Linux Day 2009 II

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Monday, 6 of September of 2010

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As I recently told you, we are approaching the Linux Day: at present, updated speech lists, locations and other info’s can be found at the ImoLUG refferring page. All the speech are in Italian therefore the event may be of little interest for English speaking readers, but I think it is interesting for everyone to get the feel of what is going on on the Linux side here in Italy, precisely in our LUG. Next, again continuing to explore the power of the WP LaTeX plugin I’ll write down the weak form of the tangential Cauchy-Riemann equation as Gaetano Fichera introduced and studied for the first time:

$\int_{\Sigma}W(\boldsymbol{z})\wedge\mathrm{d}({\mu(\boldsymbol{z},\bar{\boldsymbol{z}})}\wedge {\mathrm{d}z_1}\wedge\dots\wedge {\mathrm{d}z_n})=0$

where

$\Omega\subset\mathbb{C}^n$ with $n>1$ is a bounded domain in the finite dimensional complex vector space and $\Sigma\subset\partial\Omega$ is a part of its boundary $\partial\Omega$ , which we assume of class $C^1$
$\boldsymbol{z}=(z_1,\dots,z_n)\in\Sigma\subset\partial\Omega$ is a point belonging to its boundary
$\mu(\boldsymbol{z},\bar{\boldsymbol{z}})$ is the differential form

${\mu(\boldsymbol{z},\bar{\boldsymbol{z}})}= \sum_{k=2}^n\sum_{h=1}^{k-1}{f_{hk}(\boldsymbol{z},\bar{\boldsymbol{z}})}\mathrm{d}\bar{\boldsymbol{z}}[h][k]$

with $\mathrm{d}\bar{\boldsymbol{z}}[h][k] = \mathrm{d}\bar{z}_1\wedge\cdots \wedge\mathrm{d}\bar{z}_{h-1}\wedge \mathrm{d}\bar{z}_{h+1}\wedge\cdots\wedge\mathrm{d}\bar{z}_{k-1}\wedge \mathrm{d}\bar{z}_{k+1}\wedge\cdots\wedge \mathrm{d}\bar{z}_n$ and $f_{hk}(\boldsymbol{z},\bar{\boldsymbol{z}})$ is an arbitrary polynomial.
$W(\boldsymbol{z})$ is a continuous function of $\boldsymbol{z}\in\Sigma\subset\partial\Omega$ .