Synthesis and characterization of novel graft copolymers by radiation induced grafting

The radiation-induced graft copolymerization of N-vinyl-2-pyrrolidone (NVP), 4-vinyl pyridine (4VP), 2-vinyl pyridine (2VP) monomers onto poly (ethylene-alt-tetrafluoroethylene) (ETFE) was investigated. The influence of synthesis conditions particularly the solvent was studied. Various solvents, such as n-propanol, isoproponol, benzyl alcohol, methanol, ethanol, cyclohexanone, tetrahydrofuran (THF), nitromethane, 1,4-dioxane and n-heptane were examined for this purpose. Graft copolymers were characterized by Fourier transform infrared (FTIR) spectroscopy, dynamic mechanical analysis (DMA), and scanning electron microscopy-energy dispersive spectroscopy (SEM-EDAX). It was found that the nature of the solvent had profound influence over the grafting reaction. Cyclohexanone, n-propanol and isoproponol for 4VP/ETFE grafting, THF and 1,4-dioxane for NVP/ETFE grafting and benzyl alcohol and methanol for 2VP/ETFE grafting were found to be the suitable solvents yielding highest graft levels. Isoproponol and n-propanol are promising in terms of both graft level and mechanical properties for 4VP/ETFE. Grafting of NVP, 4VP and 2VP onto ETFE were verified through FTIR spectroscopy. Storage modulus and glass transition temperature of the copolymers were found to increase as graft level increased. Surface profile of representative films was also investigated by viewing the distribution of elemental nitrogen using SEM-EDAX. Results indicated that copolymers of 4VP, NVP and 2VP are considerably different from each other. 4VP based copolymers exhibited relatively more homogenous grafting over the surface compared to NVP and 2VP based copolymers

Counting fixed points and rooted closed walks of the singular map x↦xxnx \mapsto x^{x^n} modulo powers of a prime

The "self-power" map $x \mapsto x^x$ modulo $m$ and its generalized form $x \mapsto x^{x^n}$ modulo $m$ are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use $p$-adic methods, primarily $p$-adic interpolation, Hensel's lemma, and lifting singular points modulo $p$, to count fixed points and rooted closed walks of equations related to these maps when $m$ is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix.Comment: 18 pages. Version 2 shortens proofs, reduces redundancy, and introduces new technique for counting rooted closed walks. Version 3 updates title to agree with journal publicatio