The definition of the nonsingularity is presented based on analyzing digital discrete chaotic sequence.

Based on the kernel estimating theory of nonlinear semiparametric models under the least-square principle,this paper proves the nonsingularity of coefficient matrix of normal equation in least-square estimator of non-linear semiparametric models under certain condition.

The new nonsingularity criteria of matrices and the equivalent representation of M-matrices are presented in this paper.

The present paper deals with the existence of the positive solution of the singular nonlinear boundary value problem:(g(u′))′=-K(x)f(u), 0<t<1; u(0)=0, u′(1)=cwhere g(s)=|s| p-2 s, p>1, c is a non negative real numbers, f(u) is non negative, right continuous, nonincreasing in (0,+∞).

The shooting method is used to infer the existence and uniqueness of the positive solution to the singular nonlinear boundary value for n order ordinary differential equations u (n) (t)+f(t,u)=0, t∈(0,1), u (k) (0)=0, 0≤k≤n-2, u′(1)=c, where c is a non negative real number; f(t,u) is non negative and continuous on (0,1) ×(0,∞) and nonincreasing in u .

Quenching for a nonlinear parabolic equation with nonlinear singular boundary condition;

In this paper, the authors present such a better method, which is essentially the combination in a clever way of two well known methods: (1) global transformation of nonlinear system; (2) multi-time scale decomposition.

Using augmented matrix, computers can select linearly independent columns(rows) of a linearly nonsingular transformation matrix and achieve structural decompositionsof linear system.