Thursday, July 15, 2004
Fractals, Fractals everywhere
I wanted to be doubly sure I understood the definition of a fractal before I posted this, so I've been doing some research into fractals. More specifically, I've been looking at fractal sequences, and I think I'm sure enough about the definition of fractal sequences to state the following:
Let Z = all positive integers greater than 0.
BigOmega(Z) is a fractal sequence.
BigOmega(Z2), Z2 = all odd positive integers greater than 0, is also a fractal sequence.
I haven't, for some reason, been able to find that stated on the 'net. So I'm stating it here without argument, but these sequences are very much self-similar. If you start with a prime p, and take the set of all all the numbers in the sequence divisible by that prime, BigOmega of that set will be the same as BigOmega(Z) + BigOmega(p) . This is true of the set of all positive integers, all even integers, and all odd integers.
Let Z = all positive integers greater than 0.
BigOmega(Z) is a fractal sequence.
BigOmega(Z2), Z2 = all odd positive integers greater than 0, is also a fractal sequence.
I haven't, for some reason, been able to find that stated on the 'net. So I'm stating it here without argument, but these sequences are very much self-similar. If you start with a prime p, and take the set of all all the numbers in the sequence divisible by that prime, BigOmega of that set will be the same as BigOmega(Z) + BigOmega(p) . This is true of the set of all positive integers, all even integers, and all odd integers.
Friday, July 09, 2004
A more formal approach to defining the factoring problem
I've been thinking more (of course) about factoring. Here's some thoughts; I'm recording them here mostly so I won't forget what I was thinking (it's more portable and permanent than paper):
Let II equal the set of all "Information". I believe there's already a well-defined "universal" set, but I don't know it's symbol. Anyway, all sets of numbers are subsets of this "Information" set. Since this is the set of all "Information", for kicks we'll throw into this set all possible logical operations, all math functions, etc. Z, the set of positive integers, is a subset of this universal set. R, the subset of rational numbers, is also a subset of this set. It seems to me that most apporaches to factoring are based on techniques applied mostly to the set Z. What if we include the set R? In R, a number like 7 has factors, because it can also correspond to 70, 700, 7000, etc. Namely, 7/2 = 3.5, which is equivalent to 7 * 5 / 10, or 35/10. It could also be equal, for that matter, to 700/20, and so on. Thus while 7 has no factors in Z besides one and itself, it _does_ have factors in R. Perhaps there is some logical or functional subset of II which, when given a member of I, determines whether it has factors in R as well as I.
I'm trying to formalize this because I've found that by symbolically representing things I sometimes see stuff that I otherwise wouldn't see. Anyway, some food for thought.
Let II equal the set of all "Information". I believe there's already a well-defined "universal" set, but I don't know it's symbol. Anyway, all sets of numbers are subsets of this "Information" set. Since this is the set of all "Information", for kicks we'll throw into this set all possible logical operations, all math functions, etc. Z, the set of positive integers, is a subset of this universal set. R, the subset of rational numbers, is also a subset of this set. It seems to me that most apporaches to factoring are based on techniques applied mostly to the set Z. What if we include the set R? In R, a number like 7 has factors, because it can also correspond to 70, 700, 7000, etc. Namely, 7/2 = 3.5, which is equivalent to 7 * 5 / 10, or 35/10. It could also be equal, for that matter, to 700/20, and so on. Thus while 7 has no factors in Z besides one and itself, it _does_ have factors in R. Perhaps there is some logical or functional subset of II which, when given a member of I, determines whether it has factors in R as well as I.
I'm trying to formalize this because I've found that by symbolically representing things I sometimes see stuff that I otherwise wouldn't see. Anyway, some food for thought.
