Compared with commercial polyolefin membranes, polyacrylonitrile (PAN) membrane prepared by electrostatic spinning has higher porosity, electrolyte uptake, thermal stability, and lithium-ion conductivity, etc. However, poor mechanical performance has largely limited the application of electrospun PAN separators. In this study, PAN/polyimide (PI) composite membrane is prepared by electrostatic spinning to improve the mechanical and electrochemical performances. Scanning electron microscopy, thermal analysis method, and electrochemical methods were used for evaluation of the electrospun composite membrane. The results show that the composite membrane possesses good thermal stability and exhibits better mechanical performance than pristine PAN membrane (increasing by 1.1 times in tension strength). The addition of PI can increase porosity and fluid absorption rate obviously. In addition, the composite membrane has high ionic conductivity (18.77 × 10−4 S/cm), wide electrochemical window (about 4.0 V), and excellent cycling performance. It can retain a discharge specific capacity of 153 mA h/g even after 50 cycles at 0.5 C. The electrospun PAN/PI membrane may be a promising candidate for lithium-ion battery separators.

Let $R$ be a dense subring of $\text{End}\left( _{D}V \right)$, where $V$ is a left vector space over a division ring $D$. If $\dim{{\,}_{D}}V\,=\,\infty $, then the range of any nonzero polynomial $f\left( {{X}_{1}},\,\ldots \,,\,{{X}_{m}} \right)$ on $R$ is dense in $\text{End}\left( _{D}V \right)$. As an application, let $R$ be a prime ring without nonzero nil one-sided ideals and $0\,\ne \,a\,\in \,R$. If $af{{\left( {{x}_{1}},\ldots ,{{x}_{m}} \right)}^{n\left( {{x}_{i}} \right)}}\,=\,0$ for all ${{x}_{1}},\,\ldots \,,\,{{x}_{m}}\,\in \,R$, where $n\left( {{x}_{i}} \right)$ is a positive integer depending on ${{x}_{1}},\,\ldots \,,\,{{x}_{m}}\,\in \,R$, then $f\left( {{X}_{1}},\,\ldots \,,\,{{X}_{m}} \right)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, d and δ non-zero derivations of R, f (x1,…, xn) a multilinear polynomial over K.If

then f(x1,…,xnis central-valued on R.

We show that the theory of elliptic integral transformations may be employed to construct iterative approximations for π of order p (p any prime). Details are provided for two, three and seven. The cubic case proves amenable to surprisingly complete analysis.