inf/sup vs min/max vs lb/ub
Ulrich Kulisch wrote: [in: A question Re: Level 1 <---> level 2
mappings; arithmetic versus applications]
Ralph Baker Kearfott schrieb:
P.S. Opinion: As someone with mathematical training, I would prefer
"min-max" to "inf-sup" since the min and the max exist,
because the mathematical objects to which we refer are closed
and bounded sets. However, we are definitely stuck with
"inf-sup," because that's
what practically everyone uses everywhere. In any case,
a "min" is an "inf" and a "max" is a "sup."
inf and sup work for interval vectors with the standard partial order
on R^n, while min and max only work for single intervals, and even only
those with finite endpoints (under our general assumption that intervals
are sets of real numbers).
The only exception is the mpty set, which in terms of infsup,
should be represented as [+inf,-inf]. Would such a representation
cause any problems?
I am also not very happy about "inf-sup". An interval is denoted by an
ordered pair [a, b].
But in mathematics, an interval _is_ a set of real numbers, and
a=inf[a,b] even when a=-inf.
The first element is the lower bound and the second
is the upper bound. So would not lower bound (lb) and upper bound (ub)
be better?
The intervals IR over the real numbers and IF over the floating-point
numbers are both (completely) ordered sets with respect to set inclusion
as an order relation. The 'tightest' enclosure of an interval A of IR in
IF is just the least upper bound (the supremum of A (sup A)) in IF. A
'valid' enlosure of an interval A of IR in IF just maps A on an upper
bound in IF.
But with the two different notions of upper bound, both introduced
in your short mail, this is confusing;
''The'' upper bound of [0,1] is 1,
while [0,2] is ''an'' upper bound of [0,1].
So we could use conventional mathematical concepts and would not have to
develop a particular interval jargon.
inf and sup _are_ the conventional mathematical jargon, and not an
invention of the interval community.
Arnold Neumaier