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Topic: A problem with harmonic functions
Replies: 4   Last Post: Jun 3, 2008 1:47 PM

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Brian Quincy Hutchings

Posts: 3,031
Registered: 12/6/04
Re: A problem with harmonic functions
Posted: Jun 3, 2008 1:47 PM
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the partials commute, and do you still need
to use the trilateral inequality?

how about the tetrahedral analogs?

> > > Yes. For example, this follows easily from another fact: If v is
> > > harmonic on R^n and int |v|^2 < infinity then v = 0. To get
> > > the first fact from the second consider v = partial derivative
> > > of u.

> > > So assume v is harmonic and square-integrable. For any
> > > r > 0

> > >  v(0) = (c/r^n) int_{|x|<r} v(r)
> > > so the Schwarz inequality shows that (with a different c)
> > >  |v(0)| <= (c/r^n) r^(n/2).
> > > Letting r -> infinity shows that v(0) = 0; similarly
> > > v(x) = 0 for other x.

> > Very nice proof. I didn't though that if u is harmonic then all its partial
> > derivative are harmonic too.

> That follows trivially from the fact that partial
> derivatives commute *assuming* you know that a
> harmonic function has derivatives of all orders.

--who killed Schroedinger's puppy?!
thus: so, what's the math for this nonconvex shape of equigonal
trigona ??

thus quoth:
Darfur is Sudan's poorest, wildest region. One of the Islamic World's
first anti-colonial
movements, known in the west as the Dervishes, burst from the wastes
of Darfur in the
1880s. Led by the fiery "Mahdi," the Dervishes drove the British
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an event immortalized in the splendid Victorian novel, Four Feathers.
The Dervishes took
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The "martyred" Gordon's death roused a storm in Britain, resulting in
a punitive army
sent up the Nile (including the young Winston Churchill) that
destroyed the Dervish army
at Omdurman. But remote Darfur remained a hotbed of rebellion.
Arms and money

well, not a moment too soon,
to decelerate the ethanol-fed dustbowl
in the Midwest ... hopefully.

otherwise, coal's a very (?) important feedstock. so,
what is the sustainable production of natural gas & natural oil?...
Atoms, Machiavelli, Machiavelli & Oil need a figure
for the venture ads!... coal, we know from its pre-fossil
to be produced catastrophically; the burial
of whole, swampy ecosystems.
Yucca Flats is a farce majeur, even if
there's any thing left to bury, by the time it'd open, or
by noon, Pacific Time.

the Brooklyn Bridge is a powerful metaphor, but
it has to be told in the context of its manifestation
as David McCullough's first production for PBS:
his first, big SALE, in other word,
to the "eastern liberal establishment" of Wall Street,
the District of Columbia and protege Ken "Civil War Jazzbo" Burns.

so, did you recalibrate the orbital constraints
in your JPL publicdomain trajectories,
to include the phasal parameter "as 'seen' from Earth?..."
was Venus closer to conjunction or opposition
to Earth, during those alleged missions?...
> Where's Venus (from orbit or surface EVA) as
> of missions A-11, A-14 and A-16?

--Harry of Darfuria, Episode 9 (Deuterocanonicos):
Prisoners of the Guantanamo Episcopate/poolcess!...

"We Cesspools are not to be sniffed at!"
?Lady Cesspool in Al Capp's "Li'l Abner" Cartoon Strip

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