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Continuity of the maximal commutators in Sobolev spaces
Xixi Jiang,Feng Liu 대한수학회 2024 대한수학회지 Vol.61 No.3
We prove the Sobolev continuity of maximal commutator and its fractional variant with Lipschitz symbols, both in the global and local cases. The main result in global case answers a question originally posed by Liu and Wang in \cite{LW}.
Transparent and Self-Powered Multistage Sensation Matrix for Mechanosensation Application
Zhang, Qian,Jiang, Tao,Ho, Donghae,Qin, Shanshan,Yang, Xixi,Cho, Jeong Ho,Sun, Qijun,Wang, Zhong Lin American Chemical Society 2018 ACS NANO Vol.12 No.1
<P>Electronic skin based on a multimodal sensing array is ready to detect various stimuli in different categories by utilizing highly sensitive materials, sophisticated geometry designs, and integration of multifunctional sensors. However, it is still difficult to distinguish multiple and complex mechanical stimuli in a local position by conventional multimodal E-skin, which is significantly important in the signals’ feedback of robotic fine motions and human–machine interactions. Here, we present a transparent, flexible, and self-powered multistage sensation matrix based on piezoelectric nanogenerators constructed in a crossbar design. Each sensor cell in the matrix comprises a layer of piezoelectric polymer sandwiched between two graphene electrodes. The simple lamination design allows sequential multistage sensation in one sensing cell, including compressive/tensile strain and detaching/releasing area. Further structure engineering on PDMS substrate allows the sensor cell to be highly sensitive to the applied pressures, representing the minimum sensing pressure below 800 Pa. As the basic combinations of compressive/tensile strains or detaching/releasing represent individual output signals, the proposed multistage sensors are capable of decoding to distinguish external complex motions. The proposed self-powering multistage sensation matrix can be used universally as an autonomous invisible sensory system to detect complex motions of the human body in local position, which has promising potential in movement monitoring, human–computer interaction, humanoid robots, and E-skins.</P><P><B>Graphic Abstract</B> <IMG SRC='http://pubs.acs.org/appl/literatum/publisher/achs/journals/content/ancac3/2018/ancac3.2018.12.issue-1/acsnano.7b06126/production/images/medium/nn-2017-06126z_0005.gif'></P><P><A href='http://pubs.acs.org/doi/suppl/10.1021/nn7b06126'>ACS Electronic Supporting Info</A></P>