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51.
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Let be an arbitrary group. It is proved that if contains a bicyclic unit , then is a nonabelian free subgroup of invertible elements.

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Multifunctional materials owing to their efficacy to perform more than one job have attracted the attention of global materials science community to resolve the issue of material crunch and reduce the work load. Here, in this work, we have demonstrated the suitability of CePO4:Sm3+ (CPOS) phosphor for white light luminophore and oxygen reduction catalyst. Raman spectroscopy confirmed the stabilization of monoclinic phase for CPOS. White light emission with correlated color temperature (CCT) favoring cool white feature in CPOS is endowed by inefficient host to dopant energy transfer (HDET). X-ray photoelectron spectroscopy (XPS) suggested presence of both Ce3+ and Ce4+ in CPOS with later in lower fraction. XPS results also showed the presence of surface oxygen in CPOS which along with mixed oxidation state of cerium on the surface may be aiding CPOS to work as electrocatalysts for oxygen reduction reaction (ORR). The best electrocatalytic performance was achieved with 1.0% Sm doped sample which has higher concentration of oxygen associated with Ce (IV) on the surface.  相似文献   
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Commercially available anhydrous zinc chloride has been found to be a highly efficient catalyst for dimethyl acetalization in high yields by treatment of aldehydes and ketones with trimethyl orthoformate in methanol-cyclohexane at reflux temperature.  相似文献   
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The iron bearing phases present in a ferromanganese nodule from the Central Indian Ocean have been determined using57Fe Mössbauer spectroscopy. The Mössbauer results have been corroborated by XRD, IR and TG-DTA studies. The Mössbauer spectrum of a ferromanganese nodule shows a broad line width which indicates the presence of more than one iron bearing paramagnetic oxide or oxyhydroxide phases where iron is present as Fe3+. -FeOOH has been distinctly characterised as one of the iron bearing phases in the nodule. Other oxyhydroxide and oxide phases of iron in the nodule have been ruled out. A typical paramagnetic doublet persists even at very high temperature which has been proposed to be due to iron(III)phosphate. Formation of solid solution of Mn2O3–Fe2O3 has been observed in the heat treated nodule at 1073 K, which has been characterised by the Mössbauer technique.  相似文献   
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Nanosilica was prepared by precipitation method and was characterized by various analytical tools. From transmission electron micrograph the silica particles are found to have almost spherical shape with a dimension of 50 nm. The surface area is found to be of 560 m2 g−1 and density 2.2 g cm−3. From thermogravimetric analysis the total silanol density in the silica is found to be 7.68 nm−2. A two tier hydration model is proposed from the results of thermogravimetric analysis. The number of reactive silanols that forms hydrogen bond with water molecules is found to be 2.48. The infrared spectral data supports the presence of hydrogen bonded silanol group and the siloxane groups in silica.  相似文献   
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