Re: Motion on interval flavors
On 07/08/2012 06:42 PM, Corliss, George wrote:
Yes, your application to Bezier and b-spline bases for polynomials is a deeper application, but am I on the right track?
That is EXACTLY the sort of example I was hoping to hear.
Pros and cons of this are all discussed in my paper on nonstandard
intervals, which I wrote a few years ago when we started this
discussion. Nothing has changed in the mean time.
On 07/09/2012 11:35 AM, Svetoslav Markov wrote:
> The historical lesson of the negative numbers teaches us
> that the correct way is to work in group structures
> whenever possible. Kaucher/modal intervals provide
> such structures.
For addition, but the multiplicative structure becomes much poorer.
In contrast, introducing negative numbers and reals adds structure
that simplifies the algebra!
> Note also that the applications of negative numbers increased
> only when mathematicians started to use them systematically and got
> used to the (group-like) properties of real numbers. The history
> teaches us that soon or later the same will happen with
> Kaucher intervals.
History teaches that the most useful structures will be used, not the
most complete ones. otherwise we'd all be using quaternions on a regular
basis. But they are used only for very special applications where they
are indeed suitable. Kaucher intervals compare to ordinary intervals
much more like quaternions to the complex numbers than like reals to
rationals.
On 07/08/2012 12:52 PM, Svetoslav Markov wrote:>
> I make the analogy with negative numbers because
> improper (Kaucher/modal) intervals are just
> like negative numbers (as their radii are negative).
>
> There is a lot in internet about the history of the acceptance
> of negative numbers, see e. g.:
>
> http://en.wikipedia.org/wiki/Negative_number
> (section "history")
When Kaucher intervals have gained the same widespread use in interval
method as negative numbers or real numbers have in ordinary numerical
methods, it is time to standardize their use.
But not earlier. Being more general and/or analogous and/or have
additional properties is not enough justification for an incorporation
into standards. Only extensive, widespread use is.