Suppose M and N are two r×r and s×s dilation matrices,respectively.LetΓM andΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-T Zr/Zr and N-T Zs/Zs,respectively....Suppose M and N are two r×r and s×s dilation matrices,respectively.LetΓM andΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-T Zr/Zr and N-T Zs/Zs,respectively.Two methods for constructing nonseparable Ω-filter banks from M-filter banks and N-filter banks are presented,where Ω is a（r＋s） ×（r＋s） dilation matrix such that one of its complete sets of representatives of distinct cosets of the quotient groups Ω-T Zr＋s/Zr＋s areΓΩ={[γT h,ζ T q] T：γh∈ΓM,ζq∈ΓN}.Specially,Ω can be [MΘ0N],whereΘis a r×s integer matrix with M-1Θbeing also an integer matrix.Moreover,if the constructed filter bank satisfies Lawton＇s condition,which can be easy to verify,then it generates an orthonormal nonseparable Ω-wavelet basis for L2（Rr＋s）.Properties,including Lawton＇s condition,vanishing moments and regularity of the new Ω-filter banks or new Ω-wavelet basis are discussed then.Finally,a class of nonseparable Ω-wavelet basis for L2（Rr＋1） are constructed and three other examples are given to illustrate the results.In particular,when M=N=2,all results obtained in this paper appeared in[1].展开更多
Solid polymer electrolytes(SPEs) have been considered as the spotlight in recent years due to their high safety, non-flammability and good flexibility. Nonetheless, high crystallinity of polymer matrix leads to low io...Solid polymer electrolytes(SPEs) have been considered as the spotlight in recent years due to their high safety, non-flammability and good flexibility. Nonetheless, high crystallinity of polymer matrix leads to low ionic conductivity at ambient conditions and retards the practical applications of SPEs. Herein, we report hybrid solid electrolytes(HSE) containing bulky LATP in poly(ethylene oxide)(PEO) matrix, which significantly enhances the electrochemical properties. LATP has been easily obtained by an accessible solid-state method. The solid electrolyte based on 20 wt% LATP in PEO polymer matrix(abbreviated as PEO-20) exhibits an ionic conductivity of 2.1 ×10-5 S·cm-1 at 30 ℃, an order of magnitude higher than 2.9 × 10-6 S·cm-1 of the pristine PEO solid electrolyte(abbreviated as PEO-0), mainly resulting from the decline of crystallinity in polymer matrix. The electrochemical window of PEO-20 can reach 4.84 V at room temperature, compared with 4.40 V for PEO-0, which could be compatible with high-voltage cathode materials.展开更多
基金Supported by the National Natural Science Foundation of China(11071152)the Natural Science Foundation of Guangdong Province(10151503101000025)
文摘Suppose M and N are two r×r and s×s dilation matrices,respectively.LetΓM andΓN represent the complete sets of representatives of distinct cosets of the quotient groups M-T Zr/Zr and N-T Zs/Zs,respectively.Two methods for constructing nonseparable Ω-filter banks from M-filter banks and N-filter banks are presented,where Ω is a（r＋s） ×（r＋s） dilation matrix such that one of its complete sets of representatives of distinct cosets of the quotient groups Ω-T Zr＋s/Zr＋s areΓΩ={[γT h,ζ T q] T：γh∈ΓM,ζq∈ΓN}.Specially,Ω can be [MΘ0N],whereΘis a r×s integer matrix with M-1Θbeing also an integer matrix.Moreover,if the constructed filter bank satisfies Lawton＇s condition,which can be easy to verify,then it generates an orthonormal nonseparable Ω-wavelet basis for L2（Rr＋s）.Properties,including Lawton＇s condition,vanishing moments and regularity of the new Ω-filter banks or new Ω-wavelet basis are discussed then.Finally,a class of nonseparable Ω-wavelet basis for L2（Rr＋1） are constructed and three other examples are given to illustrate the results.In particular,when M=N=2,all results obtained in this paper appeared in[1].
基金the Fujian Provincial Science and Technology Program (No. 2018H0042,2019T3017)Putian Municipal Science and Technology Program (2019HJSTS009)the DNL Cooperation Fund,CAS (DNL180308)。
文摘Solid polymer electrolytes(SPEs) have been considered as the spotlight in recent years due to their high safety, non-flammability and good flexibility. Nonetheless, high crystallinity of polymer matrix leads to low ionic conductivity at ambient conditions and retards the practical applications of SPEs. Herein, we report hybrid solid electrolytes(HSE) containing bulky LATP in poly(ethylene oxide)(PEO) matrix, which significantly enhances the electrochemical properties. LATP has been easily obtained by an accessible solid-state method. The solid electrolyte based on 20 wt% LATP in PEO polymer matrix(abbreviated as PEO-20) exhibits an ionic conductivity of 2.1 ×10-5 S·cm-1 at 30 ℃, an order of magnitude higher than 2.9 × 10-6 S·cm-1 of the pristine PEO solid electrolyte(abbreviated as PEO-0), mainly resulting from the decline of crystallinity in polymer matrix. The electrochemical window of PEO-20 can reach 4.84 V at room temperature, compared with 4.40 V for PEO-0, which could be compatible with high-voltage cathode materials.
