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What is the average number of intersections for two infinite curves confined to a plane?
How long is a piece of string?What is the average number of intersections for two infinite curves confined to a plane?
How do you define a random curve? A random walk is discrete, but random curve?
No, when you just "choose" an element out of the set of all curves it isn't "random" in the probabilistic sense. First of all can we even define a probability measure on that set? How big is it? It might be bigger than the set of reals.
It may turn out that infinite sets cannot be averaged or proportioned, but if they can, I believe one of your answers is the correct number.
No, when you just "choose" an element out of the set of all curves it isn't "random" in the probabilistic sense. First of all can we even define a probability measure on that set? How big is it? It might be bigger than the set of reals.
The problem with the step size approach is that the limit might not be a curve, so there's no sense of talking about "intersecting" itself.
The set of all curves in R^2 is probably larger than the set of all reals. And the upper bound is the size of the set of all functions from R to R.
The average of any (finite or infinite) set of numbers that are non-negative and contains at least 1 infinity is equal to infinity.