Quanta Magazine nails it yet again. The connection to turbulence is also very interesting.
Only question is what the C ^ x,y notation is about, as they only explained one axis. Anyone know?
Only question is what the C ^ x,y notation is about, as they only explained one axis. Anyone know?
"""
Let Ω be an open set in Rn, 0<α≤1, and k a nonnegative integer. The (uniform) Holder spaces Ck,α(Ω) consist of functions whose k−th order derivatives are uniformly Holder continuous with exponent α in Ω. The(local) Holder spaces Ck,α(Ω) consist of functions whose k−th order derivatives are locally Holder continuous with exponent α
"""
The Holder continuity condition looks sorta like the limit definition of a derivative with an exponent on the denominator.
This is actually stronger than the epsilon-delta definition of continuity, as noted in the wiki article -- "For any α > 0, the condition implies the function is uniformly continuous."
Normally, you'd have something like
For all ϵ > 0, there exists δ(x, ϵ) > 0 such that
if |x - x₀| < δ, then |f(x) - f(x₀)| < ϵ.
(namely, at any given point, there's a ball of nearby points such that all their corresponding outputs are close together.)
But, with absolute continuity, δ is purely a function of ϵ, namely, that ball doesn't change size even as we move further away.
For instance, f(x)=x is absolutely continuous, while f(x)=x^2 is not. (for the former case, we can use δ = ϵ/2; for the latter case, we'd need a δ that looks something like 2ϵ·x)
In this case, since the Holder condition depends only on the distance between x and y, it automatically implies absolute continuity.
[1] https://en.m.wikipedia.org/wiki/Uniform_continuity [2] https://en.m.wikipedia.org/wiki/Absolute_continuity
(left for posterity so your comment makes sense)
I think this was at the end of the analysis textbook too, in one of those "looking ahead future topics" sections.
Just don't ask me to explain it :D