The present paper deals with the average case complexity of the shift—invariant problem. The main aim is to give a new proof of the upper bound of average error of finite element method. Our method is based on the te...The present paper deals with the average case complexity of the shift—invariant problem. The main aim is to give a new proof of the upper bound of average error of finite element method. Our method is based on the techniques proposed by Heinrich (1990). We also point out an essential error regarding the proof of the upper bound in A. G. Werschulz (1991).展开更多
This paper puts forward a complex inner product averaging method for calculating normal form of ODE. Compared with conventional averaging method, the theoretic analytical process has such simple forms as to realize co...This paper puts forward a complex inner product averaging method for calculating normal form of ODE. Compared with conventional averaging method, the theoretic analytical process has such simple forms as to realize computer program easily. Results can be applied in both autonomous and non-autonomous systems. At last, an example is resolved to verify the method.展开更多
In 2018,Petersen and Wilson introduced the notion of dynamical intricacy and average sample complexity for dynamical systems of Z-action,based on the past works on the notion of intricacy in the research of brain netw...In 2018,Petersen and Wilson introduced the notion of dynamical intricacy and average sample complexity for dynamical systems of Z-action,based on the past works on the notion of intricacy in the research of brain network and probability theory.If one wants to take into account underlying system geometry in applications,more general group actions may need to be taken into consideration.In this paper,we consider this notion in the case of amenable group actions.We show that many basic properties in the Z-action case remain true.We also show that their suprema over covers or partitions are equal to the amenable topological entropy and the measure entropy,using the quasitiling technique in the theory of the amenable group.展开更多
Two new perovskites,(Bi_(0.5)A_(0.5))(Sc_(0.5)Nb_(0.5))O_(3)where A=K^(+)and Na^(+),have been made phase pure for the first time.The synthesis procedure that prevented the formation of pyrochlore impurities is describ...Two new perovskites,(Bi_(0.5)A_(0.5))(Sc_(0.5)Nb_(0.5))O_(3)where A=K^(+)and Na^(+),have been made phase pure for the first time.The synthesis procedure that prevented the formation of pyrochlore impurities is described.B-site ordering in A=K^(+)is successfully achieved by prolonged heating.Combined Rietveld refinement on synchrotron X-ray diffraction and neutron diffraction data is used to determine the complex average structure of these materials.A=Na^(+)crystallizes in the tetragonal P4/mbm space group and has disordered a^(0)a^(0)c^(+)rotations of BO6 octahedra.A=K^(+)crystallizes in the cubic Pm[3 with combining macron]m space group,but shows local A-site displacements in the[100]direction.Both materials are highly disordered but illustrate the potential of(A)NbO_(3)-BiScO_(3)solid solutions as useful piezoelectrics or relaxors.展开更多
文摘The present paper deals with the average case complexity of the shift—invariant problem. The main aim is to give a new proof of the upper bound of average error of finite element method. Our method is based on the techniques proposed by Heinrich (1990). We also point out an essential error regarding the proof of the upper bound in A. G. Werschulz (1991).
文摘This paper puts forward a complex inner product averaging method for calculating normal form of ODE. Compared with conventional averaging method, the theoretic analytical process has such simple forms as to realize computer program easily. Results can be applied in both autonomous and non-autonomous systems. At last, an example is resolved to verify the method.
基金supported by National Natural Science Foundation of China(Grant No.11701231)supported by National Natural Science Foundation of China(Grant Nos.11801584 and 11871228)+1 种基金National Science Foundation of Jiangsu Province(Grant No.BK20170225)Science Foundation of Jiangsu Normal University(Grant No.17XLR011)。
文摘In 2018,Petersen and Wilson introduced the notion of dynamical intricacy and average sample complexity for dynamical systems of Z-action,based on the past works on the notion of intricacy in the research of brain network and probability theory.If one wants to take into account underlying system geometry in applications,more general group actions may need to be taken into consideration.In this paper,we consider this notion in the case of amenable group actions.We show that many basic properties in the Z-action case remain true.We also show that their suprema over covers or partitions are equal to the amenable topological entropy and the measure entropy,using the quasitiling technique in the theory of the amenable group.
文摘Two new perovskites,(Bi_(0.5)A_(0.5))(Sc_(0.5)Nb_(0.5))O_(3)where A=K^(+)and Na^(+),have been made phase pure for the first time.The synthesis procedure that prevented the formation of pyrochlore impurities is described.B-site ordering in A=K^(+)is successfully achieved by prolonged heating.Combined Rietveld refinement on synchrotron X-ray diffraction and neutron diffraction data is used to determine the complex average structure of these materials.A=Na^(+)crystallizes in the tetragonal P4/mbm space group and has disordered a^(0)a^(0)c^(+)rotations of BO6 octahedra.A=K^(+)crystallizes in the cubic Pm[3 with combining macron]m space group,but shows local A-site displacements in the[100]direction.Both materials are highly disordered but illustrate the potential of(A)NbO_(3)-BiScO_(3)solid solutions as useful piezoelectrics or relaxors.
