Re: Motion P1788/M0013.04 - Comparisons
John Pryce wrote:
P1788
This is the third or fourth time I ask this question, but I don't recall ever having an answer.
Concerning the relation "is interior to", (\subset in Kulisch's notation), what's the result of
[1,oo] is interior to [0,oo]
?
Someone please say which of these is right:
(1) It is false, because the endpoints of the "inside" set _must_ be different from those of the "outside" one.
(2) It is true, because what we want is "topologically interior to".
Fronm a mathematical point of view (since motion 3 says intervals are
sets), version (2) is the right explanation.
(3) I couldn't care less, because this relation is only useful for bounded intervals.
Or, of course, something else.
Answers from those with experience in writing interval software especially welcome.
From usage in algorithms, one needs the interiorness check for some
existence tests. Usually these also require boundedness, and the above
would not matter.
But in view of Hadamard's theorem, which says that for the standard
Newton operator on the unbounded box [-inf,inf]^n, no condition is
required to ensure existence, I believe that it is possible to extend
some of the existence theorems to unbounded situations.
In this case, the topological definition is a necessity.
Arnold Neumaier