This paper provides a method for pricing options in the constant elasticity of variance(CEV) model environment using the Lie-algebraic technique when the model parameters are time-dependent.

The paper first introduces some important concepts on the historical relational scheme based on TNF, including time dependence, 2TNF，3TNF and so on, and then gives the algorithm and proof decomposi ng from TNF to 3TNF for the Historical Relational Scheme, this resolves data redundancy and TNF abnormal in the historical relational scheme.

The paper first introduces some important concepts on the hi storical relational scheme based on TNF, including time dependence, th e largest complete time dependence set and so on, and then gives the a lgorithm and its proof for finding the largest complete time dependenc e set.

In some recent tidal potentials, the amplitudes of the tidal waves include time terms, it means that the amplitudes are time dependent, so they can not be counted as fully harmonic developments.

In this paper, the system which consists of a reliable machine, an unreliable machine and a storage buffer is considered, and the existence of a unique positive time-dependent solution of the model corresponding to this system has been proved.

By the use of total probability principle,we obtain an ordinary differential equation of a bulk service queueing model,then using C0-semigroup theory,we prove the existence of a unique time-dependent solution of the model,laying a foundation for analyzing the model′s dynamic solution and stability.

we prove that there exists a unique nonnegative time-dependent solution of the system by using strongly continuous semigroup theory.