A structural question

*Organic*

Hi,

Take for example these 5 equations:

Code:

     1                1                1
    +            2 = +            2 = + 
     1                1                1
4 = +       4 = +            4 = +      
     1                1                1
    +                +            2 = + 
     1                1                1


         1                     1
        +                 2 = + 
     3 = 1           3 = +     1
4 = +   +       4 = +           
         1                     1
                                
         1                     1

Set A = {{x1},{x2},{x3},{x4}}, where each x# is some number.

Now, let us say that the above equations represent some cardinal's equation-trees of set A.

Let us say that any cardinal which is > 1 is the continuous side of the cardinal's equation-tree.

Let us say that any cardinal which is = 1 is the discrete side of the cardinal's equation-tree.

x#' stands for dummy variable of xor(|{x1}|,|{x2}|,|{x3}|,|{x4}|) ,
and we get 9 variations:

Code:

                                
               1 is xor(x1',x2',x3',x4') (1:16)
              + 
               1 is xor(x1',x2',x3',x4') (1:16) 
          4 = + 
               1 is xor(x1',x2',x3',x4') (1:16)
              + 
               1 is xor(x1',x2',x3',x4') (1:16)
                
                
                
               1 is xor(x1',x2') (1:8)
          2 = + 
               1 is xor(x1',x2') (1:8)
     4 = +      
               1 is xor(x1',x2',x3',x4') (1:16)
              + 
               1 is xor(x1',x2',x3',x4') (1:16)
                
                

               1 is x1' (1:4)
          2 = + 
               1 is x1' (1:4)
     4 = +      
               1 xor(x1',x2',x3',x4') (1:16)
              + 
               1 xor(x1',x2',x3',x4') (1:16)
                
                
                                
               1 is xor(x1',x2') (1:8)
          2 = + 
               1 is xor(x1',x2') (1:8)
     4 = +      
               1 is xor(x1',x2') (1:8)
          2 = + 
               1 is xor(x1',x2') (1:8)
                
                

               1 is x1' (1:4)
          2 = + 
               1 is x1' (1:4)
     4 = +      
               1 is xor(x1',x2') (1:8) 
          2 = + 
               1 is xor(x1',x2') (1:8) 
                
                

               1 is x1' (1:4)
          2 = + 
               1 is x1' (1:4)
     4 = +      
               1 is x1' (1:4)
          2 = + 
               1 is x1' (1:4)

               
    (From here each x#' has 3 elements)            
                                                                           
               1 is xor(x1',x2',x3') (1:9)
              + 
          3 =  1 is xor(x1',x2',x3') (1:9)
     4 = +    + 
               1 is xor(x1',x2',x3') (1:9)
                
               1 is |{x4}| (1:1)


 
    (From here each x#' has 2 elements)                        

               1 is xor(x1',x2') (1:4) 
          2 = + 
     3 = +     1 is xor(x1',x2') (1:4) 
4 = +           
               1 is |{x3}| (1:1)
                
               1 is |{x4}| (1:1)

                
                
               1 is |{x1}| (1:1)
          2 = + 
     3 = +     1 is |{x2}| (1:1)
4 = +           
               1 is |{x3}| (1:1)
                
               1 is |{x4}| (1:1)

As you can see above, the quantity in each cardinal's equation-tree is being kept, while the structural symmetry-degree and the information's clarity-degree of each tree are changed.

My question is:

What mathematical branch deals with this kind of information's structures ?

Organic

Hurkyl

While your post as a whole is gibberish, your first code section resembles a classic type of counting problem. (i.e. combinatorics)

*Organic*

Hello Hurkyl,

Please be more specific.

Organic

Hurkyl

Although I don't think it's what you're looking for, your first code segment looks vaguely like the opening part of http://mathworld.wolfram.com/PartitionFunctionP.html

*Organic*

Hi Hurkyl,

I am looking for some mathematical branch that deals with the
structural side of these partitions.

For example:

Code:

     1                1                1
    +            2 = +            2 = + 
     1                1                1
4 = +       4 = +            4 = +      
     1                1                1
    +                +            2 = + 
     1                1                1


         1                     1
        +                 2 = + 
     3 = 1           3 = +     1
4 = +   +       4 = +           
         1                     1
                                
         1                     1

Let us say that numeral 4 represents the integral (the sum) side of each equation-tree, where the numerals 1111 represents its differential (difference) side.

The total quantity of each equation-tree is being kapt during the
structural changes.

So, I ask:

What mathematical branch deals with the structural changes of these forms (where their quantity is being kept during their structural changes).

