The temporal development of a single mode Rayleigh-Taylor instability consists of three stages: the linear, free fall and terminal velocity regimens. The purpose of this paper is to report on new phenomena observed in the approach to terminal velocity. Our numerical study shows an unexpected nonuniform approach to terminal velocity. The nonuniformity applies especially to the spikes, which are fingers of heavy fluid falling into the light fluid, but it also applies to the rising bubblesof light fluid. For spikes especially, our results call into question the meaningfulness of a terminal velocity for moderate values of the Atwood number A. After a short time period of pseudo-terminal plateau, the spike velocity increases to a significantly higher maximum, followed by a decrease. This phenomena appears to be due to a slow evolution in the shape of the spike and bubble. We find a relation between the spike (bubble) acceleartion and the tip curvature. In correlation with an increase in the spike velocity, the main body of the spike becomes narrower and the tip curvature increases. Our numerical results are by the Front Tracking method. The very late time simulations considered here required stabilization by a small value for the viscosity, so that the compressible Navier-Stokes equations govern the dynamics.

The temporal development of a single mode Rayleigh-Taylor instability consists of three stages: the linear, free fall and terminal velocity regimens. The purpose of this paper is to report on new phenomena observed in the approach to terminal velocity. Our numerical study shows an unexpected nonuniform approach to terminal velocity. The nonuniformity applies especially to the spikes, which are fingers of heavy fluid falling into the light fluid, but it also applies to the rising bubblesof light fluid. For spikes especially, our results call into question the meaningfulness of a terminal velocity for moderate values of the Atwood number A. After a short time period of pseudo-terminal plateau, the spike velocity increases to a significantly higher maximum, followed by a decrease. This phenomena appears to be due to a slow evolution in the shape of the spike and bubble. We find a relation between the spike (bubble) acceleartion and the tip curvature. In correlation with an increase in the spike velocity, the main body of the spike becomes narrower and the tip curvature increases. Our numerical results are by the Front Tracking method. The very late time simulations considered here required stabilization by a small value for the viscosity, so that the compressible Navier-Stokes equations govern the dynamics.

In this note, we present a method of constructing the homogenized operator for a general sequence of differential operators. As an example, we construct the homogenized operator for a sequence of linear parabolic operators.

In this note, we present a method of constructing the homogenized operator for a general sequence of differential operators. As an example, we construct the homogenized operator for a sequence of linear parabolic operators.

We analyze mathematical models governing planar flow of chemical reaction from unburnt gases to burnt gases in certain physical regimes in which diffusive effects such as viscosity and heat conduction are significant. These models can be then formulated as the Navier-Stokes equations for exothermically reacting compressible fluids. We first establish the existence and dynamic behavior, including stability, regularity, and large-time behavior, of global discontinuous solutions of large oscillation to the Navier-Stokes equations with constant adiabatic exponent γ and specific heat c_v. Our approach for the existence and regularity is to combine the difference approximation techniques with the energy methods, total variation estimates, and weak convergence arguments to deal with large jump discontinuities; and for large-time behavior is an a posteriori argument directly from the weak form of the equations. The approach and ideas we develop here can be applied to solving a more complicated model where γ and c_v vary as the phase changes; and we then describe this model in detail and contrast the results on the asymptotic behavior of the solutions of these two different models. We also discuss other physical models describing dynamic combustion.

We analyze mathematical models governing planar flow of chemical reaction from unburnt gases to burnt gases in certain physical regimes in which diffusive effects such as viscosity and heat conduction are significant. These models can be then formulated as the Navier-Stokes equations for exothermically reacting compressible fluids. We first establish the existence and dynamic behavior, including stability, regularity, and large-time behavior, of global discontinuous solutions of large oscillation to the Navier-Stokes equations with constant adiabatic exponent γ and specific heat c_v. Our approach for the existence and regularity is to combine the difference approximation techniques with the energy methods, total variation estimates, and weak convergence arguments to deal with large jump discontinuities; and for large-time behavior is an a posteriori argument directly from the weak form of the equations. The approach and ideas we develop here can be applied to solving a more complicated model where γ and c_v vary as the phase changes; and we then describe this model in detail and contrast the results on the asymptotic behavior of the solutions of these two different models. We also discuss other physical models describing dynamic combustion.

In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jin and Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weakly parabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion.We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system for the Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme, and those obained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and by the extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments show that the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equation obtained by the DSMC, for a range of Mach numbers for hypersonic flows, than those obtained by the other hydrodynamic systems.

