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Jul
31
comment SO$(4)$ (& SO$(n)$) characterization?
For the case of SO(4), see this MO discussion, of which the current question actually seems to be a duplicate: mathoverflow.net/questions/37136/…
Jul
31
comment SO$(4)$ (& SO$(n)$) characterization?
Some trivial comments. First, in order to embed a finite group $G$ into $O(n)$, it is enough to embed it into $GL(n,\mathbb{R})$: if you have the latter embedding, start with any scalar product on $\mathbb{R}^n$, and use averaging over $G$ to turn it into an invariant scalar product, and you have an embedding into $O(n)\subseteq SO(n+1)$. Second, $G$ always embeds into $S_{|G|}$ by Cayley's theorem, and hence into $SO(|G|+1)$. These observations suggest to me that a general answer to your question is probably too much to hope for...
Jul
16
revised Reference request for translating from Top to C*-alg
improved formatting a bit
Jul
16
comment Reference request for translating from Top to C*-alg
I've improved the answer's formatting a bit. FWIW, 'covariant' in the Pedersen quote should be 'contravariant'.
Jul
16
suggested approved edit on Reference request for translating from Top to C*-alg
Jul
10
comment Why is the identity element of a group denoted by $e$?
For German speakers: am I missing something, or does Burkhardt's Encyklopädie really define a "group" as a cancellative semigroup, and then claim that the existence of the unit and inverses follow?
Jun
24
comment A probability distribution in n dimensional space which its projection on any line is a uniform distribution?
Addendum to my comments: now I understand that Will achieves something equivalent by considering a random projection. Clever!
Jun
24
comment A probability distribution in n dimensional space which its projection on any line is a uniform distribution?
Hence we work wlog with a measure of the form $f(r)\, d\Omega\, dr$, where $d\Omega$ is the volume element of the sphere (and $f$ may be distributional), i.e. an integral over the uniform measures on spheres for varying radius of the sphere. For these measures, it should be a straightforward calculation to determine whether there exists some $f$ for which $f(r) \, d\Omega\, dr$ satisfies the OP's requirement.
Jun
24
comment A probability distribution in n dimensional space which its projection on any line is a uniform distribution?
Using Will's observation on mean and covariance, one can also assume the measure to be symmetric under rotations without loss of generality. The reason is that having projection uniform on $[-1,1]$ is a property which is invariant under taking mixtures of measures. Hence if some measure has this property, then so does its convolution with a random rotation (chosen from the Haar measure on $O(n)$).
Jun
24
comment A probability distribution in n dimensional space which its projection on any line is a uniform distribution?
@Salavati: do you know the answer for the two-dimensional case? A positive answer in three dimensions would imply a positive answer in two dimensions by projecting the measure onto the plane. So I think that it will help to consider the two-dimensional case first.
May
29
comment Reference for an unbiased definition of a symmetric monoidal category
I had asked this question on the nForum about a year ago: nforum.ncatlab.org/discussion/3101/symmetric-monoidal-category/…
Mar
24
revised Dissolution of Tensors
fixed tag, typo and latex
Mar
24
suggested approved edit on Dissolution of Tensors
Mar
9
comment Numbers greater than Skewes's whose existence can be found in number theoretic proofs
This seems close to the "eventual counterexamples" question: mathoverflow.net/questions/15444/…
Mar
6
comment Abstract connectedness
@MikeShulman: yes, I strongly agree about the minimality of the separoid axioms! It feels like they're missing something important...
Mar
6
comment Abstract connectedness
The notion of "separoid" may be related: en.wikipedia.org/wiki/Separoid "Given a topological space, we can define a separoid saying that two subsets are separated if there exist two disjoint open sets which contains them (one for each of them)." But possibly it captures something slightly different from what you're looking for; I haven't thought about it.
Mar
3
comment Base of a cone in a vector space: can one always choose a convex base?
@WillieWong: sure, done!
Mar
3
revised Base of a cone in a vector space: can one always choose a convex base?
added example due to request in the comments
Mar
1
revised Base of a cone in a vector space: can one always choose a convex base?
small improvements
Feb
28
revised Base of a cone in a vector space: can one always choose a convex base?
yet another correction