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From: Han de Bruijn on 8 Jul 2010 03:38
On Jul 7, 9:26 am, Han de Bruijn <umum...(a)gmail.com> wrote:
> 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).

Paper extended with subsection "Fuzzyfied Straight Line":

http://hdebruijn.soo.dto.tudelft.nl/jaar2010/dikte/document.pdf

And accompanying software:

http://hdebruijn.soo.dto.tudelft.nl/jaar2010/dikte/Project6.exe

Everything is Mathematics, what else .. :

http://hdebruijn.soo.dto.tudelft.nl/jaar2010/index.htm

Han de Bruijn
From: Han de Bruijn on 2 Aug 2010 04:21
On Jul 6, 12:00 pm, 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.

Updated with "Continuing Circular":
http://hdebruijn.soo.dto.tudelft.nl/jaar2010/dikte/document.pdf
http://hdebruijn.soo.dto.tudelft.nl/jaar2010/dikte/project2.exe

At "The Special Theory of Continuity":
http://hdebruijn.soo.dto.tudelft.nl/jaar2010/index.htm#STC

Han de Bruijn
From: Han de Bruijn on 2 Aug 2010 04:26
On Aug 2, 10:21 am, Han de Bruijn <umum...(a)gmail.com> wrote:
> On Jul 6, 12:00 pm, 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.
>
> Updated with "Continuing Circular":
> http://hdebruijn.soo.dto.tudelft.nl/jaar2010/dikte/document.pdf
> http://hdebruijn.soo.dto.tudelft.nl/jaar2010/dikte/project2.exe

Sorry. Make that (case sensitive):
http://hdebruijn.soo.dto.tudelft.nl/jaar2010/dikte/Project2.exe

> At "The Special Theory of Continuity":
> http://hdebruijn.soo.dto.tudelft.nl/jaar2010/index.htm#STC

Han de Bruijn
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