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From: Han de Bruijn on 4 Aug 2010 07:08
On Aug 2, 10:26 am, Han de Bruijn <umum...(a)gmail.com> wrote:
> 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

Updated with Fuzzy Optics, combs of Cauchy and Triangle distributions.

Han de Bruijn
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