The objective of this paper is to introduce elementary discrete reflected backward equations and to give a simple method to discretize in time a (continuous) reflected backward equation. A presentation of numerical simulations is also described.
The long time behavior of solutions of the generalized Hasegawa-Mima equation with dissipation term is considered. The existence of global attractors of the periodic initial value problem is proved, and the estimate of the upper bound of the Hausdorff and fractal dimensions for the global attractors is obtained by means of uniform a priori estimates method.
In this paper, an optimal control problem governed by semilinear parabolic equation which involves the control variable acting on forcing term and coefficients appearing in the higher order derivative terms is formulated and analyzed. The strong variation method, due originally to Mayne et al to solve the optimal control problem of a lumped parameter system, is extended to solve an optimal control problem governed by semilinear parabolic equation, a necessary condition is obtained, the strong variation algorithm for this optimal control problem is presented, and the corresponding convergence result of the algorithm is verified.
In this paper, we implement alternating direction strategy and construct a symmetric FVE scheme for nonlinear convection--diffusion problems. Comparing to general FVE methods, our method has two advantages. First, the coefficient matrices of the discrete schemes will be symmetric even for nonlinear problems. Second, since the solution of the algebraic equations at each time step can be inverted into the solution of several one-dimensional problems, the amount of computation work is smaller. We prove the optimal $H^1$-norm error estimates of order $O(\Delta t^2+h)$ and present some numerical examples at the end of the paper.
In a paper by Zhang and Chen et al.(see [11]), a conjecture was made concerning the minimum number of colors $\chi_{at}(G)$ required in a proper total-coloring of $G$ so that any two adjacent vertices have different color sets, where the color set of a vertex $v$ is the set composed of the color of $v$ and the colors incident to $v$. We find the exact values of $\chi_{at}(G)$ and thus verify the conjecture when $G$ is a Generalized Halin graph with maximum degree at least $6$. A generalized Halin graph is a 2-connected plane graph $G$ such that removing all the edges of the boundary of the exterior face of $G$ (the degrees of the vertices in the boundary of exterior face of $G$ are all three) gives a tree.
In this paper, we focus our attention on the precise asymptotics of error variance estimator in partially linear regression models, $y_i=x_i^\tau \beta+g(t_i)+\e_i,1\le i \le n$, $ \{\e_i, i=1,\cdots, n\}$ are $i.i.d$ random errors with mean 0 and positive finite variance $\sigma^2$. Following the ideas of Allan Gut and Aurel Sp$\breve{a}$taru$^{[7,8]}$ and Zhang$^{[21]}$, on precise asymptotics in the Baum-Katz and Davis laws of large numbers and precise rate in laws of the iterated logarithm, respectively, and subject to some regular conditions, we obtain the corresponding results in partially linear regression models.
Consider a storage model fed by a Markov modulated Brownian motion. We prove that the stationary distribution of the model exits and that the running maximum of the storage process over the interval $[0, t]$ grows asymptotically like $\log t$ as $t\rightarrow \infty$.
Quantile regression is gradually emerging as a powerful tool for estimating models of conditional quantile functions, and therefore research in this area has vastly increased in the past two decades. This paper, with the quantile regression technique, is the first comprehensive longitudinal study on mathematics participation data collected in Alberta, Canada. The major advantage of longitudinal study is its capability to separate the so-called cohort and age effects in the context of population studies. One aim of this paper is to study whether the family background factors alter performance on the mathematical achievement of the strongest students in the same way as that of weaker students based on the large longitudinal sample of 2000, 2001 and 2002 mathematics participation longitudinal data set. The interesting findings suggest that there may be differential family background factor effects at different points in the mathematical achievement conditional distribution.
This paper studies estimation and serial correlation test of a semiparametric varying-coefficient partially linear EV model of the form $Y=X^{\tau}\beta+Z^{\tau}\alpha(T)+\varepsilon, \xi= X+\eta$ with the identifying condition $E[(\varepsilon,\eta^{\tau})^{\tau}]=0,$ Cov$[(\varepsilon,\eta^{\tau})^{\tau}]=\sigma^{2}I_{p+1}$. The estimators of interested regression parameters $\beta$ , and the model error variance $\sigma^{2}$, as well as the nonparametric components $\alpha(T)$, are constructed. Under some regular conditions, we show that the estimators of the unknown vector $\beta$ and the unknown parameter $\sigma^{2}$ are strongly consistent and asymptotically normal and that the estimator of $\alpha(T)$ achieves the optimal strong convergence rate of the usual nonparametric regression. Based on these estimators and asymptotic properties, we propose the $V_{N,p}$ test statistic and empirical log-likelihood ratio statistic for testing serial correlation in the model. The proposed statistics are shown to have asymptotic normal or chi-square distributions under the null hypothesis of no serial correlation. Some simulation studies are conducted to illustrate the finite sample performance of the proposed tests.
In this paper we mainly study the ruin probability of a surplus process described by a piecewise deterministic Markov process (PDMP). An integro-differential equation for the ruin probability is derived. Under a certain assumption, it can be transformed into the ruin probability of a risk process whose premiums depend on the current reserves. Using the same argument as that in Asmussen and Nielsen$^{[2]}$, the ruin probability and its upper bounds are obtained. Finally, we give an analytic expression for ruin probability and its upper bounds when the claim-size is exponentially distributed.
In this paper, the existence and uniqueness of time-periodic generalized solutions and time-periodic classical solutions to a class of parabolic type equation of higher order are proved by Galerkin method.
In this paper, the property of practical input-to-state stability and its application to stability of cascaded nonlinear systems are investigated in the stochastic framework. Firstly, the notion of (practical) stochastic input-to-state stability with respect to a stochastic input is introduced, and then by the method of changing supply functions, (a) an (practical) SISS-Lyapunov function for the overall system is obtained from the corresponding Lyapunov functions for cascaded (practical) SISS subsystems.
We establish an improved GP iterative algorithm for the extrapolation of band-limited function to fully 3-dimensional image reconstruction by the convolution-backprojection algorithm. Numerical experiments demonstrate that the image resolving power of IGP algorithm is better than that of the original GP algorithm for noisy data.
Using qualitative analysis and numerical simulation, we investigate the number and distribution of limit cycles for a cubic Hamiltonian system with nine different seven-order perturbed terms. It is showed that these perturbed systems have the same distribution of limit cycles. Furthermore, these systems have 13, 11 and 9 limit cycles for some parameters, respectively. The accurate positions of the 13, 11 and 9 limit cycles are obtained by numerical exploration, respectively.