Re: P1788: Our first formal motion has entered its discussion period
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1. Ulrich Kulisch wrote:
Concerning StandardizedIntervalNotation I would like to make two remarks:
(in this mail R stand for the set of real numbers and I for an interval
set.)
1. Of course basic to all considerations is the set of extended reals R*.
Interval arithmetic deals with closed and connected sets of real numbers
(and nothing else). If an interval is bounded it is written as [a, b]
with a, b elements of R. If it is unbounded it is written as (-oo, a] or
[b, +oo) with a, b elements of R or (-oo, +oo) where the parantheses
indicate that the bounds -oo and +oo are not elements of the interval.
The set of all such intervals should be denoted by IR. Then
{IR, +, -, *, /} is an exception free calculus.
So what do the proposed notations like IR* or *IR^n really mean? If we
really consider intervals of IR* we would have to allow intervals like
[-oo,-oo] and [+oo, +oo].
These are in IR^* but not in IR. Standard interval arithmetic is for IR;
so there is no conflict here.
2. Of course I am aware that the bounds of an interval a frequently are
denoted by 'a\underline' and 'a\overline'. But I would plead
alternatively also to allow the notation a_1 and a_2 for the bounds
of the interval a.
This gives unnecessary conflicts once interval vectors are considered.
The notation with an index can easily be written in
any typing system. This are the coordinates of the interval in the two
dimensional plane and in a program this is usually the notation of the
interval anyhow.
Not usually. Many programming systems (e.g. Intlab) use a.inf and a.sup.
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2. Nate Hayes wrote;
> The proper/improper property of Kaucher intervals coincides with the
> existential/universal property of modal intervals and the +/- property of
> directed intervals. So the following table summarizes standard
> interpretations:
>
> [a,b] [b,a] Description
> proper improper Kaucher intervals (Kaucher, Goldsztejn,
et. al.)
> existential universal Modal intervals (M. Sainz, Gardenes, et. al.)
> + - Directed intervals (Popova, Markov, et.
> al.)
The discussion in Section 5 of
A. Neumaier,
Computer graphics, linear interpolation, and nonstandard intervals,
http://www.mat.univie.ac.at/~neum/papers.html#nonstandard
(contains the most recent version from January 3, 2009)
shows that no deviation from the proposed notation standard is needed to
discuss nonstandard intervals.
It is unacceptable to enforce the modal notation (used in a minority
of papers only), which adds unnecessary primes to the ordinary
interval notation, upon the whole interval community.
Arnold Neumaier