Can This Unique Mathematical Operation Be Solved?

[Perfectly Innocent]

This post is a sincere request for help. I assume that the following problem can be represented by a differential equation and that only a mathematician can solve it.

Let A={real numbers x such that |x|<1}
Let S={real numbers Z such that 1 < Z < infinity}

Define ^ on A by the rule x^y = (x+y)/(1+xy).
It follows that (A, ^) is an Abelian group:

x^y=y^x
x^0=x
x^(-x)=0
x^(y^z)=(x^y)^z

I'm looking for a function from AxS->S (also written ^) such that, for any x, y, in A and any Z in S:

x^(y^Z) = (x^y)^Z

0^Z=Z
x^Z > Z if x>0
x^Z < Z if x<0

Thanks

 

[AKG]

This post is a sincere request for help. I assume that the following problem can be represented by a differential equation and that only a mathematician can solve it.

Let A={real numbers x such that |x|<1}
Let S={real numbers Z such that 1 < Z < infinity}

Define ^ on A by the rule x^y = (x+y)/(1+xy).
It follows that (A, ^) is an Abelian group:

x^y=y^x
x^0=x
x^(-x)=0
x^(y^z)=(x^y)^z

I'm looking for a function from AxS->S (also written ^) such that, for any x, y, in A and any Z in S:

x^(y^Z) = (x^y)^Z

0^Z=Z
x^Z > Z if x>0
x^Z < Z if x<0

Thanks

I don't have an answer, but I think I have a start:

x^Z = Z + x(Z - 1)

if x = 0, then:
0^Z = Z + 0(Z - 1) = Z, as required

if x < 0, then:
x^Z = Z + x(Z - 1)

since Z > 1, Z-1 > 0, and thus x(Z-1) < 0, therefore x^Z < Z, as required. Also, x must be greater than -1, so Z + x(Z - 1) must be greater than 1, therefore the value of the operation is greater than one, making it an element of S, as required.

It can be shown easily that it works in the case where x > 0.

x^(y^Z)
= x^[Z + y(Z - 1)]
= [Z + y(Z - 1)] + x(Z + y(Z - 1) - 1]
= Z + yZ - y + xZ + xyZ - xy - x

(x^y)^Z
= [(x + y)/(1 + xy)]^Z
= Z + ... (it's not going to work)

At first, I thought it wouldn't work for this condition. Then, after doing it I thought it did work, because for some reason I thought you wanted x^(y^Z) = y^(x^Z). But I figured you wanted associativity, and realized I wasn't showing that, and realized what I was showing wasn't what you wanted. After looking at what you said, I also realized what the first bit was for (defining x^y). If this helps any, then there you go, but it might not. I might think about this some more and see if I can come up with something.

 

[M98Ranger]

for your help

Hello, I am not a mathematician but I can offer some insights on this problem. Firstly, it seems that the equation you are looking for is related to a concept called the "power tower," which is a sequence of exponential operations where each subsequent exponent is the result of the previous operation. For example, 2^(3^(4^5)) would be a power tower with four levels.

In this case, it appears that you are looking for a function that follows the same rules as a power tower, but with a different operation (^) instead of exponentiation. This may be difficult to solve without a deeper understanding of abstract algebra and group theory.

As for the statement that only a mathematician can solve this equation, while it may require advanced mathematical knowledge, I believe that anyone with a strong understanding of algebra and group theory could potentially solve it. It may just take some time and effort to work through the problem.

If you are seeking help with this equation, I would recommend reaching out to a mathematician or someone with a strong background in abstract algebra for assistance. They may be able to provide more specific guidance and insights on how to approach this problem. Best of luck in finding a solution.