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4h
comment Partial Sum of the Binomial Theorem
mathoverflow.net/questions/93744/… mathoverflow.net/questions/55585/…
8h
awarded  Nice Answer
10h
comment (Non)existence of mirrors with more than two foci
It is difficult to have a multipurpose boundary, although I did construct examples of that. Part of the idea here was to make sure each part of the boundary is visible from at most two points at a time. If you shine light from $x_1$ at something directing light from $x_2$ to $x_3$, you typically need another curve to refocus that.
10h
revised (Non)existence of mirrors with more than two foci
Fixed picture.
15h
revised (Non)existence of mirrors with more than two foci
Added picture.
15h
answered (Non)existence of mirrors with more than two foci
1d
comment Is there a generalization of Polya urns to continuous outcome event?
N. Makarov talked about trying to formalize the following: Start with a compact subset of the plane, and let a small particle drift in from infinity under Brownian motion. Let it stick where it first hits, and add it to the set. Then release another particle, etc. Take a limit of this process as the size of the particle goes to 0. This is similar to a Pólya urn model because the harmonic measure may tend to expand where the particle hits. I believe he proposed it as a problem worthy of study.
1d
comment Are numbers fundamental mathematical entities?
By the way, I have heard that the Ancient Greeks viewed geometry as more fundamental than arithmetic. To prove that two regions have the same area, we show that the areas are the same number, but they found geometric dissections from one region to the other, which is always possible for polygons of the same area. (See Hilbert's 3rd problem.) It's a good enough story that I pass it on without verifying it.
2d
revised Thales' semicircle theorem in higher dimensions
Mainly fixed typos.
2d
comment Thales' semicircle theorem in higher dimensions
When people ask how to visualize things in higher dimensions, I think it is good to mention problems like this showing that we struggle to visualize things in $3$ dimensions.
2d
answered Thales' semicircle theorem in higher dimensions
Mar
27
comment Solid angles of a tetrahedron
@Joseph O'Rourke: Thanks, I fixed them.
Mar
27
revised Solid angles of a tetrahedron
fixed expired picture links
Mar
24
revised Is there a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$?
Added details
Mar
18
comment Expected value of the minimum with limited independence
Nice construction. Those should be $\frac{k+1}{k(k+1)}$ and $\frac{(k+1)^2}{k(k+1)}$ on the left hand sides. Also, in the middle, $\frac{1}{k+1}$ should be $\frac{1}{k}$.
Mar
15
comment Expected value of the minimum with limited independence
@Dorothy: I don't know. The argument that James Martin gave in the comments on his answer can be restated as that $E[X] =1+\sum_{k=1}^n P(N_k = 0) \le 1 + \sum_{k=1}^n \frac{1}{k}-\frac{1}{n} = H_n \approx \log n + \gamma.$ The estimate $P(N_k =0 ) \le \frac{1}{k} - \frac{1}{n}$ can only be improved by a constant factor since if you define the partition $\mu_k = k+k+...+k$ then the distribution $\frac{k-2}{k-1} \mu_1 + \frac{1}{k-1} \mu_k$ is pairwise independent and has $P(N_k = 0) \sim \frac{1}{ek}.$ I haven't found a distribution so that for many different $k$, $P(N_k=0)$ is large.
Mar
14
revised Expected value of the minimum with limited independence
added 32 characters in body
Mar
14
answered Expected value of the minimum with limited independence
Mar
12
comment can you use Bayes' rule twice?
I don't think the naive Bayesian classifier assumes independence, only conditional independence.
Mar
12
comment can you use Bayes' rule twice?
Your assumption was not only that $X$ and $Y$ are independent, but also that they are conditionally independent given $C$. Those are very different and neither implies the other.