Similarly, if we calculate the discrete problem of the left hand side equation in (3.2) using the same central

difference operator by taking [[xi].sub.i] = [u.sup.2.sub.i], where u is the central difference computed solution for equation (2.1), we observe that LHS is negative wherever the solution is smooth enough and positive where we have boundary layers (or oscillation in the computed solution of equation (2.1)).

Divided

difference operator. In this section, we observe a new property of polynomials [f.sub.i,j] in (1.2) related with the divided

difference operator defined by Bernstein-Gelfand-Gelfand and Demazure.

Then A is a difference ring with a

difference operator [delta].

The

difference operator is represented by I; while the coefficients to be estimated are represented by [beta], y and I'; the variable whose time series properties are analyzed is represented by Y and error term at time is represented by Iut Phillip-Perron (PP) test proposes a non-parametric technique of adjusting for higher order autocorrelation in a series and is centered on AR (1) process:

Something like a '

difference operator' is required, to relate difference to difference.

The forward

difference operator is denoted by [DELTA].

Korhonen, "Nevanlinna theory for the

difference operator," Annales Academic Scientiarum Fennicx Mathematica, vol.

where [delta], [alpha], p, b, and c: Z (integer number set) [right arrow] [R.sup.+] (nonnegative real number set) are all bounded sequences, a : Z [right arrow] (0,1) is also a bounded sequence and inf a > 0, m is a nonnegative integer, and [DELTA] is the first-order former

difference operator and obtained the following.

In [3], for the linear discrete-time system with previewable reference signal and disturbance signal, the augmented error system is constructed by using the

difference operator and the preview controller is designed.

where [eta] [member of] [0, 1], v, [beta], [epsilon], [delta] [member of] (0, 1) with [beta], [delta] [less than or equal to] v [less than or equal to] [epsilon], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is fractional

difference operator. We give the following assumptions:

where ([c.sub.n]) [subset] (0, [infinity]), ([f.sub.n]) [contains as member] H are p-periodic sequences for a positive integer p and [DELTA] is the

difference operator defined as usual, i.e., [DELTA][u.sub.n] = [u.sub.n+1] - [u.sub.n].

In order to prove Theorem 3.3 we give an explicit formula for the divided

difference operator. With this we work by induction on the length function l(w) to define simultaneously the action of the simple reflection [s.sub.i].