Abstract:
In this article, a cell culture microchip was fabricated on the SU-8 mold based on polymer-MEMS process. In the microchip, the cell culture area was separated with microchannel by a microgap, which kept the cell culture area independent, but also regulated the micro-environment of extracellular matrix by the microfluidic flow. The cell culture microchip provided a new platform for cell research.

Abstract:
We study the existence, uniqueness, and continuous dependence on initial data of the solution to the Cauchy problem and steady state solutions of a nonlocal Cahn-Hilliard equation on a bounded domain.

Abstract:
Meshless method is a type of promising numerical approach. But for the method, the convergence is still lack of common theoretical explanations, and the technique of numerical implementation also remains to be improved. It is worth noting that a kind of uniformly defined intervention point is used in many existing schemes. Therefore, the intervention-point principle is proposed. The viewpoint is likely to give a reasonable explanation for the inaccuracy and instability of the collocation method. Based on the principle, a design process for a new scheme was demonstrated. Some initial numerical tests were also offered. The results have revealed the intervention point to take effect on convergence, suggested a construction concept using intervention point for meshless collocation method, and presented a new scheme of meshless method for application.

Abstract:
Let $R$ be an associative ring with identity. A unital right $R$-module $M$ is called strongly finite dimensional if Sup$\{{\rm G.dim} (M/N) | N\leq M\} < +\infty$. Properties of strongly finite dimensional modules are explored. It is also proved that: (1)If $R$ is left $F$-injective and strongly right finite dimensional, then $R$ is left finite dimensional. (2) If $R$ is right $F$-injective, then $R$ is right finite dimensional if and only if $R$ is semilocal. Thus the Faith-Menal conjecture is true if $R$ is strongly right finite dimensional. Some known results are obtained as corollaries.

Abstract:
A ring $R$ with an involution * is called (strongly) *-clean if every element of $R$ is the sum of a unit and a projection (that commute). All *-clean rings are clean. Va${\rm \check{s}}$ [L. Va${\rm \check{s}}$, *-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras, J. Algebra 324 (12) (2010) 3388-3400] asked whether there exists a *-ring that is clean but not *-clean and whether a unit regular and *-regular ring is strongly *-clean. In this paper, we answer both questions by several examples. Moreover, some characterizations of unit regular and *-regular rings are provided.

Abstract:
Let $R$ be a ring, a right ideal $I$ of $R$ is called small if for every proper right ideal $K$ of $R$, $I+K\neq R$. A ring $R$ is called right small injective if every homomorphism from a small right ideal to $R_{R}$ can be extended to an $R$-homomorphism from $R_{R}$ to $R_{R}$. Properties of small injective rings are explored and several new characterizations are given for $QF$ rings and $PF$ rings, respectively.

Abstract:
A ring $R$ is said to be $n$-clean if every element can be written as a sum of an idempotent and $n$ units. The class of these rings contains clean ring and $n$-good rings in which each element is a sum of $n$ units. In this paper, we show that for any ring $R$, the endomorphism ring of a free $R$-module of rank at least 2 is 2-clean and that the ring $B(R)$ of all $\omega\times \omega$ row and column-finite matrices over any ring $R$ is 2-clean. Finally, the group ring $RC_{n}$ is considered where $R$ is a local ring. \vskip 0.5cm {\bf Key words:}\quad 2-clean rings, 2-good rings, free modules, row and column-finite matrix rings, group rings.

Abstract:
Let $R$ be a ring. $R$ is called a right countably $\Sigma$-C2 ring if every countable direct sum copies of $R_{R}$ is a C2 module. The following are equivalent for a ring $R$: (1) $R$ is a right countably $\Sigma$-C2 ring. (2) The column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_{\mathbb{N}}(R)$ is a right C2 (or C3) ring. (3) Every countable direct sum copies of $R_{R}$ is a C3 module. (4) Every projective right $R$-module is a C2 (or C3) module. (5) $R$ is a right perfect ring and every finite direct sum copies of $R_{R}$ is a C2 (or C3) module. This shows that right countably $\Sigma$-C2 rings are just the rings whose right finitistic projective dimension r$FPD(R)$=sup\{$Pd_{R}(M)|$ $M$ is a right $R$-module with $Pd_{R}(M)<\infty$\}=0, which were introduced by Hyman Bass in 1960.