In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator po...In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator potential V（x） ∈ C^1 （R^n, L （H1） ）, where L （H1 ） is the space of all bounded linear operators on the Hilbert space H1, while AAu is the biharmonic differential operator and△u=-∑i,j=1^n 1/√detg δ/δxi[√detgg-1（x）δu/δxj]is the Laplace-Beltrami differential operator in R^n. Here g（x） = （gij（x）） is the Riemannian matrix, while g^-1 （x） is the inverse of the matrix g（x）. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation Au = - △△u ＋ V（x） u （x） = f（x） in the Hilbert space H where f（x） ∈ H as an application of the separation approach.展开更多
In this work, a separated-electrode piezoelectric sensor（SEPS）, constructed by a naked quartz crystal mounted between two electrodes, is reported for applications in a corrosive gaseous phase. The response of SEPS w...In this work, a separated-electrode piezoelectric sensor（SEPS）, constructed by a naked quartz crystal mounted between two electrodes, is reported for applications in a corrosive gaseous phase. The response of SEPS was measured by an impedance analysis method. It was shown that SEPS has an excellent frequency stability because its quality factor is in the order of 10^5. The SEPS can be operated even with the electrode gap in air larger than 1 cm. Compared with a conventional quartz crystal microbalance, the resonant frequency of the SEPS is independent of the mass change in the electrode. The SEPS was applied to monitor the adsorption of iodine on quartz surface and zeolitic-imidazolate framework-8（ZIF-8） film as well as in the transfer of iodine between two ZIF-8 films. The SEPS offers the advantages of easy preparation, corrosion-resistant and convenience in combination with mass and optical measurements.展开更多
文摘In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator potential V（x） ∈ C^1 （R^n, L （H1） ）, where L （H1 ） is the space of all bounded linear operators on the Hilbert space H1, while AAu is the biharmonic differential operator and△u=-∑i,j=1^n 1/√detg δ/δxi[√detgg-1（x）δu/δxj]is the Laplace-Beltrami differential operator in R^n. Here g（x） = （gij（x）） is the Riemannian matrix, while g^-1 （x） is the inverse of the matrix g（x）. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation Au = - △△u ＋ V（x） u （x） = f（x） in the Hilbert space H where f（x） ∈ H as an application of the separation approach.
基金financial support by National Natural Science Foundation of China(Nos.21175084,21275091)the Opening Fund of Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research(Hunan Normal University),Ministry of Education(No.KLCBTCMR2001-01)Research Fund for the Doctoral Program of Higher Education of China(No.20113704110003)
文摘In this work, a separated-electrode piezoelectric sensor（SEPS）, constructed by a naked quartz crystal mounted between two electrodes, is reported for applications in a corrosive gaseous phase. The response of SEPS was measured by an impedance analysis method. It was shown that SEPS has an excellent frequency stability because its quality factor is in the order of 10^5. The SEPS can be operated even with the electrode gap in air larger than 1 cm. Compared with a conventional quartz crystal microbalance, the resonant frequency of the SEPS is independent of the mass change in the electrode. The SEPS was applied to monitor the adsorption of iodine on quartz surface and zeolitic-imidazolate framework-8（ZIF-8） film as well as in the transfer of iodine between two ZIF-8 films. The SEPS offers the advantages of easy preparation, corrosion-resistant and convenience in combination with mass and optical measurements.
