Collatz-Matrix Equations(Concept by me)

[Unity+]

I finally wrote a paper on this concept:

 

http://www.pdfhost.net/index.php?Action=Download&File=6481ecefe3c97fab4b92b14a459e5dbf

 

It is just a rough draft, so it may need revising.

 

If you don't want to download it, just go here to read it:

 

https://play.google.com/books/reader?printsec=frontcover&output=reader&id=z31EAAAAAEAJ&pg=GBS.PA0

Edited August 13, 2013 by Unity+

[Unity+]

Once people are done reading, just bring up anything that needs to be discussed with the concept.

[Unity+]

Here is a new concept to bring up, called Trans-dimensional equations.

 

Though I haven't formulated any equations to carry what they are supposed to conduct, but here is the general idea:

 

 

Using exponentiation of 0, this Collatz-Matrix equation can be formed:

 

 

From this operation, a one dimensional Collatz-Matrix equation is formed.

 

EDIT: There is an error on page 9 of my paper. Instead of [math]b_{f}[/math], it should be [math]u_{f}[/math].

Edited August 14, 2013 by Unity+

[Unity+]

It seems I had made a few mathematical errors within the paper, but they have been fixed. Here is the new PDF file containing the "Introduction to Colllatz Theory" paper: http://www.pdfhost.net/index.php?Action=Download&File=2872637595973355a892e3b2aeb83d7d

 

Is no one paying attention to this thread anymore?

[Unity+]

Here is some more new concepts.

 

Multiple variable Collatz-Matrix equations

 

Like with regular equations involving multiple variables, Collatz-Matrix equations also are allowed to have multiple variables, except there are different rules involved. These rules involve dimensional aspects and the variables' connections with these dimensions. For example, the amount of variables must be smaller than the amount of dimensions within the Collatz-Matrix equation(However, this rule only applies to dimensions higher than 1. For one dimensional Collatz-Matrix equations, which can be simplified to just an equation of any of the parameters, there can be more variables involved).

 

For example, for a regular Collatz-Matrix equation with only 2 dimensions there can only be one variable because 1 variable is less than 2 dimensions. For a three dimensional Collatz-Matrix equation, there can be the maximum of 2 variables, which are x and y.

 

 

And then the default Collatz-Matrix equation is

 

 

Then there is the determinant equation that is arrived from these types of Collatz-Matrix equations. Here is an example of one from the Collatz-Matrix equation above.

 

 

The more variables involved the larger the determinant equations that are derived from the Collatz-Matrix equations.

 

 

Each increase in dimensional squares is denoted by

 

 

A dimensional square is the parameters that define a dimension. For example, here is one dimensional square(the 2nd dimension).

 

 

Where the following would represent the determinant equation.

 

 

Where q is the amount of dimensional squares within the Collatz-Matrix equation. This is referred to as the Collatzian mean.

Edited August 20, 2013 by Unity+

[Unity+]

I made some mathematical errors within the math above. Here are the corrections:

 

Default Collatz-Matrix equations:

 

 

And here is the determinant equation:

 

 

And here it is completed:

 

 

Edited August 21, 2013 by Unity+

[Unity+]

Could people please at least read the paper? I wrote the paper so the information would be easier to understand. Is there something wrong with my mathematical concept?

[Unity+]

I noticed that my original Google document was not correctly inputted. Here is the correct one:

 

https://docs.google.com/document/d/1yjlLCOkOYPwPciwDtBB6rdrdjXnWe2DjLt2JKnOA2d0/edit?usp=sharing

[Unity+]

Here is some information on dimensional squares:

 

A dimensional square, like cube sets, should be treated as a set, where the set is defined by [math]\partial_{n}[/math]. More specifically, the partial sets would be defined by , where a represents the two parameters [math]a_{f}[/math] and [math]u_{f}[/math], while b represents the two parameters [math]b_{f}[/math] and [math]v_{f}[/math]. The partial set notation for the first two parameters is and the notation for their inverse is .

There is a way to analyze the parameters of a dimensional, which is to divide the two partial sets of the dimensional square.

