I have received the following list of =
interesting and=20
worthwhile challenges for the interval community, from Professor Rafael =
de la=20
Llave at UTexas.

I pass them along here, hoping that some of you =
might pick=20
up on one or another of them. For further details, you might contact=20

--------------------------------

One possibility would be to think hard about how to represent the =
data=20

that people who solve functional equations use.

For = example,=20 the renormalization theory abounds with those. Many of

which have = been left=20 on shaky ground since they were only

done numerically and the = numerics is=20 sometimes suspect.

The current implementations of these things could = be=20

certainly improved.

In harmonic analysis, there are = several=20 "maximal operators" which

would be useful to bound accurately. (Once = you get=20 good bounds on

those, you can start applying contraction=20 arguments)

In dynamical systems, one often is interested in = knowing=20 whether

some concrete features are present in the system and to = obtain=20

quantitative estimates of them (e.g. intersection of manifolds that=20

determine capture etc.)

Another problem that I would=20 like a lot to see

solved is the implementation of = the=20 Ziglin criterion

(or Morales-Ramis-Simo) that just looking at = properties=20 of

a periodic orbit can conclude that the system is not=20 integrable.

Recently, there are many results estimating Lyapunov=20 exponents of

certain dynamical systems. Some particularly important = examples=20

are those dynamical systems that appear in number theory.

Just = the=20 existence of those Lyapunov exponents, implies

that certain continued = fraction algorithms work.

I point out to some recent success = stories=20 which I would like to

see repeated: The proof of the double bubble=20 conjecture and

the proof of the existence of the Lorenz attractor =

(the=20 later is more exemplary since there was a serious cooperation =

between the=20 mathematician and the people who implemented the package

that was = used)=20

that people who solve functional equations use.

For = example,=20 the renormalization theory abounds with those. Many of

which have = been left=20 on shaky ground since they were only

done numerically and the = numerics is=20 sometimes suspect.

The current implementations of these things could = be=20

certainly improved.

In harmonic analysis, there are = several=20 "maximal operators" which

would be useful to bound accurately. (Once = you get=20 good bounds on

those, you can start applying contraction=20 arguments)

In dynamical systems, one often is interested in = knowing=20 whether

some concrete features are present in the system and to = obtain=20

quantitative estimates of them (e.g. intersection of manifolds that=20

determine capture etc.)

Another problem that I would=20 like a lot to see

solved is the implementation of = the=20 Ziglin criterion

(or Morales-Ramis-Simo) that just looking at = properties=20 of

a periodic orbit can conclude that the system is not=20 integrable.

Recently, there are many results estimating Lyapunov=20 exponents of

certain dynamical systems. Some particularly important = examples=20

are those dynamical systems that appear in number theory.

Just = the=20 existence of those Lyapunov exponents, implies

that certain continued = fraction algorithms work.

I point out to some recent success = stories=20 which I would like to

see repeated: The proof of the double bubble=20 conjecture and

the proof of the existence of the Lorenz attractor =

(the=20 later is more exemplary since there was a serious cooperation =

between the=20 mathematician and the people who implemented the package

that was = used)=20

-------------------

best regards to all,

Ramon Moore

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=9This could be your ad=21**
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=9Click here to e-mail us your contact info.

This ad is being sent in compliance with Senate Bill 1618=
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You have recently visited our web site, referral or affiliate sit=
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