NarcyChadlite
Aesthete, scholar, Meghan's husband
- Apr 11, 2021
- 764
Today I have been trying to discover a family of mathematical functions which spit out the number of days someone will be banned for based on their ban percentage.
Chapter 1. Building the function
The argument of the function is the ban percent (50% ban, 70% ban, 90% ban, 100% ban, you got the idea). For simplicity purposes, I am going to convert percentages into their decimal forms (namely, 0.5, 0.7, 0.9, 1). In this sense, the domain can only be the interval from 0 to 1 (you cannot be issued a 200% ban or a -10% ban). The range, on the other hand, is the entire real line. Mathematically, a 100% ban results in a permaban - ergo, an infinity of days.
According to the nigger of neets.me, @Haruhi Suzumiya haruhifag, here is how the ban system works:
0% ban - of course, 0 days. It makes no sense to be banned for 0% ban, so the origin of the Cartesian system is included in the graph of the function. (Relation 1)
70% ban - 2 days ban
90% ban - 4 days ban
100% ban - permaban
So it makes sense to write the following mathematical relations (essentially, a system of functional equations):
f(0) = 0 (Condition 1, Relation 1)
f(0.7) = 2 (Condition 2)
f(0.9) = 4 (Condition 3)
lim{x->1}f(x)=inf (Condition 4, Relation 2)
2. Remarkable solutions
The first relation (f(0)=0) tells us an incredibly important aspect about the structure of our function. When x=0 (0% ban), f(x)=0 (banned for 0 days). If I wish to express my function as a fraction, this means that x MUST appear in the numerator. (Relation 2)
The second relation (the permaban case, where a user is banned for infinite days for a 100% ban) is also incredibly important.
At first, I thought the function must fundamentally be an exponential. But the condition made above calls for a vertical asymptote.
That's right, the line x=1 is a vertical asymptote to the graph of our function. So my function must be fundamentally rational!
Vertical asymptotes (pointing, in this case, towards positive infinity for the left limit) usually occur upon dividing by 0. This means that my function has the term (1-x) in the denominator! If x=1, 1-x =0, my function gets divided by 0 yielding positive infinity! (Relation 3)
From relations 2, and 3, our function must have this structure:
Now, of course, this is not enough. Our function so far only satisfies conditions 1 and 2. It does have a vertical asymptote and it does include the origin point, but its graph does not cross the points (0.7, 2), (0.9, 4) (conditions 2 and 3), as my Desmos graph shows:
We shall further build upon this family of functions.
Chapter 3. Interpolation
Mathematically, graphing a function using only a system of points is known as interpolation (the graph must go through all specified points). This would have easily been achieved had our function been a polynomial. Unfortunately, that is not the case, since it is rational. There are 3 methods at the forefront of polynomial interpolation - Lagrange, Newton, and Neville. I have been trying using those interpolations myself, but to no avail. By crossing out the (1-x) term (and compensating for this by dividing by 1-x in the expression of the resulting polynomial), all three methods returned the same interesting result -
After the rearrangement made above it becomes quite obious that upon dividing by (1-x), the function turns into the linear function f(x)=x. It was to be expected. This method DOES NOT work.
After experimenting a bit with these functions, I came to the conclusion that an exponential might be the way to go.
Namely,
Now it might seem we got even further from our previous result! However, this expression gives us the freedom to manipulate the graph in multiple ways.
For simplicity purposes, I will purposefully try to structurally adjust this function and discover a constant a for which
It's quite complicated to work out the correct values from the very beginning (namely, f(0.7)=2, f(0.9)=4). If I manage to solve an exponential equation and find the correct a for which that proportionality law holds true, I am almost done!
We could, then, write the function so as to exactly get f(0.7)=2 and f(0.9)=4:
And THAT gentleman and gentleman (bcos femoids should rot in hell) is the FINAL form of our function. We now shall move onto calculating a and b.