If the parameters were the Collatz parameters, the following would occur.

This can be become a function, which will describe the properties of the first partial square compared to the second partial square, or the inverse. It will also describe the cube set that encompasses these two partial squares.

[math]\Upsilon_{\mathbb{W}}(a,b,u,v)=\frac{\partial_{a\bigsqcup u}}{\partial_{b\bigsqcup v}}[/math]

The following would be a representation of a function from the above example of the Collatz parameters.

[math]\Upsilon_{\mathbb{W}}(\frac{x}{2},\frac{x-1}{3},3x+1,2x)=\frac{\partial_{a\bigsqcup u}}{\partial_{b\bigsqcup v}}[/math]

[math]\Upsilon_{\mu }(x)=\frac{-2x-1}{2x^{2}-2x}[/math]

 

This function would output the following graph.

In this graph, the parabola closest to the top represents the first partial set of the set of parameters. The lower parabola represents the inverse parameters. This function will apply to all cube sets.[math]\Upsilon_{\mathbb{W}}(x)=\Upsilon_{\Im}(x)[/math]

This shows the commonality of cube sets. This sets a basic blueprint for all number systems that exist.

Edited August 24, 2013 by Unity+

[Unity+]

Now, dimensional squares and cube sets can be combined into one concept known as complex cubes. This occurs when two dimensional squares are combined into one cube set, which is the whole number dimensional square and another dimensional square. For example, the whole number dimensional square and the imaginary dimensional square can be combined into a cube set.

 

Each dimension also can be represented by the amount of cube sets.

[math]\pm \chi _{d} \left ( \Upsilon _{\mu _{1}},\Upsilon _{\mu _{2}},\Upsilon _{\mu _{3}},...\Upsilon _{\mu _{d-1}} \right )[/math]

 

This is a simplified version of the representation of a dimensional Collatz-Matrix equation. Here is the expanded version to specifically show the many number systems that exist.

 

 

This relates to complex numbers:

 

[math]a+bi[/math]

 

Where there is the real side to the number and the imaginary side.

 

For example, if there were to be two cube sets multiplied together belonging to different systems, here would be the outcome.

 

[math](a+bi)(a+bj)=a^{2} + abi + abj +b^{2}k[/math]

 

Where the last variable would be the outcome of the number types in the two formulas for complex numbers.

Edited August 24, 2013 by Unity+

[Unity+]

There are also what are known as super-dimensions, which are the dimensions that exist outside of the geometric dimensions.

 

In this case, [math]\psi[/math] represents the next super-dimension. With this super-dimension, squaring will result in the following.

There an infinite amount of super-dimensions. The way to convert a Quadratic Formation to a Quadratic Formation is to use the following sets.

If this is true, then the following must also be true.

 

For simplicity, since there are an infinite amount of super-dimensions, a super-dimension can be referred to as [math]\Psi_{i}[/math], where [math]i[/math] represents the index of the super-dimension.

Edited August 27, 2013 by Unity+

[Unity+]

A continuation on with this work....

 

If [math]-\chi_{3}(\Upsilon _{\mathbb{C}}\left \langle \partial _{\mathbb{W}}|\partial _{\Im } \right \rangle)\circ -\chi_{3}(\Upsilon _{\mathbb{C}}\left \langle \partial _{\mathbb{W}}|\partial _{\Im } \right \rangle)\rightarrow \psi _{3}(\Upsilon _{\mathbb{C}}\left \langle \partial _{\mathbb{W}}|\partial _{\Im } \right \rangle)[/math], then [math]\psi _{3}(\Upsilon _{\mathbb{C}}\left \langle \partial _{\mathbb{W}}|\partial _{\Im } \right \rangle)=\chi _{2}(\Upsilon _{\mathbb{W}}\left \langle \partial _{\mathbb{W}}|\partial _{\mathbb{W}} \right \rangle)[/math].

