How Many Transitional States Exist in an Element with n Sub-Elements?

[stratman]

Set + Number = "Sumber"

Set+Number = "Sumber"


The content of this thread is based on ideas proposed in the Complementary Association Theory, which was posted in several threads in this forum by Organic.

Consider the following computation task:

Given a graphical representation of an Element, comprised of n sub-elements. How many transitional states are there from the initial state where all n sub-elements are indistinguishable to the final state where each sub-element can be uniquely identified?

For an Element comprised of 1 sub-element the task is simple, there is exactly one state. For example given an "arc" representation model, the graphic representation of this state would be:

http://62.219.145.154/s1.png


For an Element comprised of 2 sub-elements, there are two transitional states. In the initial state we have 2 indistinguishable sub-elements, and in the final state we make one sub-element uniquely identifiable by marking it with an underline:

http://62.219.145.154/s2.png

For an Element comprised of 3 sub-elements, there are three transitional states. In the initial state, again we have 3 indistinguishable sub-elements, then in the second state, we associate two sub elements into one, by drawing an arc around them and thereby making them uniquely identified. In the third and final state we make one of the sub elements in the associated twin sub-elements uniquely identified by underlining it:

http://62.219.145.154/s3.png

For an Element comprised of 4 sub-elements, there are 9 transitional states:

http://62.219.145.154/s4.png


For any n > 0, the following algorithm calculates the number of transitional states:

ASSOCIATION_LEVEL(int n):Vector

1 V© new vector
2 if n = 1 then
3 V©V + the single root of level 1
4 if n = 2 then
5 V©V + the two roots of level 2
6 else
7 if n >2 then do
8 For each partition vector P in n do:
9 ROOTS(P,V)
10 return V

ROOTS(vector P,vector V)

1 roots ©[]
2 for each element in P do:
3 if element is a root then
4 roots© roots + element
5 else
6 roots©ASSOCIATION_LEVEL(integer value of element)
7 return horizontal alignment of the Cartesian product of P and draw new arc around them


When considering the graphical representation of a compound element comprised of n sub-elements, one may find properties in this element which belong to sets, as the ability to contain other elements. Also, properties that belong to numbers can be identified in these elements, properties as quantity and order. But, there are also properties that can be found in neither numbers nor sets, the most important one is structure.

I am interested in learning what applications if any can be applied to these compound structured elements.



Stratman.


P.S. I have implemented a small Java Applet, grphically displaying all states for elements until n=8, as you see this function grows quite rapidly, and this computation depends on the power/memory of the running computer. For those interested I can send this Applet.

 

[Hurkyl]

For the record; you've drawn 10 "transitions" in your png for 4 sub-elements (the 5-th and 6-th ones cannot be distinguished from each other).


Also, you are missing a good many "transitions"; for instance, in the 3 sub-element case, you are missing the "transition" where you have the blue arc under the red arc, and you underline the element outside the blue arc.



I would like to point out that the connection between these and numbers is very superficial; it's about as strong of a connection as there is between wallets and numbers. (you can assign to a wallet a number (how much money is inside))

In any case, these are interesting combinatorial objects, but I can't say that I'm aware of any application for them. (They do seem somewhat related to partitions, though)

 

[stratman]

Hurkyl,

Thank you for your post. You are right, in the element containing 4 sub-elements there are 10 tranisition states. The fifth and sixth states are equivalent and differ only in the order in which their two sub-elements are associated. In Organic's theory only one of these two states is considered, this is why I have originally stated that there are 9 states.

Regarding the 3 sub-elements, here there are only three states, the graphic representation is less important. In the initial state there are three identical sub elements, then you group two of them into one, and you end up with two distinguishable elements. You then only need to be able to distinguish between the two sub-elements grouped together, and you achive that by marking each of them by some unique notation.

I agree that this is some sort of extension to partitions, but still think that each such tranisiton set, which Organic refers to as "association level" is conceptually associated with a natural number.

Startman.

 

[master_coda]

Originally posted by stratman
I agree that this is some sort of extension to partitions, but still think that each such tranisiton set, which Organic refers to as "association level" is conceptually associated with a natural number.

I have to agree with Hurkyl. Associating these sets with natural numbers is somewhat arbitrary and superficial. You're basically just assigning numbers to these sets.


I'll have to think about these for a bit. I'm not sure what kind of use they would have, but it takes a bit of time to think of something like that.

 

[Hurkyl]

I think you need to specify better what you mean by "transition state"; I can't imagine, at the moment, a meaning that is consistent with your usage.

For instance, sometimes order matters, sometimes it doesn't. Why?

Why, for instance, isn't there a 4-element state where two elements are grouped under a blue arc and a third one is underlined? (I'm talking about the graphical representation because it seems the clearest way to describe what I'm thinking)

Why aren't you putting marks on the arcs to distinguish them from other arcs, like you do with the "sub-elements"?


Your pseudocode has the same problem; I assume "Vector" means java.util.Vector, but I don't know what a "root", a "partition vector", and I have no clue what the return statement for "ROOTS" is supposed to mean.

And, incidentally, the characters ?© are gibberish on my display (the first one is a square, the second one is a copyright symbol... though it seems I couldn't copy-paste the first one); I seem them in lines 1, 3, and 5 of "ASSOCIATION_LEVEL", and on lines 1, 4, and 6 of "ROOTS".

 

[stratman]

Hurkyl, the best description for these "transition states" is provided through the algorithm. I will try to clarify: making a sub element unique is first done by trying to group it with other elements, only when no such grouping is possible we turn to making unique by marking it with some identifiably symbol. This is why your suggestion for the 4 sub-elements will not work.

As to the algorithm, a few clarifications:

1. Vector may very well be java.util.Vector, but in general it is a dynamic array structrue.

2. A root is a single "transition state", a graphical representation of a structured element containing n sub-elements.

3. A partition vector is a vector holding all the partitions of a natural number n. For example the partition vector of 2 would be [[1,1,1],[1,2],[3]].

4. The "©" signs were miscopied from Word, they should be replaced with an assignment notation ":=".

5. ROOTS disassembles a partition vector, coverting each integer in the vector to its corresponding ordered group of transitional states, and then storing it in V, which is passed as a parameter.

Stratman.

 

[Organic]

Hi Hurkyl,

You wrote:

Also, you are missing a good many "transitions"; for instance, in the 3 sub-element case, you are missing the "transition" where you have the blue arc under the red arc, and you underline the element outside the blue arc.

What you call a 3 sub-element case is a colorful representation of:

{1,1,1} = (1*3)
{{1,1},1} = ((1*2)+1)
{{{1},1},1} = (((1)+1)+1)

Your extra sub-element is {{{1},1},{1}} but it is identical to {{{1},1},1} from
multiplication <-- --> addition transition point of view.

Another examples:

By your argument we have to add {{1}} to 1 sub-element case and {{1},{1}} to 2 sub-element case, which is:

{1,1} = (1*2)
{{1},1} = ((1)+1)



This is a main idea in my theory, which is: fading transition between multiplication and addition (multiplication and addition are complementary concepts).

If you understand the above, then please read again:

http://www.geocities.com/complementarytheory/ET.pdf (where you are a participator in the first 9 lines)

And also (including its sub-links):

http://www.geocities.com/complementarytheory/AHA.pdf

Thank you,


Organic