Linux Day 2009 II
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:
where
with
is a bounded domain in the finite dimensional complex vector space and
is a part of its boundary
, which we assume of class
is a point belonging to its boundary
is the differential form
with
and
is an arbitrary polynomial.
is a continuous function of
.
![{\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]](http://s.wordpress.com/latex.php?latex=%7B%5Cmu%28%5Cboldsymbol%7Bz%7D%2C%5Cbar%7B%5Cboldsymbol%7Bz%7D%7D%29%7D%3D%20%5Csum_%7Bk%3D2%7D%5En%5Csum_%7Bh%3D1%7D%5E%7Bk-1%7D%7Bf_%7Bhk%7D%28%5Cboldsymbol%7Bz%7D%2C%5Cbar%7B%5Cboldsymbol%7Bz%7D%7D%29%7D%5Cmathrm%7Bd%7D%5Cbar%7B%5Cboldsymbol%7Bz%7D%7D%5Bh%5D%5Bk%5D&bg=ffffff&fg=000000&s=1 "{\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]")
And now let’s listen to Watchtower :
Watchtower The Eldritch
Date: 11 October 2009
Categories: Chronicle, Linux, Mathematics
is a point belonging to the 
while 
is a point belonging to its boundary
is the ![U(\boldsymbol{z}-\boldsymbol{\zeta})=-\frac{(n-1)!}{(2\pi i)^n} \sum_{k=1}^n (-1)^{k-1}\frac{\bar{\zeta}_k-\bar{z}_k}{|\boldsymbol{z}-\boldsymbol{\zeta}|^{2n}}\mathrm{d}\bar{\boldsymbol{\zeta}}[k]\wedge\mathrm{d}\boldsymbol{\zeta}](http://s.wordpress.com/latex.php?latex=U%28%5Cboldsymbol%7Bz%7D-%5Cboldsymbol%7B%5Czeta%7D%29%3D-%5Cfrac%7B%28n-1%29%21%7D%7B%282%5Cpi%20i%29%5En%7D%20%5Csum_%7Bk%3D1%7D%5En%20%28-1%29%5E%7Bk-1%7D%5Cfrac%7B%5Cbar%7B%5Czeta%7D_k-%5Cbar%7Bz%7D_k%7D%7B%7C%5Cboldsymbol%7Bz%7D-%5Cboldsymbol%7B%5Czeta%7D%7C%5E%7B2n%7D%7D%5Cmathrm%7Bd%7D%5Cbar%7B%5Cboldsymbol%7B%5Czeta%7D%7D%5Bk%5D%5Cwedge%5Cmathrm%7Bd%7D%5Cboldsymbol%7B%5Czeta%7D&bg=ffffff&fg=000000&s=1 "U(\boldsymbol{z}-\boldsymbol{\zeta})=-\frac{(n-1)!}{(2\pi i)^n} \sum_{k=1}^n (-1)^{k-1}\frac{\bar{\zeta}_k-\bar{z}_k}{|\boldsymbol{z}-\boldsymbol{\zeta}|^{2n}}\mathrm{d}\bar{\boldsymbol{\zeta}}[k]\wedge\mathrm{d}\boldsymbol{\zeta}")
![\mathrm{d}\bar{\boldsymbol{\zeta}}[k] = \mathrm{d}\bar{\zeta}_1\wedge\cdots \wedge\mathrm{d}\bar{\zeta}_{k-1}\wedge \mathrm{d}\bar{\zeta}_{k+1}\wedge\cdots\wedge \mathrm{d}\bar{\zeta}_n](http://s.wordpress.com/latex.php?latex=%5Cmathrm%7Bd%7D%5Cbar%7B%5Cboldsymbol%7B%5Czeta%7D%7D%5Bk%5D%20%3D%20%5Cmathrm%7Bd%7D%5Cbar%7B%5Czeta%7D_1%5Cwedge%5Ccdots%20%5Cwedge%5Cmathrm%7Bd%7D%5Cbar%7B%5Czeta%7D_%7Bk-1%7D%5Cwedge%20%5Cmathrm%7Bd%7D%5Cbar%7B%5Czeta%7D_%7Bk%2B1%7D%5Cwedge%5Ccdots%5Cwedge%20%5Cmathrm%7Bd%7D%5Cbar%7B%5Czeta%7D_n&bg=ffffff&fg=000000&s=0 "\mathrm{d}\bar{\boldsymbol{\zeta}}[k] = \mathrm{d}\bar{\zeta}_1\wedge\cdots \wedge\mathrm{d}\bar{\zeta}_{k-1}\wedge \mathrm{d}\bar{\zeta}_{k+1}\wedge\cdots\wedge \mathrm{d}\bar{\zeta}_n")
.
is a continuous function of 