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The following two theorems are obtained. (1) A mapping fis A-continuous if and only if it is semi-continuous and C-continuous. (2) A mapping f is A-continuous if and only if it is spr-continuous and LC-continuous.

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Reaction Kinetics, Mechanisms and Catalysis
Haiying Wang
Yucheng Yang
Jianhong Wei
Ling Le
Yang Liu
Chunxu Pan
Pengfei Fang
Rui Xiong
, and
Jing Shi


Nitrogen-doped titanium dioxide (TiO2) nanotube arrays were synthesized by anodization in ethylene glycol electrolyte and annealing in ammonia at 500 °C. Detailed analysis showed that the nitrogen-doped titania nanotubes were pure anatase of ordered structure, with a crystallite size of 8.5 nm. The doping nitrogen atoms were induced on the interstitial sites and substitutional sites and the ratio of oxygen vacancies increased to 27.15 %, resulting in an add-on peak in the absorption spectrum and extended the absorption from 387 to 618 nm. The photocatalytic activity of the nitrogen-doped TiO2 nanotubes was evaluated by photocatalytic degradation of methyl blue under visible light irradiation. Significant improvement of photocatalytic activity under visible light irradiation was observed. We assumed the nitrogen doping induced the effect produced by nitrogen atoms, Ti3+ cations and oxygen vacancies and the size effect of the TiO2 crystallite should be responsible for the effective photocatalytic activity.

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