 

If this is the case, then [math]-\psi _{3}(\Upsilon _{\mathbb{C}}\left \langle \partial _{\mathbb{W}}|\partial _{\Im } \right \rangle)\circ -\psi _{3}(\Upsilon _{\mathbb{C}}\left \langle \partial _{\mathbb{W}}|\partial _{\Im } \right \rangle)\rightarrow \chi _{2}(\Upsilon _{\mathbb{W}}\left \langle \partial _{\mathbb{W}}|\partial _{\mathbb{W}} \right \rangle)[/math].

 

Then, these statements must also be true.

 

[math]\chi _{2}(\Upsilon _{\mathbb{W}}\left \langle \partial _{\mathbb{W}}|\partial _{\mathbb{W}} \right \rangle)\setminus -\psi _{3}(\Upsilon _{\mathbb{C}}\left \langle \partial _{\mathbb{W}}|\partial _{\Im } \right \rangle) \rightarrow -\psi _{3}(\Upsilon _{\mathbb{C}}\left \langle \partial _{\mathbb{W}}|\partial _{\Im } \right \rangle)[/math]

 

[math]\psi _{3}(\Upsilon _{\mathbb{C}}\left \langle \partial _{\mathbb{W}}|\partial _{\Im } \right \rangle)\setminus -\psi _{3}(\Upsilon _{\mathbb{C}}\left \langle \partial _{\mathbb{W}}|\partial _{\Im } \right \rangle) \rightarrow -\psi _{3}(\Upsilon _{\mathbb{C}}\left \langle \partial _{\mathbb{W}}|\partial _{\Im } \right \rangle)[/math]

 

The paper linked gives more description of the relevance of this work to Collatz-Matrix equations.

Edited August 27, 2013 by Unity+

[ADreamIveDreamt]

Here are some examples of the applied speculation:

 

 

 

 

 

I like your work but it is easier to Understand it in my design.

[Unity+]

I like your work but it is easier to Understand it in my design.

This is a bit vague. Please explain what the meaning of this diagram is.

[Unity+]

Here is some more work dealing with integrals and relevance to Collatz-Matrix equations.

 

Here is a Collatz-Matrix equation:

 

[math]C(x)_{k\times d}\begin{Bmatrix} \frac{x}{2} &\frac{x-1}{3} \\ 3x+1& 2x \end{Bmatrix},s(k_{p},d_{p})[/math]

 

With a Quadratic Formation:

 

[math]\chi _{d}(\Upsilon _{\mu}\left \langle \partial _{n_{1}}|\partial _{n_{2}} \right \rangle)[/math]

 

And to integrate the dimensional aspect of a Collatz-Matrix equation, the following must be done.

 

 

With a two-dimensional Collatz-Matrix equation the following must be done.

 

 

This could also be represented as the following.

 

This type of integration, however, only applies to proper and complete Collatz-Matrix equations.

 

There can also be an analysis of the matrix solutions for a Collatz-Matrix equation:

 

This gives an analysis for both the size of the matrix solutions and the last elements within a matrix solution for a given Collatz-Matrix equation.

Edited August 30, 2013 by Unity+

[Unity+]

Here is some work dealing with a problem that might be solved for if there is an equation to determine how many matrix solutions there are for a given Collatz-Matrix equation.

 

 

where...

 

 

Of course, the problem is one would need to evaluate the Collatz-Matrix equation in order to use this.

 

This is proven false when x is bigger than 1. The parameter of this equation is x must be equal to 1 in order for this equation to work.

 

Here is a pattern I noticed.

 

So, here is a Collatz-Matrix equation:

 

 

 

For, example, an equation can be derived from using definite integration of the 2nd dimension with matrix solutions the size of 2x2.

 

Edited September 2, 2013 by Unity+

[Unity+]

I made a mathematical error.

 

The actual result would be:

 

 

Here is something interesting. Using the equation derived from the definite integral, the following can be done.

 

 

Next, a must be evaluated. Then, the following is true.

 

 

Where C represents the constant that is associated with the size of the Collatz-Matrix equation.

Edited September 3, 2013 by Unity+

[Unity+]

The last equation only works for a 2x2 Collatz-Matrix equation with s(1,1).

