Fourier Time

As time goes by, the various tools become more familiar. It takes the brain some time and energy to establish the models for managing methods. It seems that I play blender, gimp, inkscape, zim, python, shell, ... like a piano now. I hardly notice that I perform a series of a dozen key strokes to get from one aspect to another. Triple reference frames with vector sums of a manifold is not an easy transform to nD → 4D → 3D → 2D, much is lost. I wish there was a better way to xform and that is a WIP. This is a Fourier transform product of the S_G(Δt&S₁δs·S₂δs)<â₁,â₃,â₃> time wing. A label for what happens in multiple simultaneous reference frames. Even ζ(s) eventually becomes familiar. This is the sum of the vector product in time and space. I really dislike the fact that time is assumed to be a real dimension. It is the relative relationship of change and so is as arbitrary as the direction North.

Another interesting thing is some thoughts about the brain in a holistic way. Here is a link to a "Research Fellow", Steven Lehar at Harvard. I have come to realise that some aspects of superconducting theory can be applied at room temperature ( actually any T°K ) and explain some really odd happening at the molecular level with DNA and proteins. I suppose that it is possible that the brain also acts as a multi-mode resonant parallel computer on top of a serial computer. Nature seems to come up with some really odd combinations. Isn't a flying parasite odd ? Plant/fungus sex is sure some kinky stuff. It seems to me that thought at a chemical level in the cell and its amplification and fine tuning, is consistent with the idea of holistic interaction and the gross process of neural action. It seems to me that the "Matrix" is a possibility, but what would the gain be to control a bunch of funky bio jerks, when a cyborg is much more manageable and effective for most purposes.

Hey, I thought of a good movie. Terminator III where people dress up as terminators to infiltrate the cyborgs. It could be told from the viewpoint of a cyber girl who is the future parent of all cyborgs. Synthia Borgia

Solving Fourier without the imaginary

This equation is the core. This could be viewed as the differential flow (in one dimension) of anything. I would write this program as:

#include <stdio.h>
int main(int argc , char * argv[]){
const int h_length=20;
float HD=0.0;
float H[h_length];
float k=2.0;
int l=0;
int i;
for (i=1;i<h_length;i++)
 {
 H[i]=0.0;
 }
H[0]=100.0; 
while (H[0] < 6.0)
{
l+=1;
printf("%d %f %f \n",l,H[0],H[h_length-1]);
 for (i=1;i<h_length;i++)
 {
 HD=(H[i-1]-H[i])/k;
 H[i]+=HD;
 H[i-1]-=HD;
 }
}
return 0;}

What this works out to is a process that describes linear transfer. What happens in general is that as time passes, each term is divided over and over again until it reaches the end of this array. That is a simulated bar of metal of some length and with a specific heat transfer rate and some time interval that goes to 0. The difference between 0 and 1 is 1/n and the difference then in the next step is 1/n^2 and so on. This may well be the wrong equation to start with. My experience with momentum and interaction in systems leads me to choose a different approach to this. What this describes is a one dimensional boundary condition which is applicable to all momentum systems, which is everything in this universe I have seen.

It gets more complex from here and that just makes for more fun. I am beginning to see some new relationships here and this looks like a number that could have some important cousin numbers. There is good reason why e^t is very meaningful. 't' is the while. Very interesting stuff. If you change the equation a little it could even describe a vampire I know.

ADDED: After implementing this and graphing the result it was obvious that it was not exactly what I wanted and there is some interaction with the FPU limits of digits. Using double and excluding FPU sign on below threshold values it becomes this done in reverse order, which is more reasonable.

for (i=h_length;i>0;i--)
{
HD=(H[i-1]-H[i])/k;
if( HD > 0.0){ H[i]+=HD; H[i-1]-=HD;}
}

It is very interesting as the graphic is very fluid and natural, as you would expect from something that simulates a natural thing.

I have a 3 dimensional version of the Fourier-like infinite series and it is my assumption that I will have an n-dimensional universal transfer equation when I am done. The spatial conversions with relativity are quite convoluted and so I will test for a while and do some repairs so I can do some more videos and have the equations, code, videos, and animated gifs on the web page as links so they don't clutter the blog.

I managed to make a major breakthrough in the mathematics of angular motion which hopefully will help to simplify the perception and use of complex angular systems like gyroscopes.

By applying any recursive function I should get various types of wave motion. My opinion is that there are two types of sines or perhaps a combined sin(2Πx)*f(x) relationship which is perhaps what is considered the "imaginary" component. Oh well, I wonder how many people will be able or even care to follow this into a new way of looking at the root mathematical algorithms.

I am not sure whether I mentioned that I had included 2 and 3 dimensional function mapping and graphical capabilities into af. If I didn't mention it, well there it is now. This particular graph is scaled in Y by 0.20. While looking at that I decided I would add difference mapping in red on top of the elements.

Fourier, entropy of junk DNA

Googling for this, returned some close matches and it is reasonable to assume that if the DNA is not compressible that it contains information, the easy, and perhaps wrong, answer. It also seems that information which is complex and repetitive would represent some type of wave on wave action that would be amenable to Fourier. It probably would be projected against several other functions that are themselves convolved.

There is some doubt on my part that a sequence that is easily compressible is actually without information. Physical systems are not integer systems and they are also 4 dimensional. An example might be wrapping a string with regularly spaced dots around a bumpy coffee can to reveal that a picture is formed. It does seem that a regular pattern, on irregular space can contain information and that is making me wonder if it really is true that compressible information is without meaning, when taken out of context. The continuous and subtle variation in bond strength of GC and AT pairs could just as well be the information measure and by simply saying that it contains no information because it can be compressed to n*(GACT) would likely not be valid.

