## Top new questions this week:

### If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that it's an unfair coin?

Consider a two-sided coin. If I flip it $1000$ times and it lands heads up for each flip, what is the probability that the coin is unfair, and how do we quantify that if it is unfair? Furthermore, ...

(probability) (statistics) (experimental-mathematics)

### Is there something between summation and integration?

Let's take a general function $f(x)$, we can do a summation like: $$\sum_{k=m}^n f(k)$$ And we can do an integration like: $$\int_a^bf(k)dk$$ The basic difference between the two operation is that ...

(algebra-precalculus) (soft-question)

### Why is the commutator defined differently for groups and rings?

The commutator of two elements in a group is defined as $[g, h] = g^{−1}h^{−1}gh.$ In a ring, the commutator of two elements is $[a, b] = ab - ba.$ I'm asking because a ring is a (abelian) group ...

(abstract-algebra) (group-theory) (ring-theory)

### What's a group whose group of automorphisms is non-abelian?

I recently attended an interview for admission to graduate programs in Mathematics. The interviewing professor asked me a question - Tell me a group whose group of automorphisms is non-abelian. ...

(abstract-algebra) (group-theory) (group-homomorphism)

### Why doesn't L'Hopital's rule work in this case?

I have a very simple question. Suppose I want to evaluate this limit: $$\lim_{x\to \infty} \frac{x}{x-\sin x}$$ It is easy to evaluate this limit using the Squeeze theorem (the answer is $1$). But ...

(calculus)

### Completion of the real numbers

On the real line $\mathbb{R}$ endowed with euclidean topology i may put different metrics, inducing the same topology, but inducing different completions. For example if one considers the standard ...

(general-topology) (metric-spaces)

### How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet ...

(real-analysis) (special-functions) (exponential-function) (pi) (infinite-product)

## Greatest hits from previous weeks:

### Surprising identities / equations

What are some surprising equations / identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic ...

(soft-question) (big-list)

### Can I use my powers for good?

I hesitate to ask this question, but I read a lot of the career advice from MathOverflow and math.stackexchange, and I couldn't find anything similar. Four years after the PhD, I am pretty sure that ...

(soft-question) (career-development)

## Can you answer these?

### Is it possible to convert into convex constraint?

I have a constraint looks like $$g^T(FP^{-1}F^T)^{-1}g>1$$ where $P\in S_{++}^{n\times n}$ $g\in\Re^{m}$ ,$F\in \Re^{m\times n}$ ,$n>m$ and $P$ is a variable. When $m=1$, it can be convert to ...

(matrix-decomposition) (semidefinite-programming)

### Random matrices, eigenvalue distribution.

I just investigated randn(1024) + 1i*randn(1024), a 1024x1024 complex valued matrix with elements both real part and imaginary part drawn from $\mathcal{N}(\mu = 0, \sigma = 1)$. I was a bit surprised ...

(probability-distributions) (eigenvalues-eigenvectors) (random-walk) (random-matrices)