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From: Han de Bruijn on 6 Jul 2010 06:00
Foreplay:

http://groups.google.nl/group/sci.math/msg/ffce208afa5b2555
Numerical Ensemble of Harmonic Oscillators

http://groups.google.nl/group/sci.math/msg/d90f07f7523b0d52
Numerical Ensemble of Exponential Decays

Quote:
What the grey valued images are all about will be explained LATER on.

Well, here and NOW, actually:

http://hdebruijn.soo.dto.tudelft.nl/jaar2010/dikte/document.pdf
Uniform Combs of Gaussians

There is quite another story about the continuity and discreteness of
curves. Any discretized curve (x_k,y_k) = (f(s_k),g(s_k)) can be made
continuous again, namely, by the following procedure:

C(x,y) = sum_k exp(-A(x,y,s_k)/2) ; s = arc length

Here A(x,y,s) = ([x-f(s)]^2 + [y-g(s)]^2)/sigma^2

Now what's the big deal of this ? The idea is that discretization, in
for example Numerical Analysis, is not really used as a means to make
things just discrete. What people actually want is the _exact_ which
is a _continuous_ solution, in the end. The discretization is nothing
but kind of a clumsy vehicle to achieve this as good as possible. The
crucial insight is: that continuity can be achieved not only exactly,
but also approximately.

If the spread of a Gaussians is chosen greater than the discretization
"error" then the discretization becomes unobservable. meaning that the
curve, within great accuracy, has become CONTINUOUS, in a fuzzy sense.

Read the article for higher precision of the above statement. Comments
and suggestions for improvement are always quite welcome.

Han de Bruijn
From: Androcles on 6 Jul 2010 06:28

"Han de Bruijn" <umumenu(a)gmail.com> wrote in message
news:6709998e-4d6c-4cca-8ee8-982f22045783(a)j4g2000yqh.googlegroups.com...
| Foreplay:
|
| http://groups.google.nl/group/sci.math/msg/ffce208afa5b2555
| Numerical Ensemble of Harmonic Oscillators
|
| http://groups.google.nl/group/sci.math/msg/d90f07f7523b0d52
| Numerical Ensemble of Exponential Decays
|
| Quote:
| What the grey valued images are all about will be explained LATER on.
|
| Well, here and NOW, actually:
|
| http://hdebruijn.soo.dto.tudelft.nl/jaar2010/dikte/document.pdf
| Uniform Combs of Gaussians
|
| There is quite another story about the continuity and discreteness of
| curves. Any discretized curve (x_k,y_k) = (f(s_k),g(s_k)) can be made
| continuous again, namely, by the following procedure:
|
| C(x,y) = sum_k exp(-A(x,y,s_k)/2) ; s = arc length
|
| Here A(x,y,s) = ([x-f(s)]^2 + [y-g(s)]^2)/sigma^2
|
| Now what's the big deal of this ? The idea is that discretization, in
| for example Numerical Analysis, is not really used as a means to make
| things just discrete. What people actually want is the _exact_ which
| is a _continuous_ solution, in the end. The discretization is nothing
| but kind of a clumsy vehicle to achieve this as good as possible. The
| crucial insight is: that continuity can be achieved not only exactly,
| but also approximately.
|
| If the spread of a Gaussians is chosen greater than the discretization
| "error" then the discretization becomes unobservable. meaning that the
| curve, within great accuracy, has become CONTINUOUS, in a fuzzy sense.
|
| Read the article for higher precision of the above statement. Comments
| and suggestions for improvement are always quite welcome.
|
| Han de Bruijn

Congratulations, you've discovered 1980s chaos theory.


From: john on 6 Jul 2010 14:43
On Jul 6, 4:00 am, Han de Bruijn <umum...(a)gmail.com> wrote:
> Foreplay:
>
> http://groups.google.nl/group/sci.math/msg/ffce208afa5b2555
> Numerical Ensemble of Harmonic Oscillators
>
> http://groups.google.nl/group/sci.math/msg/d90f07f7523b0d52
> Numerical Ensemble of Exponential Decays
>
> Quote:
> What the grey valued images are all about will be explained LATER on.
>
> Well, here and NOW, actually:
>
> http://hdebruijn.soo.dto.tudelft.nl/jaar2010/dikte/document.pdf
> Uniform Combs of Gaussians
>
> There is quite another story about the continuity and discreteness of
> curves. Any discretized curve (x_k,y_k) = (f(s_k),g(s_k)) can be made
> continuous again, namely, by the following procedure:
>
> C(x,y) = sum_k exp(-A(x,y,s_k)/2)  ; s = arc length
>
> Here  A(x,y,s) = ([x-f(s)]^2 + [y-g(s)]^2)/sigma^2
>
> Now what's the big deal of this ? The idea is that discretization, in
> for example Numerical Analysis, is not really used as a means to make
> things just discrete. What people actually want is the _exact_ which
> is a _continuous_ solution, in the end. The discretization is nothing
> but kind of a clumsy vehicle to achieve this as good as possible. The
> crucial insight is: that continuity can be achieved not only exactly,
> but also approximately.
>
> If the spread of a Gaussians is chosen greater than the discretization
> "error" then the discretization becomes unobservable. meaning that the
> curve, within great accuracy, has become CONTINUOUS, in a fuzzy sense.
>
> Read the article for higher precision of the above statement. Comments
> and suggestions for improvement are always quite welcome.
>
> Han de Bruijn

Does your theory say
that smaller and smaller harmonics
of things must exist?
And that, in fact, the original
is actually *made from* these smaller harmonics?
Mine does.
john
From: Han de Bruijn on 7 Jul 2010 03:09
On Jul 6, 12:28 pm, "Androcles" <Headmas...(a)Hogwarts.physics_z> wrote:

[ .. snip original posting .. ]
>
> Congratulations, you've discovered 1980s chaos theory.

Alas, your comment is not even wrong.

Han de Bruijn

From: Han de Bruijn on 7 Jul 2010 03:26
On Jul 6, 8:43 pm, john <vega...(a)accesscomm.ca> wrote:
> On Jul 6, 4:00 am, Han de Bruijn <umum...(a)gmail.com> wrote:
>
[ .. snip original posting .. ]
>
> Does your theory say
> that smaller and smaller harmonics
> of things must exist?
> And that, in fact, the original
> is actually *made from* these smaller harmonics?
> Mine does.

No. In my theory the spread of the Gausssians is chosen such that all
harmonics disappear. The result for sigma = Delta/(2.Pi).alpha, with
alpha = sqrt(2.ln(1/epsilon)) is: sigma.sqrt(2.Pi).1/Delta , where
(Delta) is the discretization and (epsilon) is the relative error in
observation. The truth of things is not in Generalities, but in tiny
Details. It's a SPECIAL theory, not a general one. (Though it can be
made quite general, perhaps, LATER).

Han de Bruijn
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