On 04/16/2013 04:47 PM, Ren? Grognard wrote:
>> From: Richard
> The only way out is to postulate that somehow H(x)^p =/= H(x) for p=/=1
> which means that even the Heaviside "function" does not behave at all
> as a regular function.
>
> More generally in the theory of generalised functions it is found that for
> consistency products are only possible with a single generalized function
> as factor. To go further one needs some form of non-standard analysis.
> (H(x)^p=/=H(x) for p=/=1 and an "infinitesimal" x.)
>
> Thanks to Daniel Lakeland to draw the attention to a less demanding form
> of non-standard analysis than the "classical" Robinson's one.
You're welcome, I've found it very useful recently for building some
mathematical models.
...
> No way then "to avoid the icky business of computing with the approximants
> in the first place" otherwise this becomes like the mathematical equivalent
> of the biblical confusion of tongues (;-)
Yes, the language of NSA can help a lot here. We should remember that
these functions were created primarily for the purpose of creating
idealizations of physical models. If we're dealing with some physical
models in which the non-idealized versions of the problem always have
some property (for example they are even functions, they are odd
functions, they have some other symmetry, they are bounded below, or
positive everywhere, etc) then we can define a family of functions with
some scale parameter which have these symmetry properties for all
standard values of the scale parameter, and then allow the scale
parameter to be infinitesimal. We can now work with these restricted
family of functions, doing our analysis in such a way as to preserve the
appropriate property that the physical model should have, while at the
same time applying the idealization of infinitesimal size or whatnot.
Sometimes this can help avoid the biblical style confusion.
For example, we could define a family of functions
H(x,e) = 1/(1+exp(-x/e)). This is a smooth continuous function for all
standard values of e and has a symmetry around x=0. It's value at x=0 is
1/2. If e is infinitesimal then H(x,e) is *one type* of Heaviside
function. One that is appropriate for idealizing behavior that is
symmetric in differences around the point of interest.
Alternatively we could define a family
H2(x,e) = 0 when x < 0 and 1-exp(-x/e) for x > 0. This is a perfectly
standard function for a standard e value, and it has a non-symmetric
rapid change beginning at x=0, a behavior appropriate for problems where
x should be thought of as time, and the function should be
non-anticipatory of the "event" that happens at t=0. For these kinds of
models, when e is infinitesimal we can define a different kind of
Heaviside function.
Problems where we multiple delta(x) * H(x) and integrate will now depend
on certain properties of the family of functions we used to define
delta(x) and H(x). So if delta(x) is symmetric and H(x) is like H2, then
the product will be defined in an infinitesimal region on the positive
side of 0 and will depend on the relative rate that delta decays and
that H2 rises. These are not good properties for some kind of general
purpose mathematical thing to be used by everyone and mean the same
thing, but they are excellent and meaningful properties for specific
models of specific physical systems.