We consider the stability of the steady state of the compressible Navier-Stokes-Poisson equations with the non-flat doping profile. We prove the global existence of classical solutions near the steady state for the large doping profile. For the small doping profile, we prove the time decay rates of the solution provided that the initial perturbation belongs to L p with 1 p < 3/2.

Summary A new fast algorithm for computing the nonparametric maximum likelihood estimate of a univariate log‐concave density is proposed and studied. It is an extension of the constrained Newton method for nonparametric mixture estimation. In each iteration, the newly extended algorithm includes, if necessary, new knots that are located via a special directional derivative function. The algorithm renews the changes of slope at all knots via a quadratically convergent method and removes the knots at which the changes of slope become zero. Theoretically, the characterisation of the nonparametric maximum likelihood estimate is studied and the algorithm is guaranteed to converge to the unique maximum likelihood estimate. Numerical studies show that it outperforms other algorithms that are available in the literature. Applications to some real‐world financial data are also given.

The low Mach limit for 1D non-isentropic compressible Navier-Stokes flow, whose density and temperature have different asymptotic states at infinity, is rigorously justified. The problems are considered on both well-prepared and ill-prepared data. For the well-prepared data, the solutions of compressible Navier-Stokes equations are shown to converge to a nonlinear diffusion wave solution globally in time as Mach number goes to zero when the difference between the states at ±∞ is suitably small. In particular, the velocity of diffusion wave is only driven by the variation of temperature. It is further shown that the solution of compressible Navier-Stokes system also has the same property when Mach number is small, which has never been observed before. The convergence rates on both Mach number and time are also obtained for the well-prepared data. For the ill-prepared data, the limit relies on the uniform estimates including weighted time derivatives and an extended convergence lemma. And the difference between the states at ±∞ can be arbitrary large in this case.(1.5) which satisfies the following equation ∂ t (P ε ) + div(u ε P ε ) + P ε divu ε = div(κ∇T ε ) + ε 2 [2µ|D(u ε )| 2 + λ|divu ε | 2 ].(1.6)

Data quality, including record inaccuracy and missingness (incompletely recorded crashes and crash underreporting), has always been of concern in crash data analysis. Limited efforts have been made to handle some specific aspects of crash data quality problems, such as using weights in estimation to take care of unreported crash data and applying multiple imputation (MI) to fill in missing information of drivers’ status of attention before crashes. Yet, there lacks a general investigation of the performance of different statistical methods to handle missing crash data. This paper is intended to explore and evaluate the performance of three missing data treatments, which are complete-case analysis (CC), inverse probability weighting (IPW) and MI, in crash severity modeling using the ordered probit model. CC discards those crash records with missing information on any of the variables; IPW includes weights in estimation to adjust for bias, using complete records’ probability of being a complete case; and MI imputes the missing values based on the conditional distribution of the variable with missing information on the observed data. Those missing data treatments provide varying performance in model estimations. Based on analysis of both simulated and real crash data, this paper suggests that the choice of an appropriate missing data treatment should be based on sample size and data missing rate. Meanwhile, it is recommended that MI is used for incompletely recorded crash data and IPW for unreported crashes, before applying crash severity models on crash data.

The simplified governing equations of applied mechanics play a pivotal role and were derived based on ingenious assumptions or hypotheses regarding the displacement fields for specific problems. In this paper, we introduce a data-driven method by the name AI-Timoshenko in honor of Timoshenko, father of applied mechanics, to automatically discover simplified governing equations for applied mechanics problems directly from discrete data simulated by the 3D finite element method. This liberates applied mechanicians from burdensome labor, including assumptions, derivation, and trial and error. The simplified governing equations for Euler-Bernoulli and Timoshenko beam theories are successfully rediscovered using the present AI-Timoshenko method, which shows that this method is capable of discovering simplified governing equations for applied mechanics problems.