Chapter 3. Exponential equation
(i wont bother posting the equations here, I used LaTeX instead):
And that is the value of a for which 2*f(0.7) = f(0.9). We have finally solved the proportionality law!!! Now we only have to solve for b and actually make the two values equal to 2, and 4 respectively!!!
Plotting the identified value for a into our function, we obtain that
f(0.7) ~ 0.234234971b = 2
f(0.9) ~ 0.468469942b = 4
We can therefore find a value for b:
b = 4.269217337...
Now here is what Desmos has to say:
It works!!! It fucking works!!
It respects all 4 necessary conditions. We can finally describe the function as
Which is lifefuel! Our mission here is done
Chapter 5. Final notes
I am not completely satisfied with this function in particular. It follows an unnatural concavity (notice how approximately f(x)=0.43 is an inflection point, the function becoming concave after starting out as convex). It does indeed follow our axiomatic rules, but it seems a bit farfetched. Nevertheless, disclosing a family of functions capable of telling us for how many days a user will be banned based on their ban percentage is truly an achievement on its own.
Chapter 6. Where next?
We shall further explore the behavior of trigonometric functions for our purposes.
The function f(x) = x/(1-x) * arcsin(ax) * b (with different values for a and b, of course) achieves quite the result (still, without crossing the points of interest):
A Taylor series might do the job too. But I have tried so far to stick to simple fundamental algebraic and trigonometric families of functions. The 3-step approach - finding remarkable solutions, interpolation/proportionality and fine-tuning the result with the coefficient b - seems quite rewarding.
Tagging the brocels:
@Atila @Neetgod2 @Looksmax Refugee II @Optimus @elzde95 @kaang @NeverEndingWinter @Eren @MilkisterMooTwo @Alexanderr @Ritalincel @Lucillian @IGiveUp @VoDkA @Lain @chudur-budur @Copexodius Maximus
Chapter 1. Building the function
The argument of the function is the ban percent (50% ban, 70% ban, 90% ban, 100% ban, you got the idea). For simplicity purposes, I am going to convert percentages into their decimal forms (namely, 0.5, 0.7, 0.9, 1). In this sense, the domain can only be the interval from 0 to 1 (you cannot be issued a 200% ban or a -10% ban). The range, on the other hand, is the entire real line. Mathematically, a 100% ban results in a permaban - ergo, an infinity of days.
f : [0; 1] -> ℝ
According to the nigger of neets.me, @Haruhi Suzumiya haruhifag, here is how the ban system works:
0% ban - of course, 0 days. It makes no sense to be banned for 0% ban, so the origin of the Cartesian system is included in the graph of the function. (Relation 1)
70% ban - 2 days ban
90% ban - 4 days ban
100% ban - permaban
So it makes sense to write the following mathematical relations (essentially, a system of functional equations):
f(0) = 0 (Condition 1, Relation 1)
f(0.7) = 2 (Condition 2)
f(0.9) = 4 (Condition 3)
lim{x->1}f(x)=inf (Condition 4, Relation 2)
2. Remarkable solutions
The first relation (f(0)=0) tells us an incredibly important aspect about the structure of our function. When x=0 (0% ban), f(x)=0 (banned for 0 days). If I wish to express my function as a fraction, this means that x MUST appear in the numerator. (Relation 2)
The second relation (the permaban case, where a user is banned for infinite days for a 100% ban) is also incredibly important.
At first, I thought the function must fundamentally be an exponential. But the condition made above calls for a vertical asymptote.
That's right, the line x=1 is a vertical asymptote to the graph of our function. So my function must be fundamentally rational!
Vertical asymptotes (pointing, in this case, towards positive infinity for the left limit) usually occur upon dividing by 0. This means that my function has the term (1-x) in the denominator! If x=1, 1-x =0, my function gets divided by 0 yielding positive infinity!
(Relation 3)From relations 2, and 3, our function must have this structure:
f(x) ~ x/(1-x)
Now, of course, this is not enough. Our function so far only satisfies conditions 1 and 2. It does have a vertical asymptote and it does include the origin point, but its graph does not cross the points (0.7, 2), (0.9, 4) (conditions 2 and 3), as my Desmos graph shows:
We shall further build upon this family of functions.