Thank you,

Organic

Hurkyl

If I understand correctly, the set of "equation-trees" for a given integer n is equivalent the set of all trees that have n leaves where the children at each node are unordered.

Trees would typically be studied in graph theory.

Once again, I strongly suggest trying to use the right terminology to describe your ideas rather than using related words incorrectly.

For example, if I understand correctly what you mean by equation-tree, I might define it as:

An equation tree is a tree that has a value at each node that satisfies the properties:

The value at each leaf is 1.
The value at every non-leaf node is the sum of the values of its children.

Furthermore, the order of the children of a node does not matter.

Something else you might be interested in studying is non-associative algebra, since it seems that the main feature you are trying to contemplate here is that (a+b)+c is not the same as a+(b+c)... though I imagine there might be a little more structure on an algebra than you are looking for.

*Organic*

Hi Hurkyl,

Code:

An equation tree is a tree that has a value at each node that satisfies the properties:

The value at each leaf is 1.
The value at every non-leaf node is the sum of the values of its children.

Thank you very much for your definition.

Let ET be an equation tree.

But again, the main idea is that the total quantity of each ET is being kept.

If we take any ET as some unique structure, which is closed under some quantity n, then through this point of view, we get several structural variations under the same quantity.

It means that by using ETs we can get more information from any given quantity of some natural number.

If I am more understood now, please read again my first post on this thread, where I try to show the connactions between semmetry, information and quantity.

If you know about some mathematical branch where I can find ETs, please tell me.

Again,Thank you very much for your ET's definition.

Organic

*Organic*

Also Please look at the attached pdf file, whare you can find a detailed representation of ETs 1 to 6.

Hurkyl

I had another thought on this...

It might be simpler if you define an equation tree as:

An equation-tree is 1 or it is an unordered list of equation trees with length 2 or greater.

For example, all the equation-trees with total quantity 4 would be: (presuming "total quantity" means the value at the root node)

(1, 1, 1, 1)
(1, 1, (1, 1))
((1, 1), (1, 1))
(1, (1, 1, 1))
(1, (1, (1, 1))

Some more help on expressing yourself:

If we take any ET as some unique structure, which is closed under some quantity n, then through this point of view, we get several structural variations under the same quantity.

I presume you mean something like:

"There can be several different equation-trees that each have the same total quantity n".

The phrase "closed under some quantity n" is gibberish. That is not how the word "closed' is used mathematically; classes of objects are "closed" under operations (e.g. "positive numbers" are closed under addition, and "closed sets" are closed under finding limit points), but "some quantity n" is not an operation.

If you know about some mathematical branch where I can find ETs, please tell me.

It depends on what you are going to do with them, really. It could fall under set theory, combinatorics, graph theory, and probably other things.

It seems to me that, thus far, you have only been interested in their definition and computing examples... you need to start defining operations on them and proving theorems! Addition of these things seems obvious, but how do you multiply them? Can you test if one is larger from their structure, without having to resort to comparing their total quantities? Can transfinite numbers be represented like this (IMHO this is the most interesting question)? While the structure does contain information, is that information useful? How many are there with a given total quantity? Can you bend the construction to include rational numbers or real numbers in a natural way?

*Organic*

Hi Hurkyl,

Thank you again for helping to address my ideas in formal definitions.

So, let's see what we have until now:

(Organic ideas, Hurkyl definitions)

An equation tree (let us call it ET) is a tree that has a value at each node that satisfies the properties:

The value at each leaf is 1.
The value at every non-leaf node is the sum of the values of its children.

Furthermore, the order of the children of a node does not matter.

There can be several different ETs that each has the same total quantity n.

An ET is 1 or it is an ordered list of ETs with length 2 or greater.

For example, all the ETs with total quantity 4 would be: (presuming "total quantity" means the value at the root node)

Code:

     1                1                1
    +            2 = +            2 = + 
     1                1                1
4 = +       4 = +            4 = +      
     1                1                1
    +                +            2 = + 
     1                1                1


         1                     1
        +                 2 = + 
     3 = 1           3 = +     1
4 = +   +       4 = +           
         1                     1
                                
         1                     1

It can also be represented as:

(1, 1, 1, 1)
(1, 1, (1, 1))
((1, 1), (1, 1))
(1, (1, 1, 1))
(1, (1, (1, 1)))

------------------------------------------------------------------------------

From this point Hurkyl, I need your help for more formal definitions of my ideas on this subject.

Thank you.

My informal (gibberish) definitions:

Several ETs with the same root-node value can be ordered by their structural property, where "structural property" stands for a combination of symmetry-degree and information's clarity-degree.