[Unity+]

I was actually able to connect Collatz-Matrix equations to music. Basically, the algorithm follows the idea that numbers represent notes, where there are 12 notes. What will occur is if an element within a matrix solution is larger than 12, it will loop itself to find a number associated with the 12 notes. Here are some samples that were made:

 

https://soundcloud.com/greggschaffter/collatzs-dream

 

https://soundcloud.com/greggschaffter/test1

 

https://soundcloud.com/greggschaffter/test2

 

More samples to come.

 

EDIT: Here is another short composition: https://soundcloud.com/greggschaffter/collatzs-nightmare

Edited September 8, 2013 by Unity+

[Unity+]

Due to school work, work on this has been slow.

 

Here is work done on indefinite integral equations of Collatz Theory:

 

 

Where gamma is the variable representing the other two variables, [math]d^{x}[/math] and [math]x^{d}[/math]. This makes this an indefinite integral.

Edited September 14, 2013 by Unity+

[Unity+]

However, there is also a way to integrate the function without the implied variables. The following would occur.

 

 

Where, the two partial squares would becoming a representation of the [math]\Upsilon_\mu[/math] function.

Edited September 14, 2013 by Unity+

[Unity+]

There are some special operations that must take place when adding Collatz-Matrix equations together.

 

Let there be the following equation:

 

 

The amount of matrix solutions for this given equation would be the following:

 

 

Where [math]S_{m}[/math] represents how many total matrix solutions that will be the outcome, [math]A_{m}[/math] is the amount of matrix solutions for the first Collatz-Matrix equation and [math]B_{m}[/math] is the amount of matrix solutions for the second Collatz-Matrix equation.

 

For example, the following could be an equation:

 

 

Where [math]B_{m}[/math] represents the set of all the matrix solutions that will be the outcome of this equation.

Edited September 17, 2013 by Unity+

[Euler's Identity]

Good stuff here, I clearly need linear algebra for any kind of mathematical completeness ha I'm just getting familiar with Differential Calculus

[Unity+]

Good stuff here, I clearly need linear algebra for any kind of mathematical completeness ha I'm just getting familiar with Differential Calculus

There are many similar concepts within Collatz Theory that involve the same concepts of Calculus, but some parts of Calculus are modified in Collatz Theory to fit the kinds of variables. For example, Integral equations involve manifold functions, such as [math]\chi _{d}\left ( \Upsilon _{\mu } \left \langle \partial _{n_{1}}|\partial _{n_{2}} \right \rangle\right )[/math], which require a modification of how integral equations are set up.

Edited September 17, 2013 by Unity+

[Unity+]

Here is a partial graph of a Collatz-Matrix equation of size [math]2\times 2[/math] if the default Collatz parameters up to [math]x=11[/math]

 

 

The only thing I notice is how the points on the x-axis have a sequential growth.

 

Here is how each point was received. So, for a [math]2\times 2[/math] matrix, a Collatz-Matrix equation produces 2 matrix solutions. Each matrix solution will produce a coordinate or point on the graph. To find the x coordinate of a point, you multiply the first diagonal elements. To get the y coordinate, you multiply the second diagonal elements.

 

Also, one can integrate Collatz-Matrix equations by doing the following.

 

 

 

 

 

The integration of this Collatz-Matrix equation can also be interpreted as this:

 

 

An interesting thing about the graph is the points on the x-axis is that they are separated by two times each Collatz number.

 

The coordinates of these points are:

 

(8,0)

(28, 0)

(60, 57)

(104, 0)

(160, 0)

(228, 0)

(308, 0)

(400, 0)

(504, 0)

(610, 0)

(748, 0)

And the difference between the x values is as follows:

8, 20, 32, 44, 56, 68, 80, 92, 104, 116

Where the difference between each of these differences is 12.

Now, if these values are divided by 2, then the following occurs:

4, 10, 16, 22, 28, 34, 40, 46, 52, 58

It becomes a sequence of Collatz numbers. This pattern would continue as the x value of the Collatz-Matrix equation increases in size.

Edited September 21, 2013 by Unity+