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Re: Motion P1788/M0013.04 - Comparisons

Arnold Neumaier wrote:

Nate Hayes wrote:

John Pryce wrote:

On 18 Sep 2010, at 23:15, Nate Hayes wrote:

I speak against this.
Ulrich's interior is better.
Note that the topological interior, i.e., "proper subset," is already expressed efficiently in terms of Ulrich's relations for intervals A,B: 
  ( A \subset B ) and not ( A == B )

Doesn't that make [2,3] interior to [1,3]?


I don't see Entire should be interior to Entire.
Well, it seems weird to me too, but there it is. You're an expert on quantified statements. Isn't it inescapable from the definition "B is a neighbourhood of each a in A" (eqn1)?
As you are so fond of quoting from George Corliss: if it even seems weird to you -- a seasoned mathemetician -- then "God help the casual user!"
The thing about definitions grounded in standard theory (and this theory has been around for roughly 100 years) is that, compared with ad-hoc definitions, you KNOW they can't lead to inconsistencies -- assuming math itself is consistent.
This is a reason I think P1788 might want to investigate sticking to compact intervals, instead.... which if you remember was one of my first choices.
Unbounded intervals are essential for applications to general global optimization and nonlinear systems solving, where for complex models 
one often doesn't know in advance bounds on all variables.
Yes, being able to initialize such unknown variables appropriately is important for these applications. I agree. I don't suggest P1788 should not provide a mechanism for this.
Keep in mind, Ian McKintosh has suggested replacing unbounded closed intervals with "overflown" compact intervals. In practice, there is not really a difference between the two. For example, +INF and -INF are replaced by +OVR and -OVR in all compuations, where OVR is some unknown finite real number larger than the magnitude of the largest floating-point number. In this way, intervals such as [1,+OVR] remain compact intervals and are not unbounded like the interval [1,+INF]. Arithmetic operations on the endpoints of compact overflown intervals is the same as in Motion 5, i.e., (-OVR) + (-OVR) = -OVR, 0 * OVR = 0, etc.
So this approach seems to retain all the important advantages of unbounded intervals without actually introducing the complications that arise from them. To the extent I can see, it has big potential to simplify P1788 a great deal, without hurting or compromising the important needs of global optimization and nonlinear programming application you mention. 

Nate Hayes