Wave Divisor Function

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Dec 2014
52
12
Netherlands
Hello,

Maybe for someone interested. Here is another video of the wave divisor function (n choose k notation).

This is the solution between: 0 and 2. All divisor waves are 1 at x=0. The divisors wave will curl up in the plane.

Interesting to see that patterns seem to evolve. Not understood completely by me. Sometimes the odd and even numbers are identified.

Video should work at 1080p.

https://drive.google.com/open?id=1S9robtEt4aAy96mV19yLgB-Sgv3zKhfi

Best regards,

Vincent Preemen
 
Dec 2014
52
12
Netherlands
Hello,

With the trigonometric notation, every number and it's divisors can be displayed as "orbitals" in the Re Im plane. See video below.

The Re axis shows the number of divisors. A pulsewidth has been selected so that the whole solution is displayed with a bandwidth. Every number to my understanding has its unique Orbital. Maybe there are patterns between divisors.

Ohh, I do not include 1 as a divisor. So solution is -1 !!!! :rolleyes:

More on orbitals in link original documents first post.

YouTube:
https://www.youtube.com/watch?v=fLhLaCf4xcM

Or Drive:
https://drive.google.com/open?id=1wbqomfTesSyGPvBdOh-P2OwPuWGEufjq

Gr,

Vince
 
Last edited by a moderator:
Dec 2014
52
12
Netherlands
Hello,

I updated the concept summary hope to make it more clear. I am a math noob but I try my best. Slide number 3 summarizes my results.

Concept Summary:
https://drive.google.com/Concept Summary 1.8.pdf

Slide 3:
Summary.jpg

Next I want to attempt to Fourier transform the wave divisor function.
Analysis showed it is also a discrete bellshape form.

I hoped for some feedback.

Best regards,

Vincent
 
Dec 2014
52
12
Netherlands
Hello,

The video shows the number of divisors on the Re axis of 46 (2, 23, 46). Divisor count 1 is excluded.

https://youtu.be/6h4M7hxTZz4

The divisor count from wave divisor function has an error. For wide wave packages (dx is big) the error is bigger. When wavepackage is smaller the divisor count is determined more accurate.

With narrow wave packages the symmetry will be improved. The influence of neighbor divisors is then less.

I also made some small updates on the concept summary:

Concept Summary Wave Divisor Function Rev 1.9 (pdf)


Best regards,

Vince
 
Dec 2014
52
12
Netherlands
Hello,

Last 3 days a crash course in Python my Holiday :). Newbe never programmed with python/Anaconda/GitHub/Mybinder before. I made a "Jupyter' file about the Wave Divisor Function.

When I understand correctly non Python/Anaconda users can open the file trough "mybinder". All code should run in the cloud (Whawww :cool:).

Jupyter File (along MyBinder):
https://mybinder.org/v2/gh/oooVincentooo/Shared/master?filepath=Wave%20Divisor%20Function%20rev%202.4.ipynb

Loading might take some time. Graphs should be interactive. In order to run the Python code I had to select in the Main menu: [Cells] $\rightarrow$ [Run All] twice. Or select [Kernell] $\rightarrow$ [Restart & Clear Output].

No guarantees it all works. It's still a hobby! Cloud not perfect yet. :spin:

Best regards,

Vince
 
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Dec 2014
52
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Jupyter File

Updated Jupyter file (fix Typos etc). For the ones who downloaded *.ipynb.

Mybinder/Cloud link remains the same. Jupyter File (along MyBinder):
https://mybinder.org/v2/gh/oooVincentooo/Shared/master?filepath=Wave Divisor Function rev 2.4.ipynb

Loading might take some time. Graphs should be interactive.

In order to run the Python code I had to select in the Main menu: [Cells] $\rightarrow$ [Run All] twice. And/Or select [Kernell] $\rightarrow$ [Restart & Clear Output].

Hoped for some feedback. Thank You,

Vincent Preemen
 
Dec 2014
52
12
Netherlands
Updated Jupyter file.

It is amazing technology Jupyter notebook. File is only 285KB! Starting to get hooked on it.

Zooming and rescaling editing data in graphs is amazing :), could not have don it in any spreadsheet.

- Updated content.
- Added section "Wave Pulse Outline".
- For interactive plots menu: [Cells]→[Run All] (one time is sufficient now)

link:
https://mybinder.org/v2/gh/oooVincentooo/Shared/master?filepath=Wave%20Divisor%20Function%20rev%202.4.ipynb

btw also cool that the link remains avtice although I pushed update Jupyter file!

Best regards,

Vince
 
Dec 2014
52
12
Netherlands
Dec 2014
52
12
Netherlands
Dirichlet's divisor problem

Hello,

With the wave divisor function I calculated the error in: "Dirichlet's divisor problem".

$ \mathcal{O}(x^{\Theta})$

Note that the wave divisor function is excluding 1 as divisor. So Dirichlet equation may look strange.

For image:
Theta.jpg

More information in Jupyter notebook:
https://mybinder.org/v2/gh/oooVincentooo/Shared/master?filepath=Wave%20Divisor%20Function%20rev%202.4.ipynb

Notebook is interactive press: [Cell] $\rightarrow$ [Run All]. First loading might take some time.

I really hoped for some input. Not sure how to interpreted it all. I do not have the capabilities to continue any further. Stretching my limited math skills to the most right now. :unsure:

Best regards,

Vince
 
Dec 2014
52
12
Netherlands
Hello,

With the wave divisor function I calculated the error in: "Dirichlet's divisor problem".

For image:
View attachment 10531

More information in Jupyter notebook:
https://mybinder.org/v2/gh/oooVincentooo/Shared/master?filepath=Wave%20Divisor%20Function%20rev%202.4.ipynb

Vince
Did some further analysis do understand less. $\Theta=0.5$ keeps popping up for low range, that's the only good thing. For higher range many questions and not reproducible. :confused:

Gr,

Vince
 
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