Abstract:
This paper is concerned with general nonlinear regression models where the predictor variables are subject to Berkson-type measurement errors. The measurement errors are assumed to have a general parametric distribution, which is not necessarily normal. In addition, the distribution of the random error in the regression equation is nonparametric. A minimum distance estimator is proposed, which is based on the first two conditional moments of the response variable given the observed predictor variables. To overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals, a simulation-based estimator is constructed. Consistency and asymptotic normality for both estimators are derived under fairly general regularity conditions.

Abstract:
The aim of this study is to define the anatomical localization of corresponding brain function area during calculating. The activating modes in brain during continuous silent calculating subtraction and repeated silent reading multiplication table were compared and investigated. Fourteen volunteers of right-handedness were enrolled in this experiment. The quite difference of reaction modes in brain area during the two modes of calculation reveal that there are different processing pathways in brain during these two operating actions. During continuous silent calculating, the function area is localized on the posterior portion of superior and middle gyrus of frontal lobe and the lobule of posterior parietal lobe (P < 0.01,T = 5.41). It demonstrates that these function areas play an important role in the performance of calculation and working memory. Whereas the activating of visual cortex shows that even in mental arithmetic processing the brain action is having the aid of vision and visual space association.

Abstract:
We present a new second-order accurate numerical method for solving matrix coefficient elliptic equation on irregular domains with sharp-edged boundaries. Nontraditional finite element method with non-body-fitting grids is implemented on a fictitious domain in which the irregular domains are embedded. First we set the function and coefficient in the fictitious part, and the nonsmooth boundary is then treated as an interface. The emphasis is on the construction of jump conditions on the interface; a special position for the ghost point is chosen so that the method is more accurate. The test function basis is chosen to be the standard finite element basis independent of the interface, and the solution basis is chosen to be piecewise linear satisfying the jump conditions across the interface. This is an efficient method for dealing with elliptic equations in irregular domains with non-smooth boundaries, and it is able to treat the general case of matrix coefficient. The complexity and computational expense in mesh generation is highly decreased, especially for moving boundaries, while robustness, efficiency, and accuracy are promised. Extensive numerical experiments indicate that this method is second-order accurate in the norm in both two and three dimensions and numerically very stable. 1. Introduction Let ( ) be an open-bounded domain with a Lipschitz continuous boundary . We consider the variable coefficient elliptic equation where refers to the spatial variable, is the gradient operator, and the right-hand side is assumed to lie in . The coefficient is a matrix that is uniformly elliptic, and its entries are continuously differentiable on . For a given function on the boundary , the Dirichlet boundary condition is prescribed as Elliptic partial differential equations are often used to construct models of the most basic theories underlying physics and engineering, such as electromagnetism, material science, and fluid dynamics. Different kinds of boundary conditions arise with the wide range of applications, such as Dirichlet boundary condition, Neumann boundary condition, and Robin boundary condition. Elliptic equation on irregular domains has been studied by many researchers and several techniques have been developed. Finite element methods use a mesh triangulation to capture the boundary [1–4]. However, in many situations, such as when the boundary is moving, the mesh generation may be both computational expensive and challenging. A more preferred method is to combine the Cartesian grid method with level-set approach [5–7] to capture the boundary.

Abstract:
We obtain the $C^{\a}$ regularity for weak solutions of a class of non-homogeneous ultraparabolic equation, with measurable coefficients. The result generalizes our recent $C^{\a}$ regularity results of homogeneous ultraparabolic equation.

Abstract:
We obtained the $C^{\a}$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form ${\ptl_t u}= \sum_{i,j=1}^{m_0}X_i(a_{ij}(x,t)X_j u)+X_0 u$. The result is proved by simplifying and generalizing our earlier arguments for the $C^{\a}$ regularity of homogeneous ultraparabolic equations.

Abstract:
The backward uniqueness of the Kolmogorov operator $L=\sum_{i,k=1}^n\partial_{x_i}(a_{i,k}(x,t)\partial_{x_k})+\sum_{l=1}^m x_l\partial_{y_l}-\partial_t$, was proved in this paper. We obtained a weak Carleman inequality via Littlewood-Paley decomposition for the global backward uniqueness.

Abstract:
We propose an approach to approximate the boundary crossing probabilities for general one-dimensional diffusion processes, and derive the convergence rate for this approximation scheme. There results are based on the explicit expression of the Laplace transforms of the first passage densities for diffusions with piecewise linear drifts. The proposed method is applied to a reliability problem where the standard degradation model based on Wiener process is extended to diffusion processes with piecewise linear drifts.

Abstract:
We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or piecewise linear boundaries in any finite interval. We then use these formulas to approximate the boundary crossing probabilities for general nonlinear boundaries. The jump process can be any integer-valued process and jump sizes can have general distributions. Moreover, the jump sizes can be even correlated and/or non-identically distributed. The numerical algorithm is straightforward and easy to implement. Some numerical examples are presented.

Abstract:
We obtained the $C^{\a}$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form $${\ptl_t u}= \ptl_x(a(x,y,t)\ptl_x u)+b_0(x,y,t)\ptl_x u+b(x,y,t)\ptl_y u,$$ which generalized our recent results on KFP equations.

Abstract:
Particle filtering algorithm has been applied to various fields due to its capacity to handle nonlinear/non-Gaussian dynamic problems. One crucial issue in particle filtering is the selection of the proposal distribution that generates the particles. In this paper, we give a novel strategy for selecting proposal distribution. Firstly, divide-conquer strategy is used, in which the particles used are divided into several parts. Afterward, different parts of particles are drawn from different proposal distributions. People can flexibly adjust how many of the particles drawn from specific proposal distributions according to their idiographic requirements. We provide simulation results that show its efficiency and performance.