The author considers the embedding problem of weighted Sobolev spaces H<sup>n</sup><sub>p</sub> in weighted L<sub>s</sub> spaces L<sub>s,r</sub>,and some sufficient cond...The author considers the embedding problem of weighted Sobolev spaces H<sup>n</sup><sub>p</sub> in weighted L<sub>s</sub> spaces L<sub>s,r</sub>,and some sufficient conditions and necessary conditions are given, when weight functions satisfy certain conditions.The author uses the results obtained to the qualitative analysis of the spectrum of 2n-order weighted differential operator,and gives some sufficient conditions and necessary conditions to ensure that the spectrum is discrete.展开更多
An in-line fiber Fabry-Perot interferometer （FPI） based on the hollow-core photonic crystal fiber （HCPCF） for refractive index （RI） measurement is proposed in this paper. The FPI is formed by splicing both ends ...An in-line fiber Fabry-Perot interferometer （FPI） based on the hollow-core photonic crystal fiber （HCPCF） for refractive index （RI） measurement is proposed in this paper. The FPI is formed by splicing both ends of a short section of the HCPCF to single mode fibers （SMFs） and cleaving the SMF pigtail to a proper length. The RI response of the sensor is analyzed theoretically and demonstrated experimentally. The results show that the FPI sensor has linear response to external RI and good repeatability. The sensitivity calculated from the maximum fringe contrast is -136 dB/RIU. A new spectrum differential integration （SDI） method for signal processing is also presented in this study. In this method, the RI is obtained from the integrated intensity of the absolute difference between the interference spectrum and its smoothed spectrum. The results show that the sensitivity obtained from the integrated intensity is about -1.34× 10^5 dB/RIU. Compared with the maximum fringe contrast method, the new SDI method can provide the higher sensitivity, better linearity, improved reliability, and accuracy, and it＇s also convenient for automatic and fast signal processing in real-time monitoring of RI.展开更多
基金Supported by the National Natural Science Fundation of Chinathe Natural Science Foundation of Inner Mongolia.
文摘The author considers the embedding problem of weighted Sobolev spaces H<sup>n</sup><sub>p</sub> in weighted L<sub>s</sub> spaces L<sub>s,r</sub>,and some sufficient conditions and necessary conditions are given, when weight functions satisfy certain conditions.The author uses the results obtained to the qualitative analysis of the spectrum of 2n-order weighted differential operator,and gives some sufficient conditions and necessary conditions to ensure that the spectrum is discrete.
基金This research is supported by the National Natural Science Foundations of China （Grant Nos. 61174018 and 61505097） and Fundamental research funds of Shandong University, China （Grant No.2014YQ009）.
文摘An in-line fiber Fabry-Perot interferometer （FPI） based on the hollow-core photonic crystal fiber （HCPCF） for refractive index （RI） measurement is proposed in this paper. The FPI is formed by splicing both ends of a short section of the HCPCF to single mode fibers （SMFs） and cleaving the SMF pigtail to a proper length. The RI response of the sensor is analyzed theoretically and demonstrated experimentally. The results show that the FPI sensor has linear response to external RI and good repeatability. The sensitivity calculated from the maximum fringe contrast is -136 dB/RIU. A new spectrum differential integration （SDI） method for signal processing is also presented in this study. In this method, the RI is obtained from the integrated intensity of the absolute difference between the interference spectrum and its smoothed spectrum. The results show that the sensitivity obtained from the integrated intensity is about -1.34× 10^5 dB/RIU. Compared with the maximum fringe contrast method, the new SDI method can provide the higher sensitivity, better linearity, improved reliability, and accuracy, and it＇s also convenient for automatic and fast signal processing in real-time monitoring of RI.
