LIMIT OF ITERATES FOR BERNSTEIN POLYNOMIALS DEFINED ON A TRIANGLE

Let f ∈ C[0,1] , and Bn(f,x) be the n-th Bernstein polynomial associated with function f. In 1967, the limit of iterates for Bn(f,x) was given by Kelisky and Rivlin. After this, Many mathematicians studied and generalized this result. But anyway, all these discussions are only for univariate case. In this paper, the main contribution is that the limit of iterates for Bernstein polynomial defined on a triangle is given completely.

Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm

Kolmogorov's exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables. This paper establishes the Kolmogorov type exponential inequalities of the partial sums of independent random variables as well as negatively dependent random variables under the sub-linear expectations. As applications of the exponential inequalities, the laws of the iterated logarithm in the sense of non-additive capacities are proved for independent or negatively dependent identically distributed random variables with finite second order moments.For deriving a lower bound of an exponential inequality, a central limit theorem is also proved under the sublinear expectation for random variables with only finite variances.

Finite element reliability analysis of slope stability

The method of nonlinear finite element reliability analysis (FERA) of slope stability using the technique of slip surface stress analysis (SSA) is studied. The limit state function that can consider the direction of slip surface is given, and the formula-tions of FERA based on incremental tangent stiffness method and modified Aitken accelerating algorithm are developed. The limited step length iteration method (LSLIM) is adopted to calculate the reliability index. The nonlinear FERA code using the SSA technique is developed and the main flow chart is illustrated. Numerical examples are used to demonstrate the efficiency and robustness of this method. It is found that the accelerating convergence algorithm proposed in this study proves to be very efficient for it can reduce the iteration number greatly, and LSLIM is also efficient for it can assure the convergence of the iteration of the reliability index.

A New Hybrid Reliability Index Definition and Its Application to the Structure Buckling Reliability Analysis of Supercavitating Projectiles

As structure buckling problems easily arise when supercavitating projectiles operate with high underwater velocity, it is necessary to perform structure buckling reliability analysis. Now it is widely known that probabilistic and non-probabilistic uncertain information exists in engineering analysis. Based on reliability comprehensive index of multi-ellipsoid convex set, probabilistic uncertain information is added and transferred into non-probabilistic interval variable. The hybrid reliability is calculated by a combined method of modified limit step length iteration algorithm(MLSLIA) and Monte-Carlo method. The results of engineering examples show that the convergence of MLSLIA is better than that of limit step length iteration algorithm(LSLIA). Structure buckling hybrid reliability increases with the increase of ratio of base diameter to cavitator diameter, and decreases with the increase of initial launch velocity. Also the changes of uncertain degree of projectile velocity and cavitator drag coefficient affect structure buckling hybrid reliability index obviously. Therefore, uncertain degree of projectile velocity and cavitator drag coefficient should be controlled in project for high structure buckling reliability.