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1d
comment On number of perfect matchings
@Turbo: The maximum number of matchings is $n! \lt n^n = 2^{n \log_2 n}$. You can have $(n/2)!$ matchings with about $n^2/4$ edges by adding disjoint edges to a complete bipartite graph with half as many vertices. $(n/2)! \sim \sqrt{\pi n} \left(\frac{n}{2e}\right)^{n/2} \gt e^{(1/2) n \ln n - n}$.
May
1
comment Do Peano curves provide a counterargument to Grothendieck's critique?
@MikhailKatz: Have you read the abstract?
May
1
comment Interesting conjectures “discovered” by computers and proved by humans?
I think it's not that interesting if you understand multiplicative functions.
Apr
30
revised What is special about polylogarithms that leads to so many interesting identities and applications?
Fixed LaTeX errors
Apr
24
comment Precise asymptotic of diophantine approximation
Given $[a_0; a_1, a_2,...]$, we're looking at $\limsup_k a_k + [0;a_{k+1},a_{k+2},...] + [0;a_{k-1},a_{k-2},...,a_1]$, right? If we let $a_{n^2+\epsilon_n}=1$ and $a_n=2$ otherwise, where each $\epsilon_n \in \{0,1\}$, that gives uncountably many reals with lim sup efficiency of $2+1/\sqrt{2}+1/(1+\sqrt{2}) = 3.121$. Maybe why this is greater than $3$ is clear from some perspective I don't have yet.
Apr
24
comment Precise asymptotic of diophantine approximation
Thanks. I'm still missing something. For large enough elements of the spectrum (maybe only greater than the critical 4.5278?), I think you can have an uncountable collection of reals with the same lim sup of approximation efficiency. Say, choose blocks A and B, and any simple continued fraction built from an infinite string of A and B blocks such that each finite pattern occurs infinitely often should have the same lim sup. I think you are saying that something prevents that lim sup from being less than $3$, that only very special sequences occur then that can't be built that way.
Apr
24
comment Precise asymptotic of diophantine approximation
@Siminore: That's not a particularly special condition on a good approximation to $\xi$. For convergents, it means that the index $n$ is even, since those are the approximations slightly below $\xi$ while the odd convergents are slightly above $\xi$. There are infinitely many even indices.
Apr
24
comment Precise asymptotic of diophantine approximation
@Siminore: Since there are infinitely many normalized errors in an interval, there must be limit points.
Apr
24
comment On number of perfect matchings
@Turbo: Yes.$ $
Apr
24
answered On number of perfect matchings
Apr
24
comment Precise asymptotic of diophantine approximation
Can't you have uncountably many $\xi$ corresponding to the same element of the Markov spectrum?
Apr
24
answered Precise asymptotic of diophantine approximation
Apr
22
comment Do Peano curves provide a counterargument to Grothendieck's critique?
@Mikhail Katz, I don't understand why you were so dismissive of my comment. I tried to explain how space filling curves arise naturally in the Cannon and Thurston paper for non-topologists, and you say "no one has bothered to summarize it..." If you want to throw out these space-filling curves that arise from important structures (surface bundles over circles) that were defined earlier, what part of mathematics do you need to destroy? Can we not look at hyperbolic structures any more, since they might produce space filling curves?
Apr
19
comment Do Peano curves provide a counterargument to Grothendieck's critique?
Yes, space-filling curves appear in the wild, even though they may seem to be contrived at first. There is a natural $S^1$ at infinity for a hyperbolic surface, and a natural $S^2$ at infinity for a hyperbolic $3$-manifold, and it might be that when you have a surface in a $3$-manifold that there is a well-defined map from the $S^1$ at infinity to the $S^2$ at infinity that is continuous and onto. The Hopf fibration is also natural even if it seems strange at first.
Apr
16
comment Expected number of changes in the sign of a rolling sum of independent normal variables
I took the liberty of adding that argument in place of the approximation.
Apr
16
revised Expected number of changes in the sign of a rolling sum of independent normal variables
Added calculation by radial symmetry.
Apr
15
comment Expected number of changes in the sign of a rolling sum of independent normal variables
When $\mu=0$, you reduce this to the probability that a two-dimensional normal distribution is within a wedge with vertex at the center. By a change of variables, that's the probability that a rotationally symmetric normal distribution is within a wedge, which has a probability of $\theta/(2\pi)$ where $\theta$ is the angle of the wedge. So, you shouldn't need to approximate this.
Apr
14
comment Models used for the Zika virus?
I'd like to add that I think the responses to the Zika virus have shown shocking levels of innumeracy, with irrelevant data given to justify telling women in entire countries to postpone having children for years (increasing chromosomal abnormalities) or to have abortions. There is a huge difference between saying that most people with a rare, terrible condition were exposed to Zika and saying that many people exposed to Zika will develop that disease, and IF the latter is true there should be data backing that up. I don't see the connection to a class on differential equations, though.
Apr
14
comment Models used for the Zika virus?
If you expect the model is the same as some previously studied viruses, what is the mathematical question? For your class, why not just make up some parameters and say, "Zika might spread like this?" If you expect that past models are inadequate, then coming up with a new model might be a mathematical problem.
Apr
14
comment Models used for the Zika virus?
Would you expect the model to be different for Zika compared with other viruses, or just a couple of parameters?