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These definitions depend on the infsup form. For those who know the theorem of embedding a semigroup in a group I attach my scan06 paper (rejected for publication). There I argue that the natural presentation of intervals is the mid-rad one. Svetoslav On 18 Mar 2009 at 2:24, Alexandre Goldsztejn wrote: > Dear collegues, > > I would like to clarify the difference between Kaucher intervals, > directed intervals and modal intervals. > > A Kaucher interval is a pair of reals. For example, [-1,1] is a proper > Kaucher interval, [1,-1] is an improper Kaucher interval. Proper > intervals are of course identified to the corresponding sets of reals, > while Kaucher arithmetic, which is defined for any proper/improper > arguments, coincides with the set interval arithmetic when restricted > to proper arguments. > > A directed interval is a pair made of an interval and a sign taken > among + or -, e.g. ([-1,1],+) or ([-1,1],-). A modal interval is very > similar to a directed interval, as it is a pair made of an interval > and a quantifier \forall or \exists, e.g. ([-1,1],\exists) or > ([-1,1],\forall). Although this is not formally defined using > equivalence classes, directed (resp. modal) intervals involving > degenerated intervals and different signs (resp. quantifiers) are > identified, e.g. the directed intervals ([1,1],+) is identified to > ([1,1],-) (resp. ([1,1],\exists) is identified to ([1,1],\forall), > which is compatible with the semantics of modal intervals). Then, > directed intervals (and modal intervals) are isomorphic to Kaucher > intervals: > > Consider a<=b > ([a,b],+) <-----------> [a,b] > ([a,b],-) <-----------> [b,a] > > Kind regards, > > Alexandre Goldsztejn > > On Tue, Mar 17, 2009 at 2:56 PM, Michel Hack <hack@xxxxxxxxxxxxxx> wrote: > > It would have helped if early on we had seen a concise, though perhaps > > incomplete, description of Kaucher intervals, as Baker just gave. Not > > even the paper that Nate pointed us to was that clear, one had to guess > > it from the way they were used. > > > > So: a Kaucher interval is an interval with Lower and Upper bounds, > > always non-empty, with an additional flag that directs how the bounds > > are to participate in monotonic arithmetic. This is useful in cases > > where backwards containment needs to be calculated. (The mode flag is > > encoded in the order in which the two bounds are given in the interval > > representation, because that leads naturally to the desired effect, at > > least in common cases -- or does it always work, obviating the need to > > test the "flag"?) > > > > Critical in the context of the Expression-Rearrangement subgroup is > > how much knowledge of applicability of Kaucher rules is needed when > > expressions are transformed. > > > > Also -- how is Empty handled in Kaucher Arithmetic? > > > > Finally -- is "Infinity as a Member" decided one way or the other in > > Kaucher Arithmetic, and if not, are the issues similar? > > > > Michel. > > > > P.S. I started using "KA" -- but that could be misinterpreted as > > "Kahan Arithmetic". Would "MA" for "Modal Arithmetic" work, > > leaving "IA" as the generic term for Interval Arithmetic? > > Sent: 2009-03-17 14:05:27 UTC > > > > > > -- > Dr. Alexandre Goldsztejn > > CNRS - University of Nantes > Office : +33 2 51 12 58 37 Mobile : +33 6 78 04 94 87 > Web: www.goldsztejn.com > Email: alexandre.goldsztejn@xxxxxxxxxxxxxx Prof. Svetoslav Markov, DSci, PhD Head, Dept. "Biomathematics", phone: +359-2-979-3704 Inst. of Mathematics and Informatics, fax: +359-2-971-3649 Bulgarian Academy of Sciences, e-mail: smarkov@xxxxxxxxxx "Acad. G. Bonchev" st., block 8, BG-1113 Sofia, BULGARIA mobile (gsm): 0885331464 Home address: 11 Mizia str, 1504 Sofia, tel. +359-2-9444651 URL: http://www.math.bas.bg/~bio/
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