By restricting the common replacement axiom schema of ZF to ∑~M-formulae,Professor Zhang Jinwen constructed a series of subsystems of Zennelo-Frankel set theory ZF and he called them ZF^M.Zhao Xi shun show that the c...By restricting the common replacement axiom schema of ZF to ∑~M-formulae,Professor Zhang Jinwen constructed a series of subsystems of Zennelo-Frankel set theory ZF and he called them ZF^M.Zhao Xi shun show that the consistency of ZF^M can be deducted from ZF.Professor Zhang Jinwen raised the question whether the consistency of ZF^M can be deducted from ZF^(M+m(M)) for some m(n)≥1.In this paper,we get a positive solution to Professor Zhang's problem.Moreover,we show that the consistency of ZF^M can be deducted from ZF^(M+3).展开更多
Some singular characteristics of analytic functions with positive definiteness are considered. To avoid these singular cases, the finite truncate condition is proposed. When the finite truncate condition is satisfied,...Some singular characteristics of analytic functions with positive definiteness are considered. To avoid these singular cases, the finite truncate condition is proposed. When the finite truncate condition is satisfied, the positive definiteness of an analytic function can be judged by the partial sum of its Taylor series, i e , by a polynomial. This discussion will be useful in the construction of Lyapunov functions for nonlinear systems.展开更多
文摘By restricting the common replacement axiom schema of ZF to ∑~M-formulae,Professor Zhang Jinwen constructed a series of subsystems of Zennelo-Frankel set theory ZF and he called them ZF^M.Zhao Xi shun show that the consistency of ZF^M can be deducted from ZF.Professor Zhang Jinwen raised the question whether the consistency of ZF^M can be deducted from ZF^(M+m(M)) for some m(n)≥1.In this paper,we get a positive solution to Professor Zhang's problem.Moreover,we show that the consistency of ZF^M can be deducted from ZF^(M+3).
文摘Some singular characteristics of analytic functions with positive definiteness are considered. To avoid these singular cases, the finite truncate condition is proposed. When the finite truncate condition is satisfied, the positive definiteness of an analytic function can be judged by the partial sum of its Taylor series, i e , by a polynomial. This discussion will be useful in the construction of Lyapunov functions for nonlinear systems.
