Mathematically, it's a question regarding the order of limits. Evaluating 0^0, is akin to considering a^b as a-->0 and b-->0, and the order in which the two limits are taken will make a difference. If 'a' approaches zero and while 'b' is still positive, the answer shall be 0. If 'b' approaches zero while 'a' is still positive, the answer shall be 1.

Pragmatically, what this means is that 0^0 is not sufficiently specified (just meaningless symbols) unless you prescribe the context/meaning with which you're using it. And with regards to defining summary statistics, the article talks about a context where the exponent scans over different (continuous) values, passing by {2,1,0} along the way.

It should be an interesting exercise to consider what happens when the exponent approaches positive or negative infinity i.e. large magnitude positive or negative numbers.

> Mathematically, it's a question regarding the order of limits. Evaluating 0^0, is akin to considering a^b as a-->0 and b-->0, and the order in which the two limits are taken will make a difference.

Mathematically, you don't take limits as a sequence of single-variable limits. You take them by constricting an n-dimensional circle (two points / circle / surface of a sphere / etc.) around the point whose limit you're interested in.

This immediately implies that there are infinite possible approaches to the (0,0) point, rather than only two as there would be if you were taking it as two single-variable limits in sequence.

What are the different limiting values for a^b (a->0, b->0) considering all paths in the a-b plane? If there are as many points generated as are generated by considering only "Manhattan" paths, then I think it's appealing to think of it as a sequence of single-variable limits.

The figure in https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero#Cont... strongly suggests that all nonnegative real numbers are limit points of the function f(x,y) = x^y as (x,y) approaches (0,0). It explicitly states that 0, 0.5, 1, and 1.5 are all limit points.

The Hamming distance is sometimes called "L0" in parallel to L_1, L_2, L_p etc., in some abuse of notation.

Isn't the hamming distance simply L_1?

Nah, L_1 is called "taxicab" or "Manhattan" distance.

https://en.wikipedia.org/wiki/Taxicab_geometry

https://en.wikipedia.org/wiki/Hamming_distance

Right, those are two different names for the L_1 distance, but why does that show that they're not the same? For binary vectors v,w we have abs(v[i] - w[i]) = v[i] xor w[i], and the L_1 distance is the sum of the former whereas the Hamming distance is the sum of the latter.

For non-binary vectors, Hamming (L_0) and L_1 are in general not identical. E.g. L_0(2) = 1 but L_1(2) = 2.

What's the L0 "norm" of [0.0, 1.0, 2.22045e-16]?

  julia> vecnorm([0.0, 1.0, eps()], 0)
  2.0

  julia> vecnorm([0.0, 1.0, eps()], 1)
  1.0000000000000002

I don't remember seeing much agreement on 0^0 so far. This line from that Wikipedia article seems pretty damning:

> There is no unanimous opinion on leaving the value generally undefined.

That is a pretty roundabout way of saying that people agree that it should be _something_, and generally disagree on what that something should be.