Sentence examples for points of random from inspiring English sources

In probabilistic analysis theory, the problem of finding random fixed points of random operators is an important issue.

Recently Beg [5, 6], Beg and Shahzad [7] and many other authors have studied the fixed points of random maps.

Furthermore, the interval of failure probability can be calculated by substituting contribution bounds of interval variables and sample points of random variables into univariate function.

Nanostructure fabrication may substantially enhance the bending strength of glasses with higher defect densities by redistributing stress from the crack initiation points of random defects.

Under Definition 4.3 we can establish the relations among supporting functionals, points of random smoothness and Gâteaux differentiability of random norms.

Throughout this article, we denote the set of all random fixed points of random mapping T by RF(T), the set of all fixed points of T by F(T for each ω ∈ Ω, respectively. Definition 2.1.

By Proposition 3.6 (x_{0}) is a point of random smoothness of (U E)).

E is said to be random smooth if each point of (S(E)) is a point of random smoothness of (U E)).

(x_{0}) is a point of random smoothness of (U E)) if and only if (Vert cdot Vert ) is Gâteaux differentiable at (x_{0}); If (x_{0}) is a point of random smoothness of (U E)) and f in (S(E^{ast})) supports (U E)) at (x_{0}), then (I_{A_{f}}G(x_{0},y)=operatorname{Re}I_{A_{f}}f y)).

Then the following statements hold: (1) (x_{0}) is a point of random smoothness of (U E)) if and only if (Vert cdot Vert ) is Gâteaux differentiable at (x_{0});   (2) If (x_{0}) is a point of random smoothness of (U E)) and f in (S(E^{ast})) supports (U E)) at (x_{0}), then (I_{A_{f}}G(x_{0},y)=operatorname{Re}I_{A_{f}}f y)).  .

The fundamental difference in the design of the following trials is the time point of random assignment.