tcepsa: (iSquared)
Would you see anything interesting if you were to do signal analysis on the prime numbers?

Is it unrealistic to think of the primes as a sort of hologram for the integers, since you can generate any integer by multiplying the necessary primes together?

Isn't it interesting, given the ease with which they can be used to create all of the other integers, that there is apparently no direct way to generate the prime numbers from themselves? (Is there a name for that?)

These thoughts brought to you by the fact that I just implemented an algorithm to find all the primes using a method inspired by the Sieve of Eratosthenes. (Start with 2 as a known prime, testing 3. Attempt to divide the test number by all of the known primes. If any of them divide it cleanly--no remainder--discard it. If none of them divide it cleanly, it is prime; add it to the collection.) This is sufficient because all of the non-prime integers are products of some collection of prime integer and therefore a number is divisible by a non-prime integer if and only if it is also divisible by a prime integer. So you can save a lot of time by not doing the checking of non-prime integers.

That having been said, I'm only up to the high 60000's. The fact that I'm doing this with a HugeInt class that I wrote myself to remove the limitation of the computer's native 4-byte integer appears to be moot, as I suspect that even if I left this to run forever I might very well die of old age before that became relevant.

~ponder~ I wonder what would happen first: me die or the computer use up all its RAM. (Annnnd we're back to signal processing and frequency analysis ~grin~)
tcepsa: (Computation Suspended)
I'm currently enjoying The Little Schemer as a way to attempt to learn more of the Lisp/Scheme programming paradigm. It's been a useful book for me so far, in that I'm 30 pages in and haven't yet had to stop and go, "Wait, WTF?!"*

Cut for rambling )

*Actually, looking at it that way, I should try to pick it up and read a single page each day. I'd be farther along now if I had done it that way... ~wry grin~

**Obligatory reference to Plato's Allegory of the Cave. It's been awhile since I read it, but I think that he was trying to take it one step further: that the physical world we observe is due to something else, like the two-dimensional shadows cast on a cave wall by three-dimensional objects with a light behind them. Whether he explicitly said so or not, the physical world that we perceive might well be such a shadow of what we call mathematics.

***L-mode is what I'm using to refer to the more logical/linguistic operations of the brain (also known as "left brained", though my understanding is that term contains implications that aren't entirely accurate).

****~blink~ I think I just reduced Understanding to Graph Theory. Or at least I made a claim of some kind of analog between the two.
tcepsa: (JuggleGeese)
It's kind of spooky that this actually makes sense to me:


A deterministic finite automaton is a structure

M = (Q, E, d, s, F) where

  • Q is a finite set, the states
  • E is a finite set, the input alphabet
  • d : Q X E -> Q is the transition function that maps each of the possible combinations of (state, input) pairs to a state.
  • s is a member of Q and known as the start state
  • F is a subset of Q and elements of F are known as accept or final states

(Kozen, Automata and Computability, Springer-Verlag New York, 1997)


I got shivers when I read the one about d : Q X E -> Q and understood it... being able to pull that kind of meaning out of such a small handful of symbols makes me more inclined than ever to try to learn more mathematical notation (and the fact that I get all tingly from reading it probably explains a bit about why I like regular expressions so much as well ^_^)
tcepsa: (Anarchy)
If I'm remembering correctly, one of the generally accepted concepts in mathematics is that two lines in a plane will never intersect if and only if they are parallel. They go on forever without crossing. However, if they are out of parallel by the tiniest amount then at some point they must cross.

What point is that?
tcepsa: (Default)
I had a thought a few seconds ago that I wanted to put in here, but I wanted to make sure that I was accurate about it. Then I realized that I didn't really know how to express it )

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