Abstract:
Before the earthquake of M=9.0 happened on 26th,Dec.2004 in Indonesia,a very strong infrasonic sound wave with a sound press being of 16.8Pa was received on 3:22 pm,19th,Dec.2004,which shows that a great earthquake should happen in the world shortly based on our experiences.We made an informal imminent forecasting that earthquakes with magnitude greater than Ms=8.1 should occur during 19th to 28th,Dec.2004 in the world.

Abstract:
Let $f: M \to M$ be a partially hyperbolic diffeomorphism with conformality on unstable manifolds. Consider a set of points with nondense forward orbit: $E(f, y) := \{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$ for some $y \in M$. Define $E_{x}(f, y) := E(f, y) \cap W^u(x)$ for any $x\in M$. Following a method of Broderick-Fishman-Kleinbock, we show that $E_x(f,y)$ is a winning set of Schmidt games played on $W^u(x)$ which implies that $E_x(f,y)$ has full Hausdorff dimension equal to $\dim W^u(x)$. Furthermore we show that for any nonempty open set $V \subset M$, $E(f, y) \cap V$ has full Hausdorff dimension equal to $\dim M$, by constructing measures supported on $E(f, y)\cap V$ with lower pointwise dimension converging to $\dim M$ and with conditional measures supported on $E_x(f,y)\cap V$. The results can be extended to the set of points with forward orbit staying away from a countable subset of $M$.

Abstract:
We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected reversible Berwald space of finite volume and bounded nonpositive flag curvature with rank at least 2 whose universal cover is irreducible is locally symmetric. Adapting the method in \cite{BBE}, \cite{BBS}, and \cite{BS2}, we will introduce an angle notion and establish a flat strip lemma, stable and unstable manifolds for the geodesic flows, Weyl Chambers and Tits Building in the sphere at infinity for the universal cover of Berwald spaces of nonpositive flag curvature.

Abstract:
Let $M$ be a smooth compact surface of nonpositive curvature, with genus $\geq 2$. We prove the ergodicity of the geodesic flow on the unit tangent bundle of $M$ with respect to the Liouville measure under the condition that the set of points with negative curvature on $M$ has finitely many connected components. Under the same condition, we prove that a non closed "flat" geodesic doesn't exist, and moreover, there are at most finitely many flat strips, and at most finitely many isolated closed "flat" geodesics.

Abstract:
We show that the set of points with non-dense forward orbit under a $C^{1+\theta}$-Anosov diffeomorphism with conformality on unstable manifolds is a winning set for Schmidt games. We also show that for a $C^{1+\theta}$-expanding endomorphism the set of points with non-dense forward orbit is a winning set for certain variants of Schmidt games. These generalize some results of J. Tseng in \cite{tseng} and \cite{tseng1} for $C^2$-expanding endomorphisms on the circle and certain Anosov diffeomorphisms on the $2$-torus.

Abstract:
Let $f: M \to M$ be a $C^{1+\theta}$-partially hyperbolic diffeomorphism. We introduce a type of modified Schmidt games which is induced by $f$ and played on any unstable manifold. Utilizing it we generalize some results of \cite{Wu} as follows. Consider a set of points with non-dense forward orbit: $$E(f, y) := \{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$$ for some $y \in M$ and $$E_{x}(f, y) := E(f, y) \cap W^u(x)$$ for any $x\in M$. We show that $E_x(f,y)$ is a winning set for such modified Schmidt games played on $W^u(x)$, which implies that $E_x(f,y)$ has Hausdorff dimension equal to $\dim W^u(x)$. Then for any nonempty open set $V \subset M$ we show that $E(f, y) \cap V$ has full Hausdorff dimension equal to $\dim M$, by using a technique of constructing measures supported on $E(f, y)$ with lower pointwise dimension approximating $\dim M$.

Abstract:
Greedy Forwarding algorithm is a widely-used routing algorithm for wireless networks. However, it can fail if network topologies (usually modeled by geometric graphs) contain voids. Since Yao Graph and Theta Graph are two types of geometric graphs exploited to construct wireless network topologies, this paper studies whether these two types of graphs can contain voids. Specifically, this paper shows that when the number of cones in a Yao Graph or Theta Graph is less than 6, Yao Graph and Theta Graph can have voids, but when the number of cones equals or exceeds 6, Yao Graph and Theta Graph are free of voids.

Abstract:
The bondage number $b(G)$ of a graph $G$ is the cardinality of a minimum edge set whose removal from $G$ results in a graph with the domination number greater than that of $G$. It is a parameter to measure the vulnerability of a communication network under link failure. In this paper, we obtain the exact value of the bondage number of the strong product of a complete graph and a path. That is, for any two integers $m\geq1$ and $n\geq2$, $b(K_{m}\boxtimes P_{n})=\lceil\frac{m}{2}\rceil$ if $n\equiv 0$ (mod 3); $m$ if $n\equiv 2$ (mod 3); $\lceil\frac{3m}{2}\rceil$ if $n\equiv 1$ (mod 3). Furthermore, we determine the exact value of the bondage number of the strong product of a complete graph and a special starlike tree.