An example of ET4:

Set A = {{x1},{x2},{x3},{x4}}, where each x# is some number.

Now, let us say that ET4 represents the cardinal's equation-trees of set A.

x#' stands for dummy variable of xor(|{x1}|,|{x2}|,|{x3}|,|{x4}|) ,
and we get 9 variations:

Code:

                                
               1 is xor(x1',x2',x3',x4') (1:16)
              + 
               1 is xor(x1',x2',x3',x4') (1:16) 
          4 = + 
               1 is xor(x1',x2',x3',x4') (1:16)
              + 
               1 is xor(x1',x2',x3',x4') (1:16)
                
                
                
               1 is xor(x1',x2') (1:8)
          2 = + 
               1 is xor(x1',x2') (1:8)
     4 = +      
               1 is xor(x1',x2',x3',x4') (1:16)
              + 
               1 is xor(x1',x2',x3',x4') (1:16)
                
                

               1 is x1' (1:4)
          2 = + 
               1 is x1' (1:4)
     4 = +      
               1 xor(x1',x2',x3',x4') (1:16)
              + 
               1 xor(x1',x2',x3',x4') (1:16)
                
                
                                
               1 is xor(x1',x2') (1:8)
          2 = + 
               1 is xor(x1',x2') (1:8)
     4 = +      
               1 is xor(x1',x2') (1:8)
          2 = + 
               1 is xor(x1',x2') (1:8)
                
                

               1 is x1' (1:4)
          2 = + 
               1 is x1' (1:4)
     4 = +      
               1 is xor(x1',x2') (1:8) 
          2 = + 
               1 is xor(x1',x2') (1:8) 
                
                

               1 is x1' (1:4)
          2 = + 
               1 is x1' (1:4)
     4 = +      
               1 is x1' (1:4)
          2 = + 
               1 is x1' (1:4)

               
    (From here each x#' has 3 elements)            
                                                                           
               1 is xor(x1',x2',x3') (1:9)
              + 
          3 =  1 is xor(x1',x2',x3') (1:9)
     4 = +    + 
               1 is xor(x1',x2',x3') (1:9)
                
               1 is |{x4}| (1:1)


 
    (From here each x#' has 2 elements)                        

               1 is xor(x1',x2') (1:4) 
          2 = + 
     3 = +     1 is xor(x1',x2') (1:4) 
4 = +           
               1 is |{x3}| (1:1)
                
               1 is |{x4}| (1:1)

                
                
               1 is |{x1}| (1:1)
          2 = + 
     3 = +     1 is |{x2}| (1:1)
4 = +           
               1 is |{x3}| (1:1)
                
               1 is |{x4}| (1:1)

These trees can also be represented as:

Code:

(1,1,1,1) <---------- Maximum symmetry-degree, 
((1,1),1,1)           Minimum information’s clarity-degree
(((1),1),1,1)
((1,1),(1,1))
(((1),1),(1,1))
(((1),1),((1),1))
((1,1,1),1)
(((1,1),1),1)
((((1),1),1),1) <------ Minimum symmetry-degree,
                        Maximum information’s clarity-degree

Please help me Hurkyl to address it in a formal way.

--------------------------------------------------------------------

I think by this kind of structural point of view on the natural numbers, maybe we can enrich our abilities to construct and explore complex relations between elements, where each natural number > 1 is the root value of several ordered ETs.

Through these ordered structures, maybe there is a way to develop a Mendeleiev-like table of complex relations between elements, which is based on combinations between symmetry-degree, information's clarity-degree and quantity.

What do you think ? (please look at the example in the next post)

*Organic*

Some example of the previous post of ETs 1 to 5 can be found in the attached pdf.

From this example I think we can learn that each ET is an "organ-like" element that can help us to construct any hierarchic model of part(s)/whole relations, and then we can find the relations between different ETs in more and more rich hierarchic and/or non-hierarchic interesting ways.

*Organic*

Hi Hurkyl,

In the attached pdf file of this post, you can find an example of using ETs to represent bases 2,3 and 4 as structural variations over scales.

Please tell me what do you think ?

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Hurkyl Recognitions: Gold Member Science Advisor Staff Emeritus	While your post as a whole is gibberish, your first code section resembles a classic type of counting problem. (i.e. combinatorics)

Hurkyl Recognitions: Gold Member Science Advisor Staff Emeritus	A structural question Although I don't think it's what you're looking for, your first code segment looks vaguely like the opening part of http://mathworld.wolfram.com/PartitionFunctionP.html