On 6 August 2014 06:34, Ralf Hemmecke <[email protected]> wrote: > ... > It reminds me that > > diracDelta : F -> F > ++ diracDelta(x) is unit mass at zeros of x. > > is a relatively unclear specification. Is there any reason (except for > the name "diracDelta") that this function has anything to do with > http://en.wikipedia.org/wiki/Dirac_delta_function ? And "function" is > perhaps not the right word. That definition goes against the meaning of > F->F (i.e. total function).
F has FunctionSpace(R), typically it is Expression(R), so all we need for diracDelta to be a total function is for diracDelta(x) to be an Expression for any Expression x. But you are right in as much as in the theory of distributions diracDelta is actually a functional, i.e. a higher-order function that operates on functions. As such it should act more like 'integrate' diracDelta(f(x),x) = f(0) for a well defined set of functions f. Instead we treat diracDelta itself like a function and write integrate(diracDelta(x)*f(x),x=%minusInfinity..%plusInfinity) = f(0) but this leaves expressions involving diracDelta in other contexts to our imagination. Shirokov and others have tried to define expressions involving diracDelta (and other distributions) as "generalized functions" in such a way that when these expressions are interpreted as functionals they act like distributions. > > OK, one might argue that we have this all over the place, simplest > example being inv: % -> % or /: (%, %) -> %, but nonetheless, > I don't really like it. > > Signatures like > > inv: % -> % throw DivideByZero > > (see Section 11.4 in http://www.aldor.org/docs/aldorug.pdf) > > would be a bit better. It is not correct to say that diraDelta(0) is undefined in the same sense as inv(0). Probably it is sufficient to leave it unevaluated. > ... > On 08/06/2014 02:07 AM, Bill Page wrote: > > And define signum by > > > > abs(x)*signum(x) = x > > Does that equality hold for x=0 of you want signum(0) to be undefined? Yes. Maybe "unevaluated" would be better than "undefined". > ... The spirit of FriCAS is to implement > things algebraically, i.e. introduce a type for distributions. > > > signum(x)*diracDelta(x) + diracDelta(x)*signum(x) = 0 > > Interesting... that algebra would be non-commutative. > > Now the question is... who is going to implement such a new domain? > Bill, do you volunteer? > Yes maybe, perhaps if other people also have some interest in this approach. But really it is about defining a domain for "generalized expressions" which in the right context might be interpreted as distributions. Bill. -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/fricas-devel. For more options, visit https://groups.google.com/d/optout.
