Blog post: Unity+: Aleph Integrals - A relationship between equations and their ratios

In an earlier post, I talked about the existence of what is called the Collatzian ratio, which is defined by d_e/d_i. This ratio is very important to using what are called Aleph Integrals.

Before explaining Aleph Integrals, here is the notation(or set of notations usable) for a Aleph Integral.

Both notation portray the same kind of solution, but in a different fashion. An Aleph Integral is used to find the relationship between the value of x to the size of a Collatz-Matrix equation and the location of the value of x in a matrix solution. In order to calculate the solution for an Aleph Integral the amount of matrix solutions must be first found, which will be used to calculate the solution. Then, the sum of the value of x divided by each quantity of matrix solutions of sizes from A_1x1 to the dimension of yxy must be found.

The following is an example of this process.

In this case, the Collatzian ratio that is involved in the Collatz conjecture, or 1/3, leads to a solution, with an Aleph Integral, of the square root of e. This may, in fact, have a connection to the solution of the Collatz conjecture because it could show that the generalization of the Collatz conjecture shows which equations would and would not work for all natural numbers. Of course, this implies that it is not the complex version(which will be discussed later) of the Collatz conjecture which involves integers with decimal values.

This has some similarity to the derivative of the product of the function and its inverse. This similarity relates to the fact that the derivative of e is itself. The constant e is its own derivative, which means there is some connection between products of functions and their inverse and the constant e.

With the derivative of 3x+1 and its inverse, the solution is double the conjugate of d_ix+d_e divided by d_i. Of course, in order for this to work the slope of the function inside these differentiation from the original function must be equal to d_i when such a method is performed.

Aleph Integrals seem to be heading towards a domain of the analysis of equations that involves determining the relationship between equations and their counterparts. For example, the derivative of the product of a function and its the inverse seems to define some form of interval, where the slope of the amount of iterations is defined by the derivative of this product.

This kind of notation addresses this bound. It states that the bound of iterations of the function and its inverse is a result of the variable r, where r will allow the bound between the function and the bound value of r.

This is all for this post. In the next post, I will be discussing patterns that occur in Hailstone sequences with the Collatz conjecture.
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