Chapter 3. Interpolation
Mathematically, graphing a function using only a system of points is known as interpolation (the graph must go through all specified points). This would have easily been achieved had our function been a polynomial. Unfortunately, that is not the case, since it is rational. There are 3 methods at the forefront of polynomial interpolation - Lagrange, Newton, and Neville. I have been trying using those interpolations myself, but to no avail. By crossing out the (1-x) term (and compensating for this by dividing by 1-x in the expression of the resulting polynomial), all three methods returned the same interesting result -
f(x) = -x^2 + x = -x(x - 1) = x(1 - x)
After the rearrangement made above it becomes quite obious that upon dividing by (1-x), the function turns into the linear function f(x)=x. It was to be expected. This method DOES NOT work.
After experimenting a bit with these functions, I came to the conclusion that an exponential might be the way to go.
Namely,
f(x) ~ x/(1-x) * e^(ax)
The base e was arbitrarily chosen (it could have been any other base and the constant a is adjusted based on the base). A is a really important coefficient. a=1 does not work by itself;Now it might seem we got even further from our previous result! However, this expression gives us the freedom to manipulate the graph in multiple ways.
For simplicity purposes, I will purposefully try to structurally adjust this function and discover a constant a for which
2*f(0.7) = f(0.9) (law of proportionality)
It's quite complicated to work out the correct values from the very beginning (namely, f(0.7)=2, f(0.9)=4). If I manage to solve an exponential equation and find the correct a for which that proportionality law holds true, I am almost done!
We could, then, write the function so as to exactly get f(0.7)=2 and f(0.9)=4:
f(x) = x/(1-x) * e^(a*x) * b
Where a and b are constants.And THAT gentleman and gentleman (bcos femoids should rot in hell) is the FINAL form of our function. We now shall move onto calculating a and b.
Chapter 3. Exponential equation
(i wont bother posting the equations here, I used LaTeX instead):
And that is the value of a for which 2*f(0.7) = f(0.9). We have finally solved the proportionality law!!! Now we only have to solve for b and actually make the two values equal to 2, and 4 respectively!!!
Plotting the identified value for a into our function, we obtain that
f(0.7) ~ 0.234234971b = 2
f(0.9) ~ 0.468469942b = 4
We can therefore find a value for b:
b = 4.269217337...
Now here is what Desmos has to say:
It works!!! It fucking works!!
It respects all 4 necessary conditions. We can finally describe the function as
Which is lifefuel! Our mission here is done
Chapter 5. Final notes
I am not completely satisfied with this function in particular. It follows an unnatural concavity (notice how approximately f(x)=0.43 is an inflection point, the function becoming concave after starting out as convex). It does indeed follow our axiomatic rules, but it seems a bit farfetched. Nevertheless, disclosing a family of functions capable of telling us for how many days a user will be banned based on their ban percentage is truly an achievement on its own.
Chapter 6. Where next?
We shall further explore the behavior of trigonometric functions for our purposes.
The function f(x) = x/(1-x) * arcsin(ax) * b (with different values for a and b, of course) achieves quite the result (still, without crossing the points of interest):
A Taylor series might do the job too. But I have tried so far to stick to simple fundamental algebraic and trigonometric families of functions. The 3-step approach - finding remarkable solutions, interpolation/proportionality and fine-tuning the result with the coefficient b - seems quite rewarding.
Anyways, thank you for reading this!
You can now calculate for yourself the amount of days you will be banned for with the function above!
You can now calculate for yourself the amount of days you will be banned for with the function above!
Tagging the brocels:
@Atila @Neetgod2 @Looksmax Refugee II @Optimus @elzde95 @kaang @NeverEndingWinter @Eren @MilkisterMooTwo @Alexanderr @Ritalincel @Lucillian @IGiveUp @VoDkA @Lain @chudur-budur @Copexodius Maximus
