Abstract:
In [Discrete Contin. Dyn. Syst. \textbf{15} (2006), no. 3, 811--818.] Xia introduced a simple dynamical density basis for partially hyperbolic sets of volume preserving diffeomorphisms. We apply the density basis to the study of the topological structure of partially hyperbolic sets. We show that if $\Lambda$ is a strongly partially hyperbolic set with positive volume, then $\Lambda$ contains the global stable manifolds over ${\alpha}(\Lambda^d)$ and the global unstable manifolds over ${\omega}(\Lambda^d)$. We give several applications of the dynamical density to partially hyperbolic maps that preserve some $acip$. We show that if $f$ is essentially accessible and $\mu$ is an $acip$ of $f$, then $\text{supp}(\mu)=M$, the map $f$ is transitive, and $\mu$-a.e. $x\in M$ has a dense orbit in $M$. Moreover if $f$ is accessible and center bunched, then either $f$ preserves a smooth measure or there is no $acip$ of $f$.

Abstract:
Let M be a closed manifold and f be a diffeomorphism on M. We show that if f has a nontrivial dominated splitting TM=E\oplus F, then f can not be minimal. The proof mainly use Mane's argument and Liao's selecting lemma.

Abstract:
We study some generic properties of partially hyperbolic symplectic systems with 2D center. We prove that $C^r$ generically, every hyperbolic periodic point has a transverse homoclinic intersection for the maps close to a direct/skew product of an Anosov diffeomorphism with a map on $S^2$ or $\mathbb{T}^2$.

Abstract:
In this paper we study the dynamical billiards on a convex 2D sphere. We investigate some generic properties of the convex billiards on a general convex sphere. We prove that $C^\infty$ generically, every periodic point is either hyperbolic or elliptic with irrational rotation number. Moreover, every hyperbolic periodic point admits some transverse homoclinic intersections. A new ingredient in our approach is that we use Herman's result on Diophantine invariant curves to prove the nonlinear stability of elliptic periodic points for a dense subset of convex billiards.

Abstract:
Let $X$ be a compact metric space and $f:X\to X$ a homeomorphism on $X$. We construct a fundamental domain for the set with finite peaks for each cocycle induced by $\phi\in C(X,R)$. In particular we prove that if a partially hyperbolic diffeomorphism is accessible, then either the set with finite peaks for the Jacobian cocycle is of full volume, or the set of transitive points is of positive volume.

Abstract:
Introduction: The cavernous sinus (CS) is a very important concept because it is not only interesting to anatomical theory but also useful to clinical medicine, especially in the field of surgery. This paper described the microsurgical anatomy of the CS with special attention to its concept that the CS was really venous sinus or plexus. Materials and Methods: Fifty CSs from 25 Chinese adult cadaver heads fixed in 10% methanal, whose artery and vein were injected with red and blue latex, respectively, dissected stepwise under the operating microscope. Results: Asymmetric and nonintegral blue latex distributed in the cavity of the CS to form a retina with various diameters and repeatedly diverged and converged were observed under the surgical microscope with magnification 5 - 25, after the lateral wall of the CS was opened by maxillary approach. Measurement of sinus included length, diameter and triangular structure of the CS. It is very important to understand the microsurgical anatomy of the CS for neurosurgeons. Conclusion: The CS was venous plexus rather than sinus. The lateral wall of the sinus had two layers, and the lateral cavity of the sinus really did exist even though it was very small. The triangles where maxillary approach passed were more important for neurosurgeons.

Abstract:
Maternal embryonic leucine zipper kinase (MELK) functions as a modulator of intracellular signaling and affects various cellular and biological processes, including cell cycle, cell proliferation, apoptosis, spliceosome assembly, gene expression, embryonic development, hematopoiesis, and oncogenesis. In these cellular processes, MELK functions by binding to numerous proteins. In general, the effects of multiple protein interactions with MELK are oncogenic in nature, and the overexpression of MELK in kinds of cancer provides some evidence that it may be involved in tumorigenic process. In this review, our current knowledge of MELK function and recent discoveries in MELK signaling pathway were discussed. The regulation of MELK in cancers and its potential as a therapeutic target were also described.

Abstract:
The self- and mutual-avoiding walk used in conventional lattice models for polymeric systems requires that all lattice sites, polymer segments, and solvent molecules (unoccupied lattice sites) have the same volume. This incorrectly accounts for the solvent entropy (i.e., size ratio between polymer segments and solvent molecules), and also limits the coarse-graining capability of such models, where the invariant degree of polymerization controlling the system fluctuations is too small (thus exaggerating the system fluctuations) compared to that in most experiments. Here we show how to properly account for the solvent entropy in the recently proposed lattice models with multiple occupancy of lattice sites [Q. Wang, Soft Matter 5, 4564 (2009)], and present a quantitative coarse-graining strategy that ensures both the solvent entropy and the fluctuations in the original systems are properly accounted for using such lattice models. Although proposed based on homogeneous polymer solutions, our strategy is equally applicable to inhomogeneous systems such as polymer brushes immersed in a small-molecule solvent.

Abstract:
For some planar Newtonian $N+3$-body problems, we use variational minimization methods to prove the existence of new periodic solutions satisfying that $N$ bodies chase each other on a curve, and the other 3 bodies chase each other on another curve. From the definition of the group action in equations $(3.1)-(3.3)$, we can find that they are new solutions which are also different from all the examples of Ferrario and Terracini (2004)$[22]$.

Abstract:
This paper is devoted to the regularity analysis of a geodesic equation in the space of Sasakian metrics. Firstly, we reduce the geodesic equation in the space of Sasakian metrics to a Dirichlet problem of degenerate complex Monge-Amp\'ere type eqution on the K\"ahler cone; secondly, we obtain a priori etimates for the above equation. These a priori estimates guarantee the existence and uniqueness of $C^{2}_{w}$ geodesic for any two points in the space of Sasakian metrics. We also give some geometric applications of the above estimates in the end of this paper.