a work session, & titles/abstracts from the arXiv. [EDITED]

edit [25 july 2006]: the reason why the computation went so well is because i was stupid and made an integration error. sorry, folks.

maybe i'm a "one good work session a day" sort of guy.

during this morning's session at the coffeehouse, i ran a computation with a few simplifying assumptions and everything worked out well ..

.. but a little too well.

looking closely, the computation doesn't depend at all on the group structure, the underlying geometry, or even key properties of the function data. as a result, it can't possibly be right.

the principle at hand: never blindly compute. most nontrivial computations require some underlying reason or "leverage" for why they should work, whether geometric or function theoretic.

i can only assume that my simplifying assumptions were too strong. i should work without assumptions first, and work geometrically for this type of leverage .. that is, if there is any.

---

of course, the word to emphasize is "should."

the internet is too addictive, the office is too distractingly quiet, and i'm not sure whether i can accomplish good mathematics tonight; i don't work as well in the evenings as i used to .. \:

so before i leave the office and try a second session, i found a few interesting preprint titles and abstracts from the arXiv and the SNS @ Pisa website.

Harmonic Univalent Mappings and Linearly Connected Domains by M. Chuaqui and R. Hernández (5 pages)

apparently there is a way to detect the univalence of a harmonic mapping, by studying how its complex dilatation relates with the linear connectivity constant of its image set.

Graphs of W^1,1-Maps with Values into S¹: Relaxed Energies, Minimal Connections, and Lifting by M. Giaquinta and D. Mucci.

i wonder: why into the 1-circle?

The sharp quantitative Sobolev inequality for functions of bounded variation by N. Fusco, F. Maggi, and A. Pratelli.

abstract: The classical Sobolev embedding theorem of the space of functions of bounded variation BV(\Rn) into Ln¢(\Rn) is proved in a sharp quantitative form.

A Generalization of Reifenberg's Theorem in R³ by G. David, T. De Pauw, and T. Toro.

the last time i heard about a reifenberg condition was at a GFT talk this past fall, as an alternative possibility to whitney flat forms in testing for lipschitz parametrizations. the last citation i saw about it was dated from 1995, in a paper by t. toro.

perhaps the lipschitz condition is too much to ask for: in the abstract, the result of the authors is formulated with bi-hölder conditions, rather than bi-lipschitz conditions.

An Isoperimetric Inequality on the l_p Balls by. S. Sodin

i can't resist hearing about isoperimetric inequalities. apparently the isoperimetric profile involves a logarithm, for when 1 < p < 2.