In [16] a visco-elastic relaxation system, called the relaxed Burnett system, was proposed by Jin and Slemrod as a moment approximation to the Boltzmann equation. The relaxed Burnett system is weakly parabolic, has a linearly hyperbolic convection part, and is endowed with a generalized entropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion.We develop a one-dimensional non-oscillatory numerical scheme based on the relaxed Burnett system for the Boltzmann equation. We compare numerical results for stationary shocks based on this relaxation scheme, and those obained by the DSMC (Direct Simulation Monte Carlo), by the Navier-Stokes equations and by the extended thermodynamics with thirteen moments (the Grad equations). Our numerical experiments show that the relaxed Burnett gives more accurate approximations to the shock profiles of the Boltzmann equation obtained by the DSMC, for a range of Mach numbers for hypersonic flows, than those obtained by the other hydrodynamic systems.

In this paper we study some nonoverlapping domain decomposition methods for solving a class of elliptic problems arising from composite materials and flows in porous media which contain many spatial scales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarse solver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domain decomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate in the presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiple-scale problems without restrictive assumptions and seems to have a convergence rate nearly independent of the aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numerical experiments which include problems with multiple-scale coefficients, as well problems with continuous scales.

In this paper we study some nonoverlapping domain decomposition methods for solving a class of elliptic problems arising from composite materials and flows in porous media which contain many spatial scales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarse solver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domain decomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate in the presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiple-scale problems without restrictive assumptions and seems to have a convergence rate nearly independent of the aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numerical experiments which include problems with multiple-scale coefficients, as well problems with continuous scales.

In this paper, a new SQP method for inequality constrained optimization is proposed and the global convergence is obtained under very mild conditions.

In this paper, a new SQP method for inequality constrained optimization is proposed and the global convergence is obtained under very mild conditions.

In the parameter tracking of time-varying systems, the ordinary method is weighted least squares with the rectangular window or the exponential window. In this paper we propose a new kind of sliding window called the multiple exponential window, and then use it to fit time-varying Gaussian vector autoregressive models. The asymptotic bias and covariance of the estimator of the parameter for time-invariant models are also derived. Simulation results show that the multiple exponential windows have better parameter tracking effect than rectangular windows and exponential ones.

In the parameter tracking of time-varying systems, the ordinary method is weighted least squares with the rectangular window or the exponential window. In this paper we propose a new kind of sliding window called the multiple exponential window, and then use it to fit time-varying Gaussian vector autoregressive models. The asymptotic bias and covariance of the estimator of the parameter for time-invariant models are also derived. Simulation results show that the multiple exponential windows have better parameter tracking effect than rectangular windows and exponential ones.

Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at most two limit cycles is false. We give the sufficient conditions for the system has at most three limit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by using numerical simulation.

Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at most two limit cycles is false. We give the sufficient conditions for the system has at most three limit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by using numerical simulation.

We first provide a simple estimate for ||A~(-1)||_∞ and ||A~(-1)||_1 of a strictly diagonally dominant matrix A. On the Basis of the result, we obtain an estimate for the smallest singular value of A. Secondly, by scaling with a positive diagonal matrix D, we obtain some simple estimates for the smallest singular value of an H-matrix, which is not necessarily positive definite. Finally, we give some examples to show the effectiveness of the new bounds.

We first provide a simple estimate for ||A~(-1)||_∞ and ||A~(-1)||_1 of a strictly diagonally dominant matrix A. On the Basis of the result, we obtain an estimate for the smallest singular value of A. Secondly, by scaling with a positive diagonal matrix D, we obtain some simple estimates for the smallest singular value of an H-matrix, which is not necessarily positive definite. Finally, we give some examples to show the effectiveness of the new bounds.

Two Armijo-type line searches are proposed in this paper for nonlinear conjugate gradient methods. Under these line searches, global convergence results are established for several famous conjugate gradient methods, including the Fletcher-Reeves method, the Polak-Ribiére-Polyak method, and the conjugate descent method.

Two Armijo-type line searches are proposed in this paper for nonlinear conjugate gradient methods. Under these line searches, global convergence results are established for several famous conjugate gradient methods, including the Fletcher-Reeves method, the Polak-Ribiére-Polyak method, and the conjugate descent method.

In this paper, we study the high-order upwind finite difference method for steady convection-diffusion problems. Based on the conservative convection-diffusion equation, a high-order upwind finite difference scheme on nonuniform rectangular partition for convection-diffusion equation is proposed. The proposed scheme is in conversation form, satisfies maximum value principle and has second-order error estimates in discrete H~1 norm. To illustrate our conclusion, several numerical examples are given.

In this paper, we study the high-order upwind finite difference method for steady convection-diffusion problems. Based on the conservative convection-diffusion equation, a high-order upwind finite difference scheme on nonuniform rectangular partition for convection-diffusion equation is proposed. The proposed scheme is in conversation form, satisfies maximum value principle and has second-order error estimates in discrete H~1 norm. To illustrate our conclusion, several numerical examples are given.

In this paper we study nonlinear Lagrangian methods for optimization problems with side constraints. Nonlinear Lagrangian dual problems are introduced and their relations with the original problem are established. Moreover, a least root approach is investigated for these optimization problems.

