Wednesday, October 19, 2016

Can We Solve a Nonlinear Equation With Many Variables? (Con)

Following to article of "Can We Solve a Nonlinear Equation with Many Variables?" posted on below link:

Let me start second example for an equation with four variables as follows:

Example (2): Solving of Sphere Equation

As you know, the sphere equation has many applications in all fields of engineering and physics. When we talk about a sphere equation, our discussion can be expanded not only macroscopic systems but also microscopic particles such as the quantum model of the Hydrogen atom. Therefore, let me start by solving of a sphere equation for a limited domain and range as follows:

Consider the sphere equation with below domain:    

If   x^2 + y^2 + z^2 = r^2     x, y,z ϵ N,            x, y,z ≤ 100         

Then the range for the radius of sphere will be:    r ϵ N,            r ≤ 173

Now, I apply previous method and get all results of "x, y, z" for "r" in given range. The number of 
results related to "r" has been presented on below graph:

In this case, total sum of possible answers is only and only equal to 4935.
Above graph shows us that there is a maximum number of answers equal to 165 for r = 99. For instance, I have brought some results on below figure:

As we can see, for r = 15 and  r = 150, the number of results are the same equal to 15 while for r = 31, we have 24 answers

We can apply this domain and range as a template for all macroscopic and microscopic numbers.