<Emphasis Type="Italic">m</Emphasis> Components-admissible solutions of systems of higher-order partial differential equations on <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Using value distribution theory and techniques in several complex variables,we investigate the problem of existence of m components-admissible solutions of a class of systems of higher-order partial differential equations in several complex variables and estimate the number of admissible components of solutions.Some related results will also be obtained.     

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

The two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.     

Extended Gauss–Markov Theorem for Nonparametric Mixed-Effects Models

The Gauss–Markov theorem provides a golden standard for constructing the best linear unbiased estimation for linear models. The main purpose of this article is to extend the Gauss–Markov theorem to include nonparametric mixed-effects models. The extended Gauss–Markov estimation (or prediction) is shown to be equivalent to a regularization method and its minimaxity is addressed. The resulting Gauss–Markov estimation serves as an oracle to guide the exploration for effective nonlinear estimators adaptively. Various examples are discussed. Particularly, the wavelet nonparametric regression example and its connection with a Sobolev regularization is presented.     

On non-Newtonian p-fluids. The pseudo-plastic case

In the following we study a class of stationary Navier-Stokes equations with shear dependent viscosity, under the non-slip (Dirichlet) boundary condition. We consider pseudo-plastic fluids. A fluid is said pseudo-plastic, or shear thinning, if in Eq. (1.1) below one has p<2. We are interested in global (i.e., up to the boundary) regularity results, in dimension n=3, for the second order derivatives of the velocity and the first order derivatives of the pressure. We consider a cubic domain Ω and impose the non-slip boundary condition only on two opposite faces. On the other faces we assume periodicity, as a device to avoid effective boundary conditions. This choice is made so that we work in a bounded domain Ω and simultaneously with a flat boundary.     

Numerical examination of the extended phase‐space volume‐preserving integrator by the Nosé‐Hoover molecular dynamics equations

This article illustrates practical applications to molecular dynamics simulations of the recently developed numerical integrators [Phys Rev E 2006, 73, 026703] for ordinary differential equations. This method consists of extending any set of ordinary differential equations in order to define a time invariant function, and then use the techniques of divergence‐free solvable decomposition and symmetric composition to obtain volume‐preserving integrators in the extended phase space. Here, we have developed the technique by constructing multiple extended‐variable formalism in order to enhance the handling in actual simulation, and by constituting higher order integrators to obtain further accuracies. Using these integrators, we perform constant temperature molecular dynamics simulations of liquid water, liquid argon and peptide in liquid water droplet. The temperature control is obtained through an extended version of the Nosé‐Hoover equations. Analyzing the effects of the simulation conditions including time step length, initial values, boundary conditions, and equation parameters, we investigate local accuracy, global accuracy, computational cost, and sensitivity along with the sampling validity. According to the results of these simulations, we show that the volume‐preserving integrators developed by the current method are more effective than traditional integrators that lack the volume‐preserving property. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Numerical simulation of tidal bore in Hangzhou Gulf and Qiantangjiang

In this paper, a new numerical model, based on a set of non‐linear shallow water equations is developed for the simulation of the formation and evolution of tidal bore in the Hangzhou Gulf and Qiantangjiang river of China. The numerical method and boundary conditions are described in detail. The method is validated against analytical solutions and experimental data. Simulation of the actual tidal bore in Hangzhou Gulf and its propagation in the Qiantangjiang river are performed. Numerical results show that this proposed method is effective for the prediction of tidal bore and current flow at the entrance of Qiantangjiang river. Copyright © 2001 John Wiley & Sons, Ltd.     

Arithmetic progressions on Pell equations

In this paper we consider arithmetic progressions on Pell equations, i.e. integral solutions (X,Y) whose X-coordinates or Y-coordinates are in arithmetic progression.     

A note on the contact angle boundary condition for Monge-Ampère equations

We give a simple proof of a result of Xinan Ma concerning a necessary condition for the solvability of the two-dimensional Monge-Ampère equation subject to the contact angle or capillarity boundary condition. Our technique works for more general Monge-Ampère equations in any dimension, and also extends to some other boundary conditions.