In this paper we study nonlinear Lagrangian methods for optimization problems with side constraints. Nonlinear Lagrangian dual problems are introduced and their relations with the original problem are established. Moreover, a least root approach is investigated for these optimization problems.

In the present paper surplus process perturbed by diffusion are considered. The distributions of the surplus immediately before and at ruin corresponding to the probabilities of ruin caused by oscillation and ruin caused by a claim are studied. Some joint distribution densities are obtained. Techniques from martingale theory and renewal theory are used.

In the present paper surplus process perturbed by diffusion are considered. The distributions of the surplus immediately before and at ruin corresponding to the probabilities of ruin caused by oscillation and ruin caused by a claim are studied. Some joint distribution densities are obtained. Techniques from martingale theory and renewal theory are used.

Under very general weight function, we discuss the convergence of Jamison-type weighted sums of pairwise negatively quadrant dependent (NQD) r.v.'s. The results on i.i.d. setting of [3] and [1] are extended and generalized. As corollaries, we obtain some results of [11].

Under very general weight function, we discuss the convergence of Jamison-type weighted sums of pairwise negatively quadrant dependent (NQD) r.v.'s. The results on i.i.d. setting of [3] and [1] are extended and generalized. As corollaries, we obtain some results of [11].

The multisymplectic structure of the nonlinear wave wquation is derived directly from the variational principle. In the numerical aspect, we present a multisymplectic nine points scheme which is equivalent to the multisymplectic Preissman scheme. A series of numerical results are reported to illustrate the effectiveness of the scheme.

The multisymplectic structure of the nonlinear wave wquation is derived directly from the variational principle. In the numerical aspect, we present a multisymplectic nine points scheme which is equivalent to the multisymplectic Preissman scheme. A series of numerical results are reported to illustrate the effectiveness of the scheme.

I reflect upon the development of nonlinear time series analysis since 1990 by focusing on five major areas of development. These areas include the interface between nonlinear time series analysis and chaos, the nonparametric/semiparametric approach, nonlinear state space modelling, financial time series and nonlinear modelling of panels of time series.

I reflect upon the development of nonlinear time series analysis since 1990 by focusing on five major areas of development. These areas include the interface between nonlinear time series analysis and chaos, the nonparametric/semiparametric approach, nonlinear state space modelling, financial time series and nonlinear modelling of panels of time series.

Some new local and parallel finite element algorithms are proposed and analyzed in this paper for eigenvalue problems. With these algorithms, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a relatively coarse grid together with solutions of some linear algebraic systems on fine grid by using some local and parallel procedure. A theoretical tool for analyzing these algorithms is some local error estimate that is also obtained in this paper for finite element approximations of eigenvectors on general shape-regular grids.

Some new local and parallel finite element algorithms are proposed and analyzed in this paper for eigenvalue problems. With these algorithms, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a relatively coarse grid together with solutions of some linear algebraic systems on fine grid by using some local and parallel procedure. A theoretical tool for analyzing these algorithms is some local error estimate that is also obtained in this paper for finite element approximations of eigenvectors on general shape-regular grids.

In this paper, a dissipative Zakharov equations are discretized by difference method. We make prior estimates for the algebric system of equations. It is proved that for each mesh size, there exist attractors for the discretized system. The bounds of the Hausdorff dimensions of the discrete attractors are obtained, and the various bounds are dependent of the mesh size.

In this paper, a dissipative Zakharov equations are discretized by difference method. We make prior estimates for the algebric system of equations. It is proved that for each mesh size, there exist attractors for the discretized system. The bounds of the Hausdorff dimensions of the discrete attractors are obtained, and the various bounds are dependent of the mesh size.

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. In this article, the maximum size of an acyclic hypergraph is determined and the number of maximum r-uniform acyclic hypergraphs of order n is shown to be

Acyclic hypergraphs are analogues of forests in graphs. They are very useful in the design of databases. In this article, the maximum size of an acyclic hypergraph is determined and the number of maximum r-uniform acyclic hypergraphs of order n is shown to be

For the weakly inhomogeneous acoustic medium in Q={(x, y, z:z>0}, we consider the inverse problem of determining the density function p(x, y). The inversion input for our inverse problem is the wave field given on a line. We get an integral equation for the 2-D density perturbation from the linearization. By virtue of the integral transform, we prove the uniqueness and the instability of the solution to the integral equation. The degree of ill-posedness for this problem is also given.

For the weakly inhomogeneous acoustic medium in Q={(x, y, z:z>0}, we consider the inverse problem of determining the density function p(x, y). The inversion input for our inverse problem is the wave field given on a line. We get an integral equation for the 2-D density perturbation from the linearization. By virtue of the integral transform, we prove the uniqueness and the instability of the solution to the integral equation. The degree of ill-posedness for this problem is also given.