The Special Theory of Discrete Continuity
in [Math]
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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 |