Collatz-Matrix Equations(Concept by me)

[Unity+]

The work suggests the following conjecture.

 

The differences of the x value for the points on the x-axis will always equal the following.

 

 

[Unity+]

There is also another way to notate a Collatz-Matrix equation.

 

 

Where each sum is representative of matrix solutions of a Collatz-Matrix equation. There are a variety of sums for a given Collatz-Matrix equation. These are called Indefinite Collatz-Matrix equations. This implies that the matrices are in an infinitely large matrix, not a finite matrix.

 

Raymond arithmetic also applies to this notation.

 

 

For example, the following would be true:

 

 

Which could also be stated as:

 

Edited September 28, 2013 by Unity+

[Unity+]

Forgot to add the following:

 

The difference between the standard notation and the sum notation is both have a different limit factor. The standard notation has what is called a Area Limit, while the sum notation has the Step Limit.

 

Area Limit -

 

Step Limit -

 

Each limit differs on what specific matrix solutions that it focuses on. For example, the Area limit focuses on 4 sets of matrix solutions, which are the solutions that use one of the four parameters first. However, the Step limit looks at each matrix solution and does not separate each one into different sets.

 

The following is the way to combine the limit types, or the Area-Step Limit.

 

Edited September 29, 2013 by Unity+

[Unity+]

There is also a way to make a new form of Conway's Game of Life, which involves the following notation.

 

 

Simply put, this is the Step-Limit notation that includes a new part called the If-Then parameters. These parameters are what can be used to modify certain elements when another element is affected. This can be used to essentially make a general form of Conway's Game of Life.

 

For example, the following makes it so that the If-Then parameters are only carried out when the variable [math]k[/math] reaches a value above [math]w[/math].

 

 

Here is the same thing within an Area-Limit Collatz-Matrix equation.

 

Edited September 29, 2013 by Unity+

[Unity+]

Here is an example of the Step-Limit notation with the Collatz parameters.

 

[Unity+]

Here is a mathematical reason why Collatz numbers are unique to their given parameters within a Collatz-Matrix equation.

 

 

 

 

This provides a way to find the value of [math]r[/math] for a given Collatz-Matrix equation.

 

However, this only applies to the Special case of the Collatz parameters(in other words, only applies to linear equations). A general form is being formulated(to apply for polynomial parameters).

Edited October 2, 2013 by Unity+

[Unity+]

Sorry for the long time since I have posted any updates on the work, but here is some more.

 

Not really much, but here is a new way to notate the matrix solutions for a Collatz-Matrix equation.

 

 

For example, if x=3...

 

 

What has been added is at the end of the notation of the solution. This is called the mean matrix, which can be found by doing the following process.

 

 

For this, x must be substituted by the value inputted into the Collatz-Matrix equation. The delta denotes that the mean matrix is merely an approximate representation of all of the matrix solutions of the Collatz-Matrix equations.

Edited November 4, 2013 by Unity+

[Unity+]

Here is some work which relates to how to get normal algebraic equations from Collatz-Matrix equations.

 

 

In this set of steps, two algebraic equation were derived, which are a curve and its tangent.

 

 

I have only done it for 2x2 Collatz-Matrix equations. I will attempt to do it with Collatz-Matrix equations of larger sizes.

 

EDIT: To clarify notation:

 

[math]\phi [\phi _{1},\phi _{2}]\phi x[/math]

 

For the matrix sum, the matrix sum is split into two matrices with two elements. In this case, the notation implies that the exponent of x is raised by 1 for the second matrix for the second matrix element.

Edited November 9, 2013 by Unity+

[Amanbir Singh Grewal]

a system of equations has a unique solution. always.

[Unity+]

a system of equations has a unique solution. always.

What point were you getting at? I know that a system of equations has a unique solution, but is there something wrong with the work?

[John]

A system of equations may have no solution, one solution, or many solutions.

 

I'll have to really dig through this thread some time to get a handle on what you're doing.

[Unity+]

A system of equations may have no solution, one solution, or many solutions.

 

I'll have to really dig through this thread some time to get a handle on what you're doing.

In the case of solutions, Collatz-Matrix equations can have one solutions or many solutions depending on the parameters of the Collatz-Matrix equations and the size of the Collatz-Matrix equations in terms of the dimensions.

 

However, I still haven't found an equation for finding the amount of solutions for a given Collatz-Matrix equation. It would be nice if I could develop one.

[Unity+]

Here is a function I would like to introduce. It is called the Collatz Phi Function.

 

 

This is how this function would be applied to the Collatz parameters.

 

 

And here is what the graphical form of this would look like:

 

 

More to be posted on this kind of function.

 

EDIT: I feel that there something that already exists like this. If anyone finds something that is exactly the same as this, please notify me.

