Let $T_n$ be the number of triangles in the random intersection graph G(n, m, p). When the mean of $T_n$ is bounded, we obtain an upper bound on the total variation distance between $T_n$ and a Poisson distribution. When the mean of $T_n$ tends to infinity, the Stein–Tikhomirov method is used to bound the error for the normal approximation of $T_n$ with respect to the Kolmogorov metric.

Area-preserving parameterization is now widely applied, such as for remeshing and medical image processing. We propose an efficient and stable approach to compute area-preserving parameterization on simply connected open surfaces. From an initial parameterization, we construct an objective function of energy. This consists of an area distortion measure and a new regularization, termed as the Tutte regularization, combined into an optimization problem with sliding boundary constraints. The original area-preserving problem is decomposed into a series of subproblems to linearize the boundary constraints. We design an iteration framework based on the augmented Lagrange method to solve each linear constrained subproblem. Our method generates a high-quality parameterization with area-preserving on facets. The experimental results demonstrate the efficacy of the designed framework and the Tutte regularization for achieving a fine parameterization.

In this paper, we establish a second-order necessary conditions for stochastic optimal control for jump diffusions. The controlled system is described by a stochastic differential systems driven by Poisson random measure and an independent Brownian motion. The control domain is assumed to be convex. Pointwise second-order maximum principle for controlled jump diffusion in terms of the martingale with respect to the time variable is proved. The proof of the main result is based on variational approach using the stochastic calculus of jump diffusions and some estimates on the state processes.

In this paper, we consider a biharmonic equation with respect to the Dirichlet problem on a domain of a locally finite graph. Using the variation method, we prove that the equation has two distinct solutions under certain conditions.

Regression analysis of interval-censored failure time data has recently attracted a great deal of attention partly due to their increasing occurrences in many fields. In this paper, we discuss a type of such data, multivariate current status data, where in addition to the complex interval data structure, one also faces dependent or informative censoring. For inference, a sieve maximum likelihood estimation procedure is developed and the proposed estimators of regression parameters are shown to be asymptotically consistent and efficient. For the implementation of the method, an EM algorithm is provided, and the results from an extensive simulation study demonstrate the validity and good performance of the proposed inference procedure. For an illustration, the proposed approach is applied to a tumorigenicity experiment.

The time series model with threshold characteristics under fully observations has been explored intensively in recent years. In this article, several methods are proposed to estimate the parameters of the self-exciting threshold integer-valued autoregressive (SETINAR(2,1)) process in the presence of completely random missing data. In order to dispose of the non-equidistance in the observed data, we research the conditional least squares and conditional maximum likelihood inference based on the p-step-ahead conditional distribution of incomplete observations; in addition, three kinds of imputation methods are investigated to deal with the missing values for estimating the parameters of interest. Multiple groups of stochastic simulation studies are carried out to compare the proposed approaches.

In this article, we prove that a quasi-isometric map between rank one symmetric spaces is within bounded distance from an f-harmonic map.