We present now a proof that the convergent power series produced by the

recursive procedure always regenerates the Taylor series in addition to a complementary one.

Thus, the

recursive procedure should not be started until the initial value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] falls into its tolerance restriction region.

[X.sub.k] can be estimated according to the following

recursive procedure:

The aforementioned meaning of the

recursive procedure denotational specification is modeled as the fixed point of the nonrecursive mapping which corresponds to the supremum and, simultaneously, the topological limit, with respect to the so-called Scott topology, of the increasing successive iterations sequence.

This outcome is sufficient in the

recursive procedure of matrix recognition, because a homomorphism of G into the permutation group Sym(k) can be computed.

The path to the base case from the final general case (created by the final recursive call) is controlled by two operational requirements of a

recursive procedure: (a) a recursive call must be completed before returning to the general case from which the call was made and (b) on the return, processing continues to the step immediately after the step that triggered the call.

3 A New

Recursive Procedure to Generate the Hilbert Series

Pattabhiramacharyulu: 'A

Recursive Procedure for two-species competing eco-system with decay and replenishment for one species', Published in Acta Ciencia Indica, Meerut, Vol.35, 2009.

If used, a

recursive procedure must have a well-defined hard coded limited depth.

with polynomials P, Q, R, S that can be computed by a

recursive procedure. This does not hold for other damaged problems, which is proved in Corollary 2.5.

This

recursive procedure ends when the subproblems are small enough, in which case, another procedure (combine) is used to solve the problem.

The lag truncation parameter is chosen using the general to specific

recursive procedure based on the t-statistic on the coefficient associated with the last lag in the estimated autoregression, see Perron (1997) for more details.