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Apr
14
comment Is a one-dimensional unstable manifold of an ODE a union of the associated equilibrium point and two full orbits?
It's possible for the stable and unstable manifolds to intersect, in what's called a "homoclinic point". This would violate #2, if I understand what you mean.
Jan
6
reviewed Approve Applications of forcing in model theory
Nov
2
awarded  Civic Duty
Oct
27
awarded  Yearling
Oct
18
reviewed Approve Linear transformation that preserves the determinant
Sep
27
reviewed Approve A family of convex bodies in Banach-Mazur position
Sep
18
reviewed Approve A problem of Keisler and Tarski
Sep
3
comment Which polynomial's roots are its coefficients?
Lubin's suggestion gives a continuous map, which is nice. The process outlined here can't be made continuous (e.g. to define it for $z^2 + e^{i\theta}$ you essentially have to choose branches of the square root function for the coefficients). On its own, that seems unnatural, but it makes sense as picking a branch of the inverse of Lubin's map. Igor's idea of looking at all permutations then corresponds to taking the full inverse.
Sep
1
awarded  Guru
Aug
30
comment Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Is it obvious that transversality arguments work for weird non-differentiable curves? In a manifold you know any curve is homotopic to a differentiable one, but showing that fact in $\mathbb{R}^3 \setminus \mathbb{Q}^3$ doesn't seem any easier than the original question.
Aug
30
awarded  Good Answer
Aug
30
awarded  Enlightened
Aug
29
comment Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
@Victor The argument takes for granted that the ambient space stays simply connected (even connected) when you remove just a finite set of points--fine in $\mathbb{R}^3$ but not $\mathbb{R}$ or $\mathbb{R}^2$.
Aug
29
awarded  Nice Answer
Aug
29
answered Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
May
26
comment John Nash's Mathematical Legacy
It's amazingly prescient, from a computer science perspective. Here's a link to the paper: rand.org/content/dam/rand/pubs/research_memoranda/2008/…
Apr
15
awarded  Nice Answer
Apr
9
answered Deep Learning / Deep neural nets for mathematician
Mar
11
comment understanding the average height of a unit hyper-semisphere
On a high-dimensional sphere most of the mass is concentrated around the equator, where the height is lowest. That's a heuristic explanation for why the average height goes to zero.
Feb
22
reviewed Approve Is the Ford disk packing optimal?