Re: Motion on interval flavors
John, p1788
My apologies for replying very late.
IMHO midrad intervals can be defined at level 1.
If at level 1 given m1 a midrad interval
we can define a reference midrad interval value m2 for f(m1)
an analog to the FTIA theorem can be asserted for midrad intervals.
Given m3 an e effective midrad interval value calculated for f(m1)
we must have
m2 \subseteq m3 .
With m2 = (r2, w2), m3 = ( r3, w3)
this gives r2 = r3 and w3 >= w2
Let me propose a tentative approach to the reference value m2.
It is based on the notion of super interval which is first defined
Since the interval set including unbounded intervals is a
lattice with respect to <=_I order relation define by
[a,b ] <=_I [c,d] iff a<= c AND b <= d
we can defined SUPER INTERVALS as interval defined on the set of intervals
A super interval S1 = [ [a, b], [c, d] ] can be defined as
S1 = { [x, y] | a <= x <= c AND b<= y <= d }
where [a,b] , [c,d] and [x, y] are interval
It should be noted that empty is not a member of any super interval.
Given a point function f we can define
f(S1) = { f([x,y]) | [x,y] \in S1}
To f(S1) we can associate a super interval FS1 = [[f1, f2] , [f3, f4] ]
with
f1 = inf_[x,y] inf( f([x, y]))
f2 = inf_[x,y] sup (f[x, y]))
f3 = sup_[x,y] inf(f[x, y]))
f4 = sup_[x,y] sup f[x, y].
Let us now come back to midrad intervals.
To a super interval S2 = [[a, b],[b,d]] such a+d = 2*b
we can associate a midrad interval mr1 = (m, w ) of midpoint m and width w
such that m = b and w = d - a
For any function defined at x = b
we have f(S2) = [[f1, f2] , [f2, f4]]
with f2 = f(b)
f1 = inf f[a,d]
f4 = sup f[a,d]
In general the condition delta = f1 + f4 - 2* f2 is not zero. So that the
result is not a midrad interval
However a midrad super interval F = [[f1bis, f2] , [f2, f4bis]]
such that if delta > 0
f1bis = f1- delta
f4bis = f'
and similarly
f1bis = f1
f4bis = f4 - delta if delta < 0
We thus have the reference a midrad reference (m1, w1) for f(S1)
with m1 =f(b) and w = f4bis - f1bis = f4 - f1 + |delta|
Since the of effective calculation of f(m, w) by an expression may need
a width correction at each step, it is expected that the effective
width will be greater that the reference width
BTW I would like to notice that super interval might be an interesting
framework for interval extensions since
The set of modal interval [b, c] structured with usual arithmetic operations
is isomorphic to the the set of super intervals of
the form
[[-inf, b], [c, +inf] ] with the same operations
This suggest that all the available extension of intervals should defined by
an association to some subset of super intervalS
The flavors associated to the extensions the would be the condition used in
the definition of corresponding super interval set.
Starting with a super interval defined as [ [a, b], [c, d] ]
we may have the flavor
modal <=> a = -inf AND d + inf
bounded <=> b # + inf AND c # - inf
number <=> b = c
centered <=> a + d = b + c
etc
Sincerely,
Dominique
Such super intervals may be of interest for interval extension since
On Sat, 23 Jun 2012 10:05:20 +0100, John Pryce wrote
> Dominique
>
> On 22 Jun 2012, at 16:26, Dominique Lohez wrote:
> > 3) The analog of the FTIA for midrad intervals requires the function
> > is defined but not necessarily continuous.
>
> Can you clarify? FTIA is primarily a Level 1 concept so midrad
> should be irrelevant. If this is a Level 2 fact, can it be stated
> for general implicit interval types?
>
> John Pryce
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