Thursday, February 02, 2006

after the dust settles ..

.. a technical issue remains in the latest edition of my thesis work. i don't see how to reconcile one bit.

it's not the collars or the tubular neighborhoods, the isotopies or the vector bundles. none of that abstract machinery is bothering me at the moment, which is surprising enough in itself. the issue actually arises from ..

.. the existence and uniqueness theorem(s) for systems of ODE.

yes, ODE. argh .. to reach this far and get stuck on ODE? [1] well, not explicit ODE; i mean the theory, but still ..



to clarify, the standard Picard Iteration theorem uses a contraction fixed-point theorem in the setting of an integral operator. boundedness of the source function for the ODE isn't enough; the function must also be Lipschitz in the "dependent variable" to make the right estimate for a contraction.

it would mean that the flow arising from the source function be a differentiable mapping with Lipschitz derivatives (C1,1-regularity). i don't want that kind of regularity, and neither does my advisor.

despite my usual inclination, i honestly hope that i've made a mistake and overlooked something .. perhaps the source function arises from something which permits the Lipschitz property so that we needn't assume it ..

.. ye gods. i just want to finish this so that i can get back to forming new results, play around with my beloved sobolev spaces. is that so much to ask?



[1] strangely enough, my other research will likely and soon encounter ODE troubles of its own .. the ugly nonlinear and non-closed form type, i fear.

this, of course, is provided that i get around to settling details about selective Steiner symmetrizations.

a boy's work is never done, is it?

1 comment:

janus said...

never mind. there are still issues with the isotopies of tubular neighborhoods.

argh.