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September 25th, 2015, 08:00 AM   #1
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More Complicate modulus algebra

As told in an old post, I figured was possible to rise with my sum several results:

- the first I show are Powers in N, than in Q than in R,

- But now I've little time to play with what I already know is possible to do:

More Complicate Modulus Algebra:

Is again a modified Sum, like my Step Sum but instead to make Fix Step 1/K, 2/K..... till A or else... we have a

New Variable Step Sum: so we now move ahead the index "y" following a known, complicate as we want, y= f(X)...

For example is possible to rise each result of the integer of type A(X)= X*(2X-1) = 2X^2-X

If we use this new Variable Step Sum where the index y = F(X):

y=2X-1 (so y= odds just...)

where uppercase X is 1,2,3,4,5,..... so the index y is

y1=2*X1-1 = 1
y2=2*X2-1 = 3
...
yn=2*Xn-1

So for example we have the variable step sum:

$\displaystyle P= 2A^2-A = \sum_{X=1 ; y=2X-1}^{X=A} 2y-1 $


That is A000384, as just a simple trick as example...

Changing the function F(X) that generate the index y is possible to make very complex m-clocks..


One just little more complicate, for example, give use the A140676:


$\displaystyle A(n) = (A140676) = \sum_{X=1 ; x=3X+1}^{X=n} 2x-1 $

so:

(A140676)= 0, 7, 20, 39, 64, 95, 132, 175, 224, 279, 340, 407, 480, 559, 64


... in some minutes of play I found A143941 where

A(n) = n*(1 + 6*n + 2*n^2)/3

etc...

So now you have an instrument that can generate billions of sequence... by the interaction of 2 simple gear.

...But you can add more and more complicate gear till you want !

So I let you digest this n-th Idiosincratic notation (where I probably find a way to let it more clear), and you can thanks to 2 days of Flu with a Laptop in the hands and silence in the emplty sweet home ...

Thanks
Ciao
Stefano
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September 27th, 2015, 08:29 AM   #2
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I call:

F(x) = CARRIER

F(X) = MODULATOR

I try with

F(X) = (sin(X))^2

F(x) = 2x-1

and joining the points I've back a sort of A.M. signal....

It works just with liner CARRIER terms, so in a squares field, since with bigger power the Carrier fell the "signal" Down or push it to diverge.

That is what I've in my as an explanation why in our square driven real world wave can exist, and while cubic or more power effect will just let our wave fell down or diverge...

That, of course, is just a thought....



Last edited by complicatemodulus; September 27th, 2015 at 08:32 AM.
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September 27th, 2015, 10:53 PM   #3
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This trick stinks lot...

X= 1/10,,,, p/10

Modulator: x= (sin(x)^2)

Carrier: 2x-1

Here the result: Modulator max /min are at same abscissa of the SUM zeros...

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September 28th, 2015, 10:23 AM   #4
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Last picture here just to show the power of simple modulator (sin dependent) / carrier (here n=3 rational develope) functions:



As told the cubic carrier let the the SUM go +/- infinite.

It looks of course like "random" distribution, like another most famous, but it is not...

A most known Riemann work seems has influence also here...

But I stop dreaming and I return to reality since I've to understand better how to use this trick to have what I want...

Thanks
Ciao
Stefano
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