In this paper we study the Kato' inequality on locally finite graph. We also
study the application of Kato inequality to Ginzburg-Landau equations on such
graphs. Interesting properties of Schrodinger equation and a Liouville type
theorem are also derived.

We introduce an extension of the P\'olya tree approach for constructing
distributions on the space of probability measures. By using optional stopping
and optional choice of splitting variables, the construction gives rise to
random measures that are absolutely continuous with piecewise smooth densities
on partitions that can adapt to fit the data. The resulting "optional P\'{o}lya
tree" distribution has large support in total variation topology and yields
posterior distributions that are also optional P\'{o}lya trees with computable
parameter values.

In this paper, we establish existence results for positive solutions to the
Lichnerowicz equation of the following type in closed manifolds -\Delta
u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where $p>1, q>0$, and $A(x)>0$,
$B(x)\geq0$ are given smooth functions. Our analysis is based on the global
existence of positive solutions to the following heat equation {ll} u_t-\Delta
u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M\times\mathbb{R}^{+}, u(x,0)=u_0,\quad
in\quad M with the positive smooth initial data $u_0$.

Suppose $X$ is a right process which is associated with a non-symmetric
Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on $L^{2}(E;m)$. For $u\in
D(\mathcal{E})$, we have Fukushima's decomposition:
$\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$. In this paper, we
investigate the strong continuity of the generalized Feynman-Kac semigroup
defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$. Let
$Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in
D(\mathcal{E})_{b}$. Denote by $J_1$ the dissymmetric part of the jumping
measure $J$ of $(\mathcal{E},D(\mathcal{E}))$.

In this paper, we extend the Reilly formula for drifting Laplacian operator
and apply it to study eigenvalue estimate for drifting Laplacian operators on
compact Riemannian manifolds boundary. Our results on eigenvalue estimates
extend previous results of Reilly and Choi and Wang.

In this paper, by the method of moving planes, we establish the monotonicity
and symmetry properties of convex solutions for Monge-Ampere systems on bounded
smooth planar domains.

In this paper, we settle the problem for time-dependent Hartree equation with
inhomogeneous boundary condition in a bounded Lipschitz domain in
$\mathbb{R}^{N}$. A global existence result is derived.

We prove that for a solution $(M^n,g(t))$, $t\in[0,T)$, where $T<\infty$, to
the Ricci flow with bounded curvature on a complete non-compact Riemannian
manifold with the Ricci curvature tensor uniformly bounded by some constant $C$
on $M^n\times [0,T)$, the curvature tensor stays uniformly bounded on
$M^n\times [0,T)$. Some other results are also presented.

In this paper, by the method of moving planes, we prove the symmetry result
which says that classical solutions of Monge-Ampere system in the whole plane
are symmetric about some point. Our system under consideration comes from the
differential geometry problem.