CNB6 A Night Thought (on serial collapsing)
A thought for today — I woke up in the night, and found it on my mind! I haven’t told you yet about some of the details you will need to understand. I’ll very briefly fill you in when I need a particular concept. All very simple.
Serial collapsing (not me):
Every cycle-number is a (0,1) sequence: e.g. 2 = 1010101010 ... I ask what happens to it if you multiply the first element by the second, and write the answer as the start element of a new sequence; then do the same with the second and third elements; and so on, thereby obtaining a new (0,1)- sequence, because the multiplications always result in a 0 or a 1.
I have called this product operation a ‘cap product’, or a Boolean Product (BP), and used the symbol Λ, or cap. The product table is: 0Λ0 = 0, 0Λ1 = 0, 1Λ0 = 0. and 1Λ1 = 1. Please learn it for future use in this Blog. It is all over the literature, of course, and I hope I haven’t used a misnomer, or mis-sign. Anyway, it is very useful when studying cycle-numbers, as you will see later. You can get used to handling it now; collapsing 2, thus.
Nearest neighbour collapse (or 1-coll.) of 2 is 1Λ0 = 0 , 0Λ1 = 0, and so on, giving the sequence oooooooo... Not very interesting! Might call it the null sequence; but it is not a cycle-number, is it? Not 0? Remember what 0 is?
Suppose we make the ‘second-nearest-neighbour collapse, or (2-coll.) … which means taking the cap products of each element with the one 2 steps further on in the sequence. That will give us another (0,1)-sequence, which might be a bit more interesting. Thus, collapsing 2 in this way, we get: 2-coll. = 1010101010… . Surprise? We got back to the original sequence. Then, obviously, if we took this step again, using 3-step collapses on 2, and again using 4-step collapses, we would get nothing new. So the cycle-number 2 ‘has only two serial collapses’. And the last one equals the one you started with!
Next question? How many serial collapses have the cycle-numbers 3, 4, 5, 6?
I’ll leave you to have a go at that one. In particular, look what happens to the series collapses of the prime cycle-numbers. You’ll learn how to use the cap product, which is the major tool in future studies.