It is constructive if we check effects of the potential coefficients and the distance of the Dirac delta functions on the reflection and transmission coefficients.

In Figure 4, by considering fix values for the potential coefficients and the energy, we change the distance between the double Dirac delta functions. It can be seen that by increasing the distance, the number of fluctuations in the reflection and transmissions increases.

For the last case, effect of the distance between the double Dirac delta functions was shown.

where [H.sub.[epsilon]] ([phi]) and [[delta].sub.[epsilon]] ([phi]) are smooth versions of Heaviside and Dirac delta functions, defined in (2) and (6).

In the above equations, [H.sub.[epsilon]] ([phi]) and [[delta].sub.[epsilon]] ([phi]) are smooth versions of the Heaviside and Dirac delta functions, which are defined in (2) and (6), respectively.

He observed that his model fills the gap between Bailey's simple stochastic epidemic model [13] (which is obtained as a special case of Kermack and McKendrik's model when the infectivity parameters f(t) and g(t) are constants) and the Reed-Frost model [12] (when the functions f(t) and g(t) are Dirac delta functions).

In our endemic model, letting a =1 and h(t) = [delta](t - 1) where [delta] is the Dirac delta function, we note that each infection increases the force of the infectivity by one unit.

where [delta](x), [delta](y) --

Dirac delta functions.

It was explained in [16] how to get [E.sup.j.sub.f] in terms of the Dirac delta functions and then use them to reduce the integrals that appear in [D.sup.(j).sub.f, b - 1].

We can use any difference scheme, but as we are working with a white noise, the Dirac delta function [delta](t - a) is expected to appear in the equations of some kernels.

The output PDF could be mathematically expressed by a string of

Dirac delta functions. Also PDF of quantizer error e could be described by means of a train of

Dirac delta functions.

Each chapter includes exercises, and Mahler supplies support information for such complex topics as

Dirac delta functions, mathematical proofs, probability theory, gradient derivatives and Gaussian identity.