Edited November 12, 2013 by Unity+

[imatfaal]

....

 

EDIT: I feel that there something that already exists like this. If anyone finds something that is exactly the same as this, please notify me.

 

You could try some of the usenet groups on maths and pure maths fora - there are some groups of heavy duty maths-geeks around. Also I would drop a line to Mathworld - Eric Weisstein used to be an avid usenetter so may well give you a decent answer

[Unity+]

 

You could try some of the usenet groups on maths and pure maths fora - there are some groups of heavy duty maths-geeks around. Also I would drop a line to Mathworld - Eric Weisstein used to be an avid usenetter so may well give you a decent answer

Thank you for the recommendation. I will contact him when I can.

 

Here is more complete notation:

 

 

I will try to find a way to shorten notation, which will still make more sense.

Edited November 13, 2013 by Unity+

[Unity+]

Here is another form of the Collatz Phi Function, which requires the input of x with an integer of n. I have also added notation for input of position and dimensional size of the Collatz-Matrix equation in the Collatz Phi Function.

 

 

And here is more specifics with the function at hand.

 

 

What this type of notation with the Collatz Phi function is saying is it is acknowledging that the a dimensional product of some number, for example 4, and since there are many factors of that specific value that could fit, this means that there are infinitely possible ways to set up the Collatz-Matrix equation that would have a product of 4. However, the notation, when addressing the Collatz-Matrix equation, will show which set of factors are being inputted into the equation.

Edited November 14, 2013 by Unity+

[Unity+]

I think I may have discovered a constant for Collatz-Matrix equations.

 

 

This applies to the contrast of amounts of matrix solutions for a specific size of Collatz-Matrix equation. For example, as the dimensions of a Collatz-Matrix equation are increased from 1-4, the following sequence occurs(for initial coordinates of (1,1) and initial x of 1):

 

1, 2, 9, 32

 

I am going to develop an equation to find the amount of matrix solutions for a given equation. It is going to take a while to do it though.

Edited November 16, 2013 by Unity+

[Unity+]

Well, I guess this constant isn't really a new constant, but it is actually the existing constant of e. However, it isn't from the above method. Here is a new type of function called the Aleph Integral, which works differently from the regular integral.

 

 

Here is another way to denote the Aleph integral:

 

 

Where [math]d_{c}[/math] is this Collatzian ratio.

 

How this works is you have a Collatz-Matrix equation that starts with the x of n and is of dimensions k and d(found in the integral of a). Knowing this, the amount of matrix solutions for a specific Collatz-Matrix equation is the value inputted into the solution. Then, divide the value of n by the amount of matrix solutions for that particular Collatz-Matrix equation. Then, repeat the process by increase the dimensions until the dimensions magnitude has equaled the value of y. EDIT: Also, b denotes position of the initial x.

 

An interesting thing is that the following is true:

 

 

I will be looking into this further.

 

Here are a few rules Aleph integrals:

 

 

Here is a popular equation that is fitted for this type of integral:

 

Edited November 22, 2013 by Unity+

[Unity+]

Here would be another way to write this Aleph integral equation:

 

 

Because the following is true:

 

Edited November 23, 2013 by Unity+

[Unity+]

There are two hypotheses that I have come up with dealing with Aleph integrals.

 

Hypothesis 1: If any Collatzian ratio is chosen for the Aleph integral equation and if the process is carried out for every kind of Collatzian ratio that exists, then for each Aleph integral there would always be a convergence to the value of [math]\sqrt{e}[/math] because this is the proportion for the quantity of matrix solutions of any given Collatz-Matrix equation. The graph of convergence is shown below.

 

 

Hypothesis 2: Iff the Collatzian ratio of [math]\frac{1}{3}[/math] is chosen for an Aleph integral, then the result will be [math]\sqrt{e}[/math]

 

I will test for each hypothesis.

 

EDIT: Here is something extra that was interesting:

 

Edited November 24, 2013 by Unity+

[Unity+]

I forgot to mention that the graph was a result of the function aleph of n.

 

[Unity+]

Since I have set up my blog, I am now mostly posting my works there instead. The blog is called "Mysteries of Math".

 

I will still post stuff here, but not as often.

[Unity+]

Alright, sorry for the long time for the next post. I am using the blog less and decided to go back to the topic.

