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bio website gottwurfelt.wordpress.com
location Atlanta, GA
age 30
visits member for 4 years, 3 months
seen 23 hours ago

Data scientist and math blogger.


1d
comment $f\left(x + \frac1x\right)= x^3+x^{-3},$ find $f(x)$
If you do the algebra, it turns out that this gives $f(y) = y^3 - 3y$, agreeing with the answer many others have given.
Oct
21
comment True or False: Basis in the space of polynomials of degree less or equal to 2014 should contain polynomial of degree 2013.
Not sure if this question will stay alive. In case it does, here's a hint: there's nothing special about 2014. Can you construct a basis in the space of polynomials of degree less than or equal to 2 that does not contain a polynomial of degree 1?
Oct
15
comment Betting system for coin toss game?
This is in fact a martingale. You can win this way if you have unlimited money and can place arbitrarily large bets - but if you have unlimited money, why are you gambling?
Oct
3
comment The geometry of a spiral made of adjacent right triangles
This is related to the "spiral of Theodorus": en.wikipedia.org/wiki/Spiral_of_Theodorus
Sep
29
comment Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$
Also, possibly relevant SE links: math.stackexchange.com/questions/217240/…, math.stackexchange.com/questions/2339/… , although I'm not sure if any of the proofs here satisfy your requirements - this problem seems very naturally to live in the complex numbers.
Sep
29
comment Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$
One thing to try: instead of doing the "special" case where you're looking at degrees, can you find a formula for $\sum_{k=1}^{n-1} \tan^2 {k \pi \over 2n}$, where the angles are in radians? Your question is the $n = 90$ case of this. If you try small integer values of $n$ it's not terribly hard to conjecture a formula.
Sep
27
comment How many non-collinear points determine an $n$-ellipse?
I don't know, but see this presentation by Sturmfels: math.berkeley.edu/~bernd/feb19.pdf or this paper: math.ucsd.edu/~njw/PUBLICPAPERS/kellipse_imaproc_toappear.pdf
Sep
23
comment How do I show that the series: $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m))) + \cdots$ converges for all real numbers $m$?
I don't have my copy at hand, but I believe that $a_k \sim \alpha \sqrt{3/k}$ is shown in de Bruijn's Asymptotic Methods in Analysis.
Sep
18
comment Proving $\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$
If I'm not mistaken, this generalizes to give $\sum_{k=1}^n \csc^2 (k\pi/(2n+1)) = 2n(n+1)$, with $n = 3$ being the original question.
Sep
15
comment Random walk on a finite square grid: probability of given position after 15 or 3600 moves
For a related blog post see John Cook's post.
Aug
15
comment Is there a name for the function $\max(x, 0)$?
It should be noted that the negative part is, in fact, a positive number. For example the negative part of -3 is 3.
Aug
5
comment Asymptotic behavior of $\sum\limits_{k=1}^n \frac{2^k}{k}$
Call your sum $S_n$. Let $T_n = S_n/2^n$; then it's not hard to show that $T_1 = 1$, $T_n = T_{n-1}/2 + 1/n$. You want to show $T_n \sim 2/(n+1)$. Not sure this helps.
Jul
16
comment Probability of Random Variable Minus Random Variable
There are quantitative bounds on the approximation such as the Berry-Esseen inequality, but those are almost surely beyond the scope of the course the poster is taking.
May
15
comment Probability of getting better than a certain score
Andre, you can always just run a more involved simulation (more trials, or compute some other intermediate results) if you want a coffee break.
May
9
comment How prove this frog can finite steps jump the point $(\frac{1}{5},\frac{1}{17})$
If I understand the problem correctly, a jump like (1/17, 4/17) isn't allowed since it's not of length 1. But you could use (8/17, 15/17) in this context.
Mar
14
comment How likely is it not to be anyone's best friend?
I wonder what, say, Facebook's data on this would look like, where "best friend" would be defined as "person most interacted with on Facebook". Of course this has nothing to do with your model.
Feb
4
comment Expected value of first head in coin toss.
Since $X$ is a discrete random variable, you should take the sum $\sum_x x f_X(x)$, not the integral.
Jan
31
comment Inverse birthday estimation
Possible duplicate of math.stackexchange.com/questions/114544/… ?
Jan
31
comment Inverse birthday estimation
Yes, it's natural log. I don't know a name for this problem, although that doesn't mean there isn't one.
Dec
31
comment What's the name of this quantity?
We have $M_n \ge \lfloor n^2/2 \rfloor$, since $dist(\sigma_n) = \lfloor n^2/2 \rfloor$ when $\sigma_n$ is the ``reverse'' permutation $n(n-1)(n-2) \ldots 1$. This is the sequence oeis.org/A007590. But I can't prove that this is an equality.