Re: flow or stock?

[Alan G Isaac <aisaac@xxxxxxxxxxxx>]

On Sun, 04 Aug 2002 15:40:05 -0500 "William B. Ryan" <william_b_ryan@xxxxxxxxxx> wrote:
> Trond, would you not agree that it is always inappropriate to model continuous
> processes in discrete time?  Isn't that the dilemma exposed 2500 hundred
> years ago by Zeno's paradoxes on motion?

Not as a logical matter.

Consider the first order differential equation
Dx=Ax
where D is the differential operator.
This has the (continuous time) definite solution
x(t)= exp{At} x0
where x0 is the initial condition.

Now consider modeling the values that will be taken at
t=1,2,3,...
(That is, consider the discrete time version of the model.)
Of course we already have the values from our solution above.
But we can also characterize this part of the solution in
terms of a difference equation that has the definite solution
x(t)= b^t x0
where a caret denotes exponentiation and b=exp{A}.  That is,
we can have a "discrete time" version of the model:
x(t) = b x(t-1)

Cheers,
Alan Isaac

PS The issue for Zeno's paradoxes is really quite different.
One natural interpretation is that Zeno is representing the
precalculus understanding of the physical world.  For example,
we now know how an infinite summation can have a well-defined, finite
sum, so it does not bother us that Achilles can catch up with the
tortoise even though it always takes "some" amount of time to cross
any finite distance.  Of course some people argue that Zeno has
pointed to a profound difficulty with the notion of the velocity
of an object at an instant.