We study a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, which is based on an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the studied algorithm for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen–Cahn equation, the Hamilton–Jacobi–Bellman equation, and a nonlinear pricing model for financial derivatives.

Recently, it was found that during the process of certain refinement of hierarchical T-meshes, some basis functions of PHT-splines decay severely, which is not expected in solving numerical PDEs and in least square data fitting since the matrices assembled by these basis functions are likely to be ill-conditioned. In this paper, we present a method to modify the basis functions of PHT-splines in the case that the supports of the original truncated basis functions are rectangular domains to overcome the decay problem. The modified basis functions preserve the same nice properties of the original PHT-spline basis functions such as partition of unity, local support, linear independency. Numerical examples show that the modified basis functions can greatly decrease the condition numbers of the stiffness matrices assembled in solving Poisson’s equation with Dirichlet boundary conditions.

Let G be a group and $G=G_1G_2$ where $G_i$ are subgroups of G. In this paper, we investigate the structure of G under the conditions that some subgroups of $G_i$ are subnormal in G.

Let f(z) be a non-constant meromorphic function of finite order, $c\in \mathbb {C}\setminus \{0\}$ and $k\in \mathbb {N}$. Suppose f(z) and $f^{(k)}(z+c)$ share 1 CM (IM), f(z) and $f(z+c)$ share $\infty $ CM. If $N(r,0;f)=S(r,f) \left( N\left( r,0;f(z)\right) +N\left( r,0;f^{(k)}(z+c)\right) =S(r,f)\right) $, then either $f(z)\equiv f^{(k)}(z+c)$ or f(z) is a solution of the following equation:

This paper presents exponential-type ratio and product estimators for a finite population mean in double sampling using information on several auxiliary variates. The proposed estimators can be viewed as a generalization over the estimators suggested by Singh and Vishwakarma (Austrian J Stat 36(3):217–225, 2007). The expressions for biases and mean square errors (MSEs) of the proposed estimators have been derived to the first degree of approximation. In addition, the expressions for minimum attainable MSEs are also investigated using the criterion for optimality of the weights. An empirical study is carried out in the support of the present study. Both theoretical and empirical findings are encouraging and support the soundness that the proposed procedures for mean estimation perform better than the usual unbiased estimators and other well-known estimators under some realistic conditions.

A new class ${ CP}_2$ groups of finite groups was characterized by using an inequality of the orders of elements. In this short paper we give a note of ${ CP}_2$ groups since ${ CP}_2$ groups is a subclass of ${ CP}$(${ EPPO}$) groups. Moreover, we discuss the structure of finite p groups contained in ${ CP}_2$ groups.