Malliavin calculus is developed in a uniform way for (possibly non separable) extensions of $L^p(W_{C_{\mathbb{F}}})$, where $W_{C_{\mathbb{F}}}$ is the Wiener measure on the space $C_{\mathbb{F}}$ of continuous functions from $[0,1]$ into any abstract Wiener Fréchet space $\mathbb{F}$ over a fixed separable Hilbert space $\mathbb{H}$. Since the continuous time line is available in $C_{\mathbb{F}}$ , we can prove the Clark- Ocone formula for these extensions, we study time-anticipating Girsanov transformations and prove that Skorohod integral processes for finite chaos levels have continuous modifications. We use a rich probability space with measure $\widehat{\Gamma }_{\mathbb{H}}$,which only depends on $\mathbb{H}$, such that for any $p\in [0,\infty [$, $ L^p\left( W_{C_{\mathbb{F}}}\right) $ can be canonically embedded into $L^p\left( \widehat{\Gamma }_{\mathbb{H}}\right) $ for any abstract Wiener Fréchet space $\mathbb{F}$ over $\mathbb{H}$.

A condition on a matroid of rank $n-2$ for the corresponding thin Schubert cell being nonempty is determined. A necessary and sufficient condition for $k$ and $n$ so that the closure of a thin Schubert cell in $G(k,n)$ is always a union of thin Schubert cells is given.

We compute weight functions (off-shell Bethe vectors) in any representation with a weight singular vector of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$ applying the method of projections of Drinfeld currents developed in {EKP}.

For each fixed $n \geq 2$ we show how the Nielsen-Thurston classification of mapping classes for a closed surface of genus $g \geq 2$ is determined by the sequence of quantum $SU(n)$-representations $(\rho_k)_{k \in {\mathbb{N}}}$. That this is the case is a consequence of the asymptotic faithfulness property proved in [A3]. We here provide explicit conditions on $(\rho_k (\phi))_{k\in {\mathbb{N}}}$, which determines the Nielsen-Thurston type of any mapping class $\phi$.

We consider semilinear heat equations on ${\mathbb{R}}^N$ and discuss the blow-up of solutions that occurs only at space infinity. We give sufficient conditions for such phenomena, and study the global profile of solutions at the blow-up time. Among other things, we establish a nearly optimal upper bound for the blow-up profile, which shows that the profile $u(x,T)$ cannot grow too fast as $|x|\to \infty$. We also prove that such blow-up is always complete.

In this paper, we consider the Cauchy problem for the damped nonlinear Schrödinger equation with superposed delta functions as initial data. The aim of this paper is to study the influence of nonlinearity and damping (absorption) on the solution. The case where the nonlinearity makes the solution unstable is of special interest, because the dissipative nature from the damping comes into play in this case.

In this paper, the existence and approximation of fixed points for two classes of systems of mixed monotone (downward and upward) multivalued operators are discussed. We present some new fixed point theorems of mixed monotone(downward and upward)operators which need not be continuous and compact. We also indicate the condition to ensure the uniqueness of the fixed point. At last we get some applications ofour theorems.

Let $M$ be a compact complex manifold and let $L \rightarrow M$ be a homorphic line bundle whose curvature form is everywhere of signature $(s_+, s_-)$. Under some conditions on the curvature form of $L$, it is show that $K \otimes L^{\otimes m}$ admits, for some $K$ and sufficiently large $m \in \mathbb{N}$, $C^{\infty}$ sections $t_0, \ldots, t_N$ such that the ratio $(t_0 : \cdots : t_N)$ embeds $M$ holomorphically in $s_+$ variables and antiholomorphically in $s_-$ variables. The result extends the Kodaira's embedding theorem as well as aresult of Matsushima for complex tori.

We give several estimates on \emph{volumes} and \emph{effective resistances} in a long-range percolation on a vertex set of a $d$-dimensional square lattice. When $d=1$, our results imply some kind of discontinuity in the long-range percolation model; more precisely, in the order of the effective resistance. Our another consequence is that, when $d \ge 2$ and $s \in (d, d+2)$, where $s$ is the parameter determining the magnitude of the range, the order of the effective resistance corresponds to the $\alpha$-stable process with $\alpha =s - d$.

We give an estimate of the \lq\lq macroscopic dimension\rq\rq of the $\ell^p$-ball with respect to the $\ell^q$-norm.

We study two objects concerning the Wiener sausage among Poissonian obstacles. The first is the asymptotics for the \emph{replica overlap}, which is the intersection of two independent Wiener sausages. We show that it is asymptotically equal to their union. This result confirms that the localizing effect of the media is so strong as to completely determine the motional range of particles. The second is an estimate on the \emph{covering time}. It is known that the Wiener sausage avoiding Poissonian obstacles up to time t is confined in some \lq clearing \rq ball near the origin and almost fills it. We prove here that the time needed to fill the confinement ball has the same order as its volume.