For a general d, Zhao [12] proved the analytic continuation of [zeta]([s.sub.1], ..., [s.sub.d]) as a function of d variables using the theory of

generalized function and Akiyama, Egami and Tanigawa [1] proved the same result by applying the classical Euler-Maclaurin formula to the index of the summation [n.sub.d].

A synchronization is

generalized function projective synchronization, which is a more general definition of chaotic synchronization, it encompasses function projective synchronization.

qualifies as a representative for a random

generalized function. We obtain an imbedding [mathematical expression not reproducible].

For the Dirac delta function, as a singular

generalized function, there are different types of approximation (see, e.g., [19, 20, 23]).

Hartley, "

Generalized function for the fractional calculus," Tech.

Among the topics are quantifying soil-root interactions, nitrogen management effects of root systems, characterizing root length density distribution in wheat using a

generalized function, modeling Lolium perenne L.

In higher dimensions we need to define it as a random

generalized function. Let [C.sup.1.sub.0](D) be the set of test functions on D, i.e.

The gamma, generalized gamma, Weibull and another type of gamma incomplete distribution are obtained as particular cases of the density

generalized function. Some properties associated with the density function and other frequently used functions such as the k-th moment, risk function, life time are derived.

A method based on the theory of the

generalized function of the Fourier transformation is used for solving the matrix equation (2) [1]:

In section 4 we give a mild generalization of this mock theta function and show this

generalized function belongs to the family of [F.sub.q] -functions.

With the help of this function we can define the Laplace transform L of a

generalized function [PHI] [member of] [F's.ub.[theta]] (N') by

Finally, as for the usual Dirac delta as a

generalized function, we show how to express a mean geometric density [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in terms of a suitable approximating sequence of classical functions.