My conclusion is that, the assumption that compressible data contains no information, is only valid in a limited context. Physical systems are continuously variable in 3 space and do not have the same unit time, binary state-wise restriction. I might guess now, that a regularly repeating pattern in a very irregular world is information because it is a difference to what would arise naturally and requires energy and work to exist.

ADDED: A good example of this would be an image interleaved on some number which would seem to be random to the eye, but if a regular repeating mask was applied to that, it would reveal a comprehensible form. The regular pattern could certainly be compressed , but in this context it would contain a great deal of information because of its association with a second set. Biological entities are required to perform work to create DNA and as such it would seem that from a naive perspective that it implies some purpose in the construction.

More Fourier

I am doing a complete review of Fourier and how it relates to my understanding of frequency decomposition of information. I designed an RF analyzer for $AGENCY at one time, long ago, and it was supposedly more elegant than any signal processor that existed in the world, however that remains to be proved as it is not open to observation for a few reasons.

I am learning SciLab and attempting to simulate the functions in my program in openGL as I observe them in MatLab, Mathematica, SciLab, and Octave.

ADDED: Now things are getting real ℜ hinky , as they say. Below is the DTFT (Discrete Time Fourier Transfer) and it includes the imaginary. Here is where things get interesting as I think I have a way that resolves all this. Symmetry is the way that I have heard discussion of why it is real and as such I don't disagree, however I believe the problem lies very deep and Q or quantity can be manipulated in a lot of strange ways, however when you add that second point as 0(Zero) or any other reference point, then you have defined dimensionality itself and you are dealing with a vector, whether you recognize that fact or not. It is all very much fun, whatever the outcome.

This is not the way I solve it and I will put up my account of how I deal with periodic signals, phase, decomposition, recomposition, identification, eigenvalues, and amplitude, etc. I think that my way of representing and utilizing a periodic is more effective and more straight forward. The DTFT is very elegant and fun to play with, but it may not be the best way. Time will tell if I am being skeptical of something I am bound to accept for its utility and application, IDK.

And here is the La^Tex, just to be consistent.

S_T(f) = \sum_{k=-\infty}^{\infty} S\left(f - \frac{k}{T}\right) \equiv \sum_{n=-\infty}^{\infty} \underbrace{T\cdot s(nT)}_{s[n]} \cdot e^{-i 2\pi f n T},

I see math in the same way that I see Python functions or Objective C++, there are methods which can be associated with objects and some objects cannot have some methods assigned without getting indeterminate results. Lisp and Forth and to some extent other languages deal with this hierarchy of function in such a way that are easier, but I don't like to sacrifice the complex utility for that level of control. It needs more complexity, and I think that it is like XML, if it doesn't work, add more XML or in this case, more complexity.

Many equations

This is mix of various linear, polar, Fourier, equations and methods. The scale of what is possible is so overwhelming sometimes. It is difficult to grasp that a single image that is 1000 by 1000 can have states of existence that exceed 2 to the millionth power. Underneath all that is :
f(x)=sin(e^x) [ In polar coordinates ]
, and that is the butterfly wing like yellow fan.

I have been doing Fourier analysis of various images I find on the web and it is interesting, what lies behind the images.

Gimp fourier

Yesterday was Fourier day and I had a lot of fun. I also managed to visit the works of Gauss, Mach and many others. I also went through an entire MIT course video lecture on linear Algebra and matrices. I did some analysis with Fourier and some open source "C" code. I also found this nice link to wikibooks and FFT with gimp. HERE . Link seems to have changed and a new one is HERE at Wikibooks. Apparently they were being neat and doing "Camel Case" on links . :)

The link shows how to apply FFT filters and allows you to manipulate frequency domain aspects of an image. I was able to remove interference noise from an image in a matter of seconds. I learned many things yesterday about frequency response of the eye, common illusion that affect perception, how to enhance images for print processing, and a better understanding of what things like "bi-cubic" and other options in filtering imply with OpenGL and image processing.

Today will be a Paul day. I discovered some new ways to apply transforms that are conceptually like Fourier, but use a different method. I am also wondering what is on the outside of a CFT ( continuous Fourier transform ). It seems to me that the transform of a continuous has an infinite progression that would go through many different convolutions, depending on the complexity of the original content. I also see that gimp with FFT filters forward and reverse are going to be a lot of fun. I discovered a lot of new ways of steganographic encoding too.

I will have to make every March 15 a Fourier day. I think that today will study Euclidian math.

I will probably start with Euclid and go through as much dimensional topology as I can tolerate.

Fourier day

Today I am playing with math : harmonics, waves, and topology. Specifically I want to have a handle on using Fourier methods and FFT in some aspects of my project. I am also working on algorithms like trees, associative matrices , normals, surfaces, vectors.... and testing some LAN protocols for my machines.

ADDED: X(f)\ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} x(t) \ e^{- i 2 \pi f t} \ dt. \ which is this in LaTex And I have an interesting link to a simple explanation of Fourier which is here: LINK and I will see how useful that is.

ADDED more , the uplink from the fourier was even more interesting as it is about neuro http://sharp.bu.edu/~slehar/. It is some interesting reading. The fourier description was instructive and suggests some applications I had not "grokked" before.

"In the fields of hell where the grass grows high Are the --_graves-- of dreams allowed to die." -- Richard Harter

And a new link if you like FFT and this looks like a winner.
http://www.archive.org/details/Lectures_on_Image_Processing