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The two-parameter exponential distribution can often be used to describe the lifetime of products for example, electronic components, engines and so on. This paper considers a prediction problem arising in the life test of key parts in high speed trains. Employing the Bayes method, a joint prior is used to describe the variability of the parameters but the form of the prior is not specified and only several moment conditions are assumed. Under the condition that the observed samples are randomly right censored, we define a statistic to predict a set of future samples which describes the average life of the second-round samples, firstly, under the condition that the censoring distribution is known and secondly, that it is unknown. For several different priors and life data sets, we demonstrate the coverage frequencies of the proposed prediction intervals as the sample size of the observed and the censoring proportion change. The numerical results show that the prediction intervals are efficient and applicable.

Let D be a bounded domain in an n-dimensional Euclidean space R n . Assume that 0 < λ1 λ2 · · · λ k · · · are the eigenvalues of the Dirichlet Laplacian operator with any order l:Then we obtain an upper bound of the (k + 1)-th eigenvalue λ k+1 in terms of the first k eigenvalues.This ineguality is independent of the domain D. Furthermore, for any l 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.

In this paper, we settle the last two open cases of non-existence of full quadratic harmonic maps from S 4 to S 5 or S 6 . Assume that there exist full quadratic harmonic maps from S 4 to S n for some integer n. As a consequence of our theorem we obtain that the sufficient and necessary condition of the existence of such maps is that n satisfy 4 ≤ n ≤ 13 and n = 5, 6.