${\mathcal F}$-Hypercyclic operators on Fr\' echet spaces
In this paper, we investigate ${\mathcal F}$-hypercyclicity of linear, not necessarily continuous, operators on Fr\' echet spaces. The notion of lower $(m_{n})$-hypercyclicity seems to be new and not considered elsewhere even for linear continuous operators acting on Fr\' echet spaces. We pay special attention to the study of $q$-frequent hypercyclicity, where $q\geq 1$ is an arbitrary real number. We present several new concepts and results for lower and upper densities in a separate section, providing also a great number of illustrative examples and open problems.
To apply continuation of potential field in spatial domain,Fredholm integral equation of the first kind has to be solved numerically.Because the coefficient matrix that comes from discretizing integral equation is not sparse and has very large scale,it requires large computer memory to save the matrix,and operation on it consumes very long time.The continuation method in spatial domain is not developed in practice.In this paper,based on the analysis of structure of coefficient matrix,we find an important fact that the coefficient matrix is a symmetric,block Toeplitz type matrix.A mathematical proof is given in the paper.By using fast algorithm that already exists to solve block Toeplitz system,we realize spatial domain upward continuation and downward continuation.Model data and aeromagnetic data test shows the feasibility and correctness of spatial continuation of potential field.