates into one point if any nonnegative solution to the Dirichlet problem for a corresponding parabolic equation with zero initial and boundary data is identically zero. We apply it in a unified way to several concrete examples to explicitly determine Martin boundaries for them.

Let $L={\sum }_{i=1}^d{X}_i\left(z\right){\partial }_{z_i}$ be a holomorphic vector field degenerating at $z=0$ such that Jacobi matrix$\left(\right(\partial X_i/\partial z_j\left)\right(0\left)\right)$ has zero eigenvalues. Consider $Lu=F(z,u)$ and let $\tilde{u}\left(z\right)$ be a formal power series solution. We study the Borel summability of $\tilde{u}\left(z\right)$, which implies the existence of a genuine solution $u\left(z\right)$ such that $u\left(z\right)\sim \tilde{u}\left(z\right)$ as $z\to 0$ in some sectorial region. Further we treat singular equations appearing in finding normal forms of singular vector fields and study to simplify $L$ by transformations with Borel summable functions.

We study the existence problem of a $2n$ dimensional Poincaré complex whose homology is isomorphic to that of $n$ dimensional complex projective space when $n=4$.

In continuation of our papers [5], [6] we complete the classification of all one-parametric selfinjective algebras over algebraically closed fields which admit simply connected Galois coverings.

In a recent paper, Isao Kiuchi and Naoki Yanagisawa studied the even power moments of the error term in the approximate functional equation for $\zeta \left(s\right)$. They got a mean value formula with an error term $O\left(T^{1/2-k\sigma }\right)$, and then they conjecture that this term could be replaced by $E_{k,\sigma }{T}^{1/2-k\sigma }(1+o(1\left)\right)$ with constant $E_{k,\sigma }$ depending on $k$ and $\sigma$. In this paper, we disprove this conjecture by showing that the error term should be $f\left(T\right)T^{1/2-k\sigma }+o\left(T^{1/2-k\sigma }\right)$ with $f\left(T\right)$ oscillating.

For a metric space $X=(X,d)$,let ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$ be the space of all nonempty closed sets in $X$ with the topology induced by the Hausdorff extended metric:$$d_H(A,B)=\max \left\{\underset{x\in B}{\sup }d(x,A),\underset{x\in A}{\sup }d(x,B)\right\}\in [0,\mathrm{\infty }].$$ On each component of ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$, $d_H$ is a metric (i.e., ). In this paper, we give a condition on $X$ such that each component of ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$ is a uniform AR (in the sense of E. Michael). For a totally bounded metric space $X$, in order that ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$ is a uniform ANR,a necessary and sufficient condition is also given. Moreover, we discuss the subspace ${\mathrm{D}\mathrm{i}\mathrm{s}}_H\left(X\right)$ of ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$ consisting of all discrete sets in $X$ and give a condition on $X$ such that each component of ${\mathrm{D}\mathrm{i}\mathrm{s}}_H\left(X\right)$ is a uniform AR and ${\mathrm{D}\mathrm{i}\mathrm{s}}_H\left(X\right)$ is homotopy dense in ${\mathrm{C}\mathrm{l}\mathrm{d}}_H\left(X\right)$.

We introduce a graphical method, called the chart description, to describe the monodromy representation of a genus one Lefschetz fibration. Using this method, we give a new and purely combinatorial proof of the classification theorem of genus one Lefschetz fibrations.

Consider a class of uniformly elliptic diffusion processes $\{X_t{\}}_{t\ge 0}$ on Euclidean spaces ${\mathbf{R}}^d$. We give an estimate of $E^{P_x}\left[\exp \left(T\mathrm{\Phi }\right(1/T{\int }_0^T{\delta }_{X_t}dt\left)\right)|X_T=y\right]$ as $T\to \mathrm{\infty }$ up to the order $1+o\left(1\right)$, where ${\delta }_{\cdot }$ means the delta measure, and $\mathrm{\Phi }$ is a function on the set of measures on ${\mathbf{R}}^d$. This is a generalization of the works by Bolthausen-Deuschel-Tamura [3] and Kusuoka-Liang [10], which studied the same problems for processes on compact state spaces.

We discuss the homogenization of stochastic partial differential equations whose coefficients are rapidly oscillating and are perturbed by a diffusion process. Such class of equations appear in nonlinear filtering problems with feedback. We specify the constant coefficients of the limit equation. The constants are essentially different from the case where the coefficients do not contain perturbed factors.

Let $T=U\left|T\right|$ be a bounded linear operator with the associated polar decomposition on a separable infinite dimensional Hilbert space. For , let $T_t=\left|T{|}^{t}U\right|T{|}^{1-t}$ and $g_T$ and $g_{T_t}$ be the principal functions of $T$ and $T_t$, respectively. We show that, if $T$ is an invertible semi-hyponormal operator with trace class commutator $\left[\right|T|,U]$, then $g_T=g_{T_t}$ almost everywhere on $\mathbf{C}$. As a biproduct we reprove Berger's theorem and index properties of invertible $p$-hyponormal operators.