10,045 reputation
13557
bio website research.microsoft.com/~cohn
location Cambridge, MA
age 40
visits member for 4 years, 9 months
seen 1 hour ago

Nov
30
comment The resolution of which conjecture/problem would advance Mathematics the most?
One somewhat objective way to answer this question is by asking about the known consequences. In that case it's probably hard to beat the Riemann hypothesis and P vs. NP, but deducing consequences directly from a single statement is a pretty limited notion of advancing mathematics. The deeper question is what the proof techniques could tell us if we had them, but that's really not easy to predict. (For example, before the elementary proof of the prime number theory, people imagined it would lead to a revolution in number theory, but that turned out not to happen.)
Oct
27
comment Proving the Irrationality of this Number
Incidentally, Schanuel's conjecture would imply that this number is transcendental.
Oct
24
comment How to visualise Bollobas' 1965 theorem?
Proof 2 in the question seems about as good an explanation as you're likely to find, and one could draw a picture to explain why the events are mutually exclusive (so it's at least kind of visual, in the same way the proof of the LYM inequality is). What more are you hoping for? If you had mentioned only Proof 1 and someone had responded with Proof 2, it would seem like a great answer to me, so I'm not sure what you're looking for. Is there some other sort of visualization you would prefer?
Oct
20
comment Who first defined quantum integers?
I believe Gauss introduced Gaussian binomial coefficients in 1811 in Summatio quarumdam serierum singularium. I just flipped quickly through it, and I couldn't see any explicit discussion of quantum integers per se, but Chris Godsil is certainly right that they are implicit since they are the special case $m$ choose $1$ (and in any case I might have missed some broader discussion).
Oct
3
comment Translative packing constant strictly larger than lattice packing constant
Yeah, I think it's very plausible that one could construct an example in a much lower dimension, and conceivable that one could prove it rigorously. I haven't thought much about how hard the difficulty of proving that a lattice packing is optimal varies with dimension (for convex bodies). The Betke-Henk paper deals beautifully with polyhedra in three dimensions, but I have no idea how hard four, five, or six dimensions would be.
Oct
3
comment Translative packing constant strictly larger than lattice packing constant
I don't know of an example, and I wouldn't be surprised if nobody knows one (but I'll be very pleased if I'm wrong about this).
Sep
30
awarded  Explainer
Aug
24
comment Time in Girard's Geometry of Interaction
My first thought upon seeing the question title was that it was going to be about mustard watches.
Jun
22
comment polynomial with rational coefficients
@ToddTrimble: I see how to do this if $f(n)$ is an integer for all $n$, but how do you handle the case where this is true for some arbitrary infinite set? (I may be missing something obvious.)
Jun
22
comment polynomial with rational coefficients
Maybe I'm missing a nice way to do it via calculus of finite differences, but that feels like overkill to me. If $f$ has degree $d$, then it's determined by its values at $d+1$ points, and polynomial interpolation preserves rationality.
May
25
awarded  Enlightened
May
24
awarded  Nice Answer
May
24
answered Can the Legendre symbol be calculated in polynomial time?
May
17
comment What is known about $\displaystyle \sum_k{a^{b^k}}$?
If $b$ is an integer and $a$ is the reciprocal of an integer, then you get a beautiful continued fraction expansion (see the articles dx.doi.org/10.1016/0022-314X(79)90040-4 and dx.doi.org/10.1016/0022-314X(82)90047-6 by Jeffrey Shallit).
Apr
23
comment Is anything known about which numbers appear in the continued fraction expansion of $\pi$?
Hopefully someone can expand on this (to make it more suitable as a formal answer), but the quick version is yes: (a) and (b) were true in 1978 and remain true today.
Apr
21
comment Why can't there be a problem both in P and NPC
You're right that the illustration you link to assumes that P ≠ NP. Note that on the P versus NP Wikipedia page, it has the caption "Diagram of complexity classes provided that P ≠ NP."
Apr
19
awarded  Enlightened
Apr
18
revised I would like to have a counter example that Peano's theorem does not apply to spaces with infinite dimension
corrected reference, added link
Apr
18
awarded  Nice Answer
Apr
18
revised I would like to have a counter example that Peano's theorem does not apply to spaces with infinite dimension
made continuity hypothesis explicit