Typically, we are able to solve
system of linear or nonlinear equations which are a set of simultaneous
equations (SE). Definitely, solving of a linear SE is very easy while we have
to use Newton's method to solve nonlinear SEs. The common case for both of them
is, to generate an equation for each variable. It means, we can solve an
equation with three variables, if we have three simultaneous equations or solving
of four variables needs to find a system of four simultaneous equations and so
on.
Can we solve a nonlinear equation with
many variables? Yes. In the special conditions, the answer is
positive.
The purpose of this
article is to present some examples which show us possibility to solve a
nonlinear equation with many variables where we have a good estimation for
limited domain and range of variables. The method applied is the same method
stated in article of "The Generating New Probability Theorems"
posted on link:
The experienced
physicists and engineers have usually the true speculation of domain and range
for the variables while they need to obtain precise amounts for the variables.
Therefore, this method can be useful for them.
At the first, I start by a nonlinear
equation with three variables then four variables and finally five
variables.
Example (1):
One of professional people asked me a question in math and
statistics group of social media as follows:
"I need some help to interpret a stock regression:
SZ is high or low size of the company
BM is high or low book to market size
R is the return of the stock
R = 5% + 6%Bm +2%SZ – 2%*BM*SZ
My question is whether I should short high size and high
book to market stocks (stocks that have both characteristics)?
I analyzed his problem by using above method as follows:
Here is my analysis:
1. To reach the maximum R, you should stay SZ the constant
in low size and then if you increase BM, you will reach the maximum R.
2. If you increase both of them (SZ and BM), you will
significantly decrease R.
3. The most important thing is about R = 0.11 because in
this case, it does not take any difference. In fact, above analysis does not
work for R = 0.11
4. If 0< SZ and BM <1, then maximum amount of R will
be always equal to 0.11 (Rmax
= 0.11)
I think that this is really a magic formula.
Example (2)
This is an example about financial
and risk management.
As you know, the basic theory which
links risk and return rate for all assets is, the Capital Asset Pricing Model.
The equation of CAPM is as follows:
Now, suppose you want to invest on an
asset in which your expected return rate (required return) is equal 12%. The
question is: What are the alternatives or scenarios for three independent
variables of risk – free, beta and market return?
Here, by applying the method stated in
this article, I have obtained 21 answers for three independent variables as
follows:
Example (3): The equation of State for an Ideal Gas
Let me tell you an example about the equation of state for an ideal
gas.
We have:
P.V = N.KB.T
Suppose we have a constant volume (V) equal to 0.03 m3
If Boltzmann’s constant (KB)
is equal to 1.38E-23 J/K, what are the answers for P and N and T?
According
to my method, I found 9 answers which are as follows:
:Where
T = temperature (K) and P = pressure (Pa) and N = number of
molecules
Example (4): Solve Circle Equation
When we open a calculus book, we can
see the signs and footprints of Pythagoras (582 B.C – 496 B.C) everywhere.
Therefore, let me start by solving of circle equation for a limited domain and
range as follows:
Consider the circle equation with
below domain:
If
x^2 + y^2 = r^2
x, y ϵ N, x, y ≤ 100
Then the range will be
r ϵ N, r ≤ 141
Now, I apply above method and get
all results of "x, y" for "r" in given range. All number of
results to "r" has been presented on below graph
In this case, total sum of possible
answers is equal to 126.
For instance, above
graph shows us, if r = 25, 50, 75, 100 then the number of results for "x, y"
are equal to 4 and if r = 65, 85 then the number of results for "x, y"
are equal to 8. The results are as follows
Now, consider the circle
equation with below domain
If
x^2 + y^2 = r^2
0 0 0 x, y ϵ N, x, y ≤ 1
Then the range will be
4 r ϵ N, r ≤ 14 1
If I apply above method, I will generate all
results of "x, y" for "r" in given range. All number of
results to "r" has been presented on below graph:
In this case, total sum of possible
answers is equal to 2068.
For instance, above graph shows us, if
r = 325, 425, 650,725,850,925,975 then the number of results for "x,
y" are equal to 14. The results are as follows:
Note: All researchers and individual people, who are interested in
having this model, don’t hesitate to send their request to below addresses:
WhatsApp: +98 9109250225
