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visits member for 4 years, 7 months
seen Sep 12 at 19:22
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Sep
12
comment Does anyone know an intuitive proof of the Birkhoff ergodic theorem?
If $f\equiv 1$ and $T=I$ we have $M_T'\equiv+\infty$, so the recurrence formula is $+\infty + f=+\infty$, which is useless. Am I missing something?
Aug
23
awarded  Nice Answer
Aug
12
revised real symmetric matrix has real eigenvalues - elementary proof
added 391 characters in body
Aug
12
revised real symmetric matrix has real eigenvalues - elementary proof
added 391 characters in body
Aug
12
revised real symmetric matrix has real eigenvalues - elementary proof
added 391 characters in body
Aug
6
revised real symmetric matrix has real eigenvalues - elementary proof
added 1 character in body
May
28
awarded  Necromancer
May
27
awarded  Yearling
May
27
answered Can the unsolvability of quintics be seen in the geometry of the icosahedron?
May
6
awarded  Popular Question
Dec
16
awarded  Nice Answer
Jul
29
awarded  Nice Question
Jun
25
awarded  Excavator
Apr
19
comment Giving $Top(X,Y)$ an appropriate topology
For the non-Hausdorff case, see ncatlab.org/nlab/show/exponential+law+for+spaces.
Mar
8
comment Topological spaces determined by generalized metric spaces
Notice that this Arens' space is slightly different from the one given in Wikipedia (en.wikipedia.org/wiki/Arens%E2%80%93Fort_space), since in this space the set $\{c\}\cup\{a_{n,m}:n,m\in\omega\}$ is not open.
Feb
20
awarded  Yearling
Jan
25
asked Nontrivial copies of SO(r) in SO(n)
Jan
24
comment Suggestions for good notation
@Ben, the index in the coordinate expression $\frac{\partial f}{\partial x^j}$ for the 1-form $df$ is clearly in the low position! In fact, this is the main reason that I see for having to put the indexes of the coordinates in the high position as we do, instead of doing everything in the opposite way, which would be better in some way: we could write $f=x_1^2+x_3$ instead of $f=(x^1)^2+x^3$.
Jan
24
comment Suggestions for good notation
Regarding differential geometry: If $f:M\to\mathbb R$ is a smooth function on a manifold and $x:M\to\mathbb R^n$ is a chart, I prefer $\left(\frac{\partial f}{\partial x}\right)_j$ or $\left(\frac\partial{\partial x}\right)_j f$ (or even $\partial_j f$ if the choice of the particular chart is clear or irrelevant). Because the notation $\frac{\partial f}{\partial x^j}$ suggests that $\frac{\partial f}{\partial g}$ could be defined using only $g$, and in fact you need to know that you are restricting to the curve along which the other coordinates $x^i$ are constant.
Jan
21
revised Why is a topology made up of 'open' sets?
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