Recent content by mXSCNT

Example, if: z=xy y=x then: the partial derivative \frac{\partial z}{\partial x} = y, treating z as a function of two variables z(x, y) = xy the total derivative \frac{dz}{dx} = 2x, treating z as a function of one variable, z(x) =z y(x). So there doesn't seem to be a way to define the...

Yeah Cool! I should have thought of that. I ran a simulation and got around 7.6 for average component size (random grids of 3000 x 3000). For grids of 300x300 it was 7.5-7.6 so it does seem to be roughly constant.

Start with an infinite 2d grid of cells, with each cell randomly colored black or white (50% chance). Now take the graph G whose nodes are the cells, and with an edge (A, B) if A and B are adjacent in the grid, and the same color. What is the average size of a connected component of G? Is it...

Yes, and yes.

Yeah, it is just like that. Thanks! Another note. f : A \rightarrow C induces a map h : B \rightarrow D. Specifically, h takes in a string in B and maps each character in the string to another single character, such that the result is always in D. I wonder if we can generalize further, to...

Hey, I was thinking of this generalization of homomorphisms. You have a language L_1 = (A, B) where A is a set of symbols and B is a set of sequences of symbols in A. Given languages L_1 = (A, B) and L_2 = (C, D) a function f: A \rightarrow C is defined to be a homomorphism of languages if...

Suppose you have a role playing game in which n players attack a boss with H hit points. Each hit reduces the boss's hit points by a certain amount. The players take turns hitting, starting with player 1, proceeding to player 2, onward to player n, and then back to player 1 again. Eventually...

I think you do want to restrict X to be [0, 1). Because if X were all the real numbers, even if you could define a map where \mu(S_{i, j}) is the same for any i, j, that wouldn't help you since you can't uniformly sample from all the real numbers. Anyway you've basically just stated an...

There's no statement that could be true or false, you have just defined some variables. Perhaps you meant to add something to it?

Where are you getting these statements from? Using my example of F, both of those statements are false. (1) is false because S_{i,j} is a countable union of intervals within [0, 1) so it is measurable. (2) is false because \sum_{i=0}^{\infty} \mu(S_{i,j}) = 1 for all j, because...

You first mistake is that the events of not having a 9, and of not having a 7, are not independent, so you can't multiply them. Your second mistake is that the the event of getting the 6 on the second roll requires no 9, 7, OR 6 on the first roll. If instead we have the single event: w = "the...

Perhaps an example would help. \pi - 3 = 0.14159... = 0.001001000011111101101010... in binary. Calculating F(\pi - 3) = m we find: a_1 = 1, d_1 = 1, r_1 = 1, m(1) = 1 a_2 = 1, d_2 = 2, r_2 = 2, m(2) = 2 a_3 = 2, d_3 = 3, r_3 = 3, m(4) = 3 a_4 = 1, d_4 = 3, r_4 = 4, m(3) = 4 a_5 = 2, d_5 =...

D_i is the set of all numbers v where (v, w) is in m_i, and R_i is the set of all numbers w where (v, w) is in m_i. They don't really need a base case, but you could say that D_0 and R_0 = the empty set.

Well you asked for it, here it is. It was a good deal harder to do than I expected. It took a few wrong turns producing invalid F's (that either weren't surjective, or weren't injective) before I found a solution. Reminder: First let b be the infinite string of binary digits representing a...

You need to clarify. Do you assume that each ball is equally likely to end up on the left or the right half of the box, and that the final positions of the balls are independent (which might NOT be a reasonable assumption since if part of the box gets too crowded, it may force balls into the...