Arnold and all
On 16 Jan Arnold Neumaier wrote (response to Prof Kulisch, and
circulated)
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More precisely, IR is the set of nonenpty, closed and bounded intervals
over R.
*IR is the set of bounded and unbounded intervals over R. So the
empty set and the set R = (-oo, +oo) are elements of *IR.
Yes.
In my paper for the proceedings of the Dagstuhl Seminar (January
2008) I used the notation (IR) for this set avoiding the *. (This
notoation could also be extended to (IR)^n.).
Yes. But it is unwise to change a long established notation; it will
not stick and only add confusion. Moore's intervals were bounded and
nonempty.
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But on 27 Oct last Arnold wrote to JDP:
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JDP: I may be wrong, but to me "Moore interval arithmetic" means one
has only bounded intervals, and no empty set.
AN: I think his interval arithmetic was closed, and included things like
4/[0,4]=[1,inf],
1 - 8/[0,4] intersect [0,2] = empty.
At least, this is needed (for x^2+7 = 0, x in [0,2]) to make the
1D interval Newton method work, which was his invention.
Of course, his implementation could have no number inf; but the
arithmetic in his theory probably had.
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It seems to me Arnold came to two different conclusions at different
times about what "Moore's model" was. Fair enough. I am probably at
least as inconsistent.
My point is that we should use common sense in what "following" the
standard notation paper means.
- Where it says "an interval, whether scalar or vector, is typeset in
bold italic" - that's pretty basic typography, and if we all follow
it, our papers will be more immediately readable both by established
intervallers and by newcomers.
- Where it says IR denotes a certain set of intervals, and *IR, a
certain larger set, let's be flexible. What one commonly needs is a
notation for "the set of all allowed intervals in whatever model I am
using". It is natural for this to be "the symbol for the underlying
number system, with an I in front", so
IR for an R-based system
and IR*, IC, etc. for other number systems.
So if some paper uses IR, while the notation paper says it should be
*IR, that's fine by me as a reader, provided each paper clearly states
what notation is being used.
More generally, I feel (as was the original intent of my motion) that
we shouldn't make ANY amendments to the notation paper, for now. My
motion asks you to take it for what it is, and to "follow" it flexibly.
Maybe I'll add to the Rationale a sentence on these lines. Unless two
people feel strongly enough to propose, and second, a formal amendment
in the next day or two, I ask that we go on to voting on the unchanged
motion.
Best wishes
John P