The Burgers’ equation is an important and basic nonlinear partial differential equation in fluid dynamics, and has been used as a model equation in other fields, such as modeling of shock waves, gas dynamics, turbulence, and large bubble structures consisting of clusters of galaxies in space. Many researchers have proposed various numerical methods for solving the Burgers’ equation, such as the...

A direct solver for large scale sparse linear system of equations is presented in this paper. As a direct solver, this method is among the most efficient direct solvers available so far with flop count as O(n logn) in one-dimensional situations and O(n) in second dimensional situation. This method has advantages over the existing fast solvers in which it can be used to handle more general situa...

Long intermediate AR models are used in Durbin's algorithms for ARMA estimation. The order of that long AR model is infinite in the asymptotical theory, but very high AR orders are known to give inaccurate ARMA models in practice. A theoretical derivation is given for two different finite AR orders, as a function of the sample size. The first is the AR order optimal for prediction with a purely...

The second moment method has been fruitfully combined with conditioning by several authors, for example Robinson and Wormald 7], 8] and Frieze and Janson 2]. We try to give a general exposition of this method, and of its relation to approximations of random variables by simpler variables. We discuss several examples where the method has been used in diierent forms. 1. The second moment method O...