The Special Theory of Discrete Continuity
in [Math]
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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 |