 

The Derivative Inverse Theorem

 

[math]\frac{\mathrm{d}}{\mathrm{d}x}u_{f}v_{f}=\frac{d_{i}rx-d_{e}r}{d_{i}}[/math]

 

[math]v_{f}=\frac{d_{i}r(x+\frac{d_{e}}{d_{i}})-d_{e}r}{d_{i}}=rx[/math]

 

[math]a_{f}=\frac{d_{i}x+d_{e}r}{d_{i}r}-\frac{d_{e}}{d_{i}}=\frac{x}{r}[/math]

 

[math]C(x)_{k\times d}\begin{Bmatrix}\frac{d_{i}x+d_{e}r}{d_{i}r}-\frac{d_{e}}{d_{i}}&\frac{x-d_{e}}{d_{i}}\\d_{i}x+d_{e}&\frac{d_{i}r(x+\frac{d_{e}}{d_{i}})-d_{e}r}{d_{i}}\end{Bmatrix},s(k_{p},d_{p})[/math]

 

Here is another part of the theorem:

 

[math]u_{f}=\frac{d_{i}r(x_{2}+\frac{2d_{e}}{d_{i}})-d_{e}r}{d_{i}r}\times \frac{d_{i}}{r}=\frac{d_{i}x_{2}r+2d_{e}r-d_{e}r}{d_{i}r}\times \frac{d_{i}}{r}=\frac{d_{i}x_{2}r+d_{e}r}{d_{i}}\times \frac{d_{i}}{r}=d_{i}x_{2}+d_{e}[/math]

 

[math]b_{f}=\frac{d_{i}x_{1}r+d_{e}r}{d_{i}r}-\frac{2d_{e}}{d_{i}}=\frac{d_{i}x_{1}r+d_{e}r-2d_{e}r}{d_{i}r}=\frac{d_{i}x_{1}-d_{e}}{d_{i}}[/math]

 

[math]C(x)_{k\times d}\begin{Bmatrix}\frac{d_{i}x+d_{e}r}{d_{i}r}-\frac{d_{e}}{d_{i}}&\frac{d_{i}x_{1}+d_{e}}{d_{i}}-\frac{2d_{e}}{d_{i}}\\\frac{d_{i}r(x_{2}+\frac{2d_{e}}{d_{i}})-d_{e}r}{d_{i}r}\times \frac{d_{i}}{r}&\frac{d_{i}r(x+\frac{d_{e}}{d_{i}})-d_{e}r}{d_{i}}\end{Bmatrix},s(k_{p},d_{p})[/math]

Edited February 20, 2014 by Unity+

[Unity+]

The Derivative Inverse Theorem produces the following equation:

 

[math]\frac{d_{i}(x_{2}+\frac{2d_{e}}{d_{i}})}{r}=\frac{d_{i}x_{2}+2d_{e}}{r}[/math]

 

More information being added.

[Unity+]

I now am applying primitive logic functions to Collatz-Matrix equations.

[math]C(\delta _{F})_{k\times d}\begin{Bmatrix} \Lambda _{a} &\Lambda _{b} \\ \Lambda _{u}& \Lambda _{v} \end{Bmatrix},s(k_{p},d_{p})[/math]

Basically, given a Fundamental theorem, or set of axioms, [math]\delta_{F}[/math] there is a set of theorems that are derived from this fundamental theorem, where the parameters [math]\Lambda _{a}[/math], [math]\Lambda _{b}[/math], [math]\Lambda _{u}[/math], and [math]\Lambda _{v}[/math] are not equations but logical statements that are added to the fundamental theorem or set of axioms.

Here is an example concept:

[math]C(\delta _{F})_{2\times 2}\begin{Bmatrix} \Lambda _{a} &\Lambda _{b} \\ \Lambda _{u}& \Lambda _{v} \end{Bmatrix},s(1,1)=\begin{bmatrix} \delta _{F} & \delta _{F}+ \Lambda _{u}\\ \delta _{F}+ \Lambda _{u} + \Lambda _{v}+\Lambda _{a}& \delta _{F}+ \Lambda _{u} + \Lambda _{v} \end{bmatrix}[/math]

In this case, the order of the logic equations matters. There are also other rules about how these work.

• All logic functions must comply with each other's rules.
• The rules of logic functions must not contradict.

That is just the start. More is coming to the concept.

 

Here is an application of the concept:

 

[math]C(A\cap B)_{2\times 2}\begin{Bmatrix} O_{a}=Q_{b}& O_{a}\in O_{b} \\ O_{a}\ni O_{b}&O_{a}\neq Q_{b} \end{Bmatrix},s(1,1)=\begin{bmatrix} A\cap B &\left (A\cap B \right )\rightarrow A\ni B_{e} \\ 0& 0 \end{bmatrix}[/math]

Therefore, the theorem arrived from this is that the elements of A and B will intersect when either some or all elements of B are within set A. The reason why order within this is important is because if it were the other way around then it would say that either some or all elements of B are within A if A and B intersect. Though, in this case, both cases are true for other instances there would be a difference.

Edited March 2, 2014 by Unity+