In this article,we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampere operators acting in different two-dimensional coordinate sections.This equation is ellipt...In this article,we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampere operators acting in different two-dimensional coordinate sections.This equation is elliptic,for example,in the class of convex functions.We show that the notion of Monge-Ampere measures and Aleksandrov generalized solutions extends to this equation,subject to a weaker notion of convexity which we call bi-planar convexity.While the equation is also elliptic in the class of bi-planar convex functions,the contrary is not necessarily true.This is a substantial difference compared to the classical Monge-Ampere equation where ellipticity and convexity coincide.We provide explicit counter-examples:classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced.We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.展开更多
For a magnetocardiography(MCG)system inside a magnetically shielded room(MSR),the residual field should be further suppressed to obtain high-quality MCG signals.In this paper,a compensation system has been developed b...For a magnetocardiography(MCG)system inside a magnetically shielded room(MSR),the residual field should be further suppressed to obtain high-quality MCG signals.In this paper,a compensation system has been developed based on the bi-planar coil and the proportional-integral-derivative(PID)controller.The bi-planar coil,derived from the target-field theory and the Tikhonov regularization method,is utilized to generate magnetic field with high uniformity in the predefined target region.And the PID controller is introduced to provide dynamic compensation current for the coil,according to the residual field change monitored by a reference SQUID magnetometer.The measurement results show that the noise suppression ratio(NSR)can reach above 20 dB in the low-frequency range from 0.1 Hz to 50 Hz.The DC component and the fluctuation of the residual field in time-domain can be respectively suppressed to 0 p T and 4 p T,indicating that this proposed compensation method is useful for the MCG measurement.展开更多
基金This article contributes to the project"Systematic multi-scale modeling and analysis for geophysical flow"of the Collaborative Research Center TRR 181"Energy Transfers in Atmosphere and Ocean"funded by the Deutsche Forschungsgemeinschaft(DFG,German Research Foundation)under project number 274762653.
文摘In this article,we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampere operators acting in different two-dimensional coordinate sections.This equation is elliptic,for example,in the class of convex functions.We show that the notion of Monge-Ampere measures and Aleksandrov generalized solutions extends to this equation,subject to a weaker notion of convexity which we call bi-planar convexity.While the equation is also elliptic in the class of bi-planar convex functions,the contrary is not necessarily true.This is a substantial difference compared to the classical Monge-Ampere equation where ellipticity and convexity coincide.We provide explicit counter-examples:classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced.We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.
基金the Open Research Fund of Anhui Key Laboratory of Detection Technology and Energy Saving Devices,Anhui Polytechnic University(Grant No.JCKJ2021A03)the Introduced Talent Research Startup Funds of Anhui Polytechnic University(Grant Nos.2021YQQ006 and 2020YQQ040)the National Natural Science Foundation of China(Grant No.62101004)。
文摘For a magnetocardiography(MCG)system inside a magnetically shielded room(MSR),the residual field should be further suppressed to obtain high-quality MCG signals.In this paper,a compensation system has been developed based on the bi-planar coil and the proportional-integral-derivative(PID)controller.The bi-planar coil,derived from the target-field theory and the Tikhonov regularization method,is utilized to generate magnetic field with high uniformity in the predefined target region.And the PID controller is introduced to provide dynamic compensation current for the coil,according to the residual field change monitored by a reference SQUID magnetometer.The measurement results show that the noise suppression ratio(NSR)can reach above 20 dB in the low-frequency range from 0.1 Hz to 50 Hz.The DC component and the fluctuation of the residual field in time-domain can be respectively suppressed to 0 p T and 4 p T,indicating that this proposed compensation method is useful for the MCG measurement.
