Sawtooth function as a Wave Function which requires two formulas and uses two domains

[Archimedes Plutonium]

Newsgroups: sci.math
Date: Mon, 13 Feb 2017 23:21:21 -0800 (PST)

Subject: sine as Y = + sqrt(1 - x^2) Re: sine as wholly a polynomial;
From: Archimedes Plutonium <***@gmail.com>
Injection-Date: Tue, 14 Feb 2017 07:21:22 +0000


sine as Y = + sqrt(1 - x^2) Re: sine as wholly a polynomial;

Alright, so, let us try it out for a test drive:

SINE function as purely a polynomial

The semicircle for positive sine is

Y = + sqrt(1 - x^2) for domain of x as 1, 5, 9, . .

the semicircle for negative sine is

Y = - sqrt(1 - x^2) for domain x as 3, 7, 11, . . .

So, we mark those integers as center of circle, then with equation we fulfill the composite function upon x values of a specific Grid.

What we end up with is the graph of the Sine Function with unit circle radius. Now if we wanted a smaller or larger radius than 1, our sine function is then a semiellipse wave, not a semicircle wave.

Now doing the same for positive Cosine

Y = + sqrt(1 - x^2) for domain of x as 0, 4, 8, . .

the semicircle for negative cosine is

Y = - sqrt(1 - x^2) for domain x as 2, 6, 10, . . .

Now a Composite function is almost like a ordinary function except it has to have two mappings, so a bit more complex. We mapp the function onto the Naturals and once the center is mapped we map the semicircle.

What this does is eliminate sine and cosine from being functions outside of polynomials, as functions as unique to themselves, odd, strange, mystical, because they are derived from having to do geometry chores-- constructing a circle, then spinning right-triangles inside the circle. Whereas a function like Y = x^3 - x^2 or Y = 4x +5 require no chores of geometry, just simply map them.

By making the trig functions as polynomials we cut away that geometry chores.

Math, like physics, always wants and yearns for unification of its objects of study. Until now, trig functions were important but alien and outsiders of all other functions.

AP

Newsgroups: sci.math
Date: Tue, 14 Feb 2017 15:11:56 -0800 (PST)

Subject: Sawtooth Wave function Y=x with Y=-x+1, compared to Sine Wave function
From: Archimedes Plutonium <***@gmail.com>
Injection-Date: Tue, 14 Feb 2017 23:11:56 +0000


Sawtooth Wave function Y=x with Y=-x+1, compared to Sine Wave function

Alright, I think that all Wave Functions will have at least two equations. As I build the Sawtooth wave function.

Just as in Sine wave, I first mapp onto the Naturals with two functions and then I mapp the two functions onto the x-axis of Grid System

In Sine it was :

SINE function as purely a polynomial

The semicircle for positive sine is

Y = + sqrt(1 - x^2) for domain of x as 1, 5, 9, . .

the semicircle for negative sine is

Y = - sqrt(1 - x^2) for domain x as 3, 7, 11, . . .

In Sawtooth it is::

Y = x for domain of x as 0, 2, 4, 6, . .
Y = -x +1 for domain of x as 1, 3, 5, . .

What that wave function looks like is this ^^^^^^^^^

AP

[Archimedes Plutonium]

Alright, so I have the Sawtooth function as a wave function:

Sawtooth Wave function Y=x with Y=-x+1, compared to Sine Wave function

Alright, I think that all Wave Functions will have at least two equations. As I build the Sawtooth wave function.

Just as in Sine wave, I first mapp onto the Naturals with two functions and then I mapp the two functions onto the x-axis of Grid System

In Sine it was :

SINE function as purely a polynomial

The semicircle for positive sine is

Y = + sqrt(1 - x^2) for domain of x as 1, 5, 9, . .

the semicircle for negative sine is

Y = - sqrt(1 - x^2) for domain x as 3, 7, 11, . . .

In Sawtooth it is::

Y = x for domain of x as 0, 2, 4, 6, . .
Y = -x +1 for domain of x as 1, 3, 5, . .

What that wave function looks like is this ^^^^^^^^^

STEP FUNCTION

Alright in Old Math, they actually could never do a step function, not with their idea of a continuum, for picture this:


______

______
0 1 2

__________________> x axis

Now that step function would be defined in Old Math as Y =1 in interval [0,1) and Y =2 in interval [1,2), and notice the interval is closed on one end and open on the other. And the reason such a function does not and cannot exist, is that there cannot be a point in which to end the function at Y=1 nor end it at Y=2.

In New Math, since we have Grids, we always know what our nearest neighbor points are. In 10 Grid the neighbor of 1 is .9 to the left and 1.1 to the right. So as we draw Y=1 it goes from 0 to .9 and then it zooms upward very steeply but not a vertical from .9 to 1 and then is a horizontal line segment from 1 to 1.9 where it zooms upward steeply to 2. So in New Math, we never need a stupid silly thing as a half closed, half open interval. Only in Old Math with their funny continuum do you need mickey mouse contraptions.

So, how does New Math have a function wave for a step function?

Well, we have the positive integer domain N on the x-axis and define Y = n where n is an element of N from a second domain of x-axis of n_k-1 to n_k

So, what that gives me is a step function fully continuous and looking like this::

__
__/
__/

Fully continuous, because, keep in mind, in New Math a present point has to connect to a successor point with a straightline segment. Only when that connection is a vertical line, is it forbidden.

Trapezoid Wave Function:

This one is interesting for it is identical to Sine, except for being semicircles we have semitrapezoids doing the periodic wavelets. Something looking like this:
__ __
/ \ / \___________> x axis
\__/ \__

similar to sine except trapezoids

So what is the Function Wave for that?

We not not worry about the diagonal steep slope lines for those come automatically in New Math, so the only thing the formula has to specify is the horizontal line segments of ___ ___

So how does one specify this sequence:

___ ___

___ ___


Here again, we have two domains we work with, for we work with the Naturals as domain and mapp the function formula of Y = n where n is an element of N the Naturals. Once we mapp the formula onto N, we then let the formula map onto the x-axis of a Grid system.

So in the above, the upper perforated or punctured line can go from 0 to 1 then 4 to 5 etc while the lower perforated line goes from 2 to 3 then 6 to 7 etc.

Once these intervals are identified the prescribing formulas are Y = n and Y = -n

Now it appears as though I can do Wave Functions with having Two formulas and two domains. I would love to have just one formula, but so many are going to need a negative formula to counterbalance the positive formula. Wave functions need two formula, ordinary functions need just one formula.

AP

[a***@gmail.com]

geshwartxnig, what?... well, iff
you ever bother to use it, you could use it for some thing

Post by Archimedes Plutonium
the semicircle for negative sine is
Y = - sqrt(1 - x^2) for domain x as 3, 7, 11, . . .
Y = x for domain of x as 0, 2, 4, 6, . .
Y = -x +1 for domain of x as 1, 3, 5, . .
What that wave function looks like is this ^^^^^^^^^
AP