We propose a deep learning-based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz Method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning. We illustrate the method on several problems including some eigenvalue problems.

A Berry–Esseen bound is obtained for self-normalized martingales under the assumption of finite moments. The bound coincides with the classical Berry–Esseen bound for standardized martingales. An example is given to show the optimality of the bound. Applications to Student’s statistic and autoregressive process are also discussed.

In this paper, we analyze the asymptotic behavior of solution sequences of the Liouville-type equation with Neumann boundary condition. In particular, we will obtain a sharp mass quantization result for the solution sequences at a blow-up point.

Let R be a prime ring of characteristic different from 2, $Q_r$ be its right Martindale quotient ring and C be its extended centroid, G be a nonzero X-generalized skew derivation of R, and S be the set of the evaluations of a multilinear polynomial $f(x_1,\ldots ,x_n)$ over C with n non-commuting variables. Let $u,v \in R$ be such that $uG(x)x+G(x)xv=0$ for all $x\in S$. Then one of the following statements holds:

In this paper, the optimum test plan and parameter estimation for 3-step step-stress accelerated life tests in the presence of modified progressive Type-I censoring are discussed. It is assumed that the lifetime of test units follows a Lomax distribution with log of characteristic life being quadratic function of stress level. The maximum likelihood and Bayesian method are used to obtain the point and interval estimators of the model parameters. The Bayes estimates are obtained using Markov chain Monte Carlo simulation based on Gibbs sampling. The optimum plan for 3-step step-stress test under modified progressive Type-I censoring is developed which minimizes the asymptotic variance of the maximum likelihood estimators of log of scale parameter at design stress. Finally, the numerical study with sensitivity analysis is presented to illustrate the proposed study.

In this paper, the properties and goodness of fit of exponential ratio type estimator proposed by Khan et al. (Sci Int (Lahore) 26(5):1897–1902, 2014) have been carried out through simulation. It has been shown that estimator is consistent; the convergence of mean and variance of the estimator are discussed. The properties of some existing estimators based on simulation study have also been discussed. Goodness of fit test on the estimator has been carried out using Kolmogorov–Smirnov, Anderson–Darling, Cramer–von Mises and Chi-square statistics. Few standard distributions such as normal, Student t, Weibull and Cauchy distributions are fitted on the estimator. The best-fitted distribution on the estimator turned out to be normal distribution.