All Questions

I find the following averaged-integral amusing and intriguing, to say the least. Is there any proof? For any pair of integers $n\geq k\geq0$, we have $$\frac1{\pi}\int_0^{\pi}\frac{\sin^n(x)}{\...

Let $f(k,n)$ be the smallest non-zero absolute value of a sum of $k$ complex $n$th roots of unity. Asking for bounds in either direction, Tao suggested that a polynomial lower bound seemed plausible ...

Let $n > 0$ be a positive integer (large) and $p > 2$ a fixed prime number. What is the probability that $$\sum_{ 1 \leq i < j \leq n} a_ia_j = 0 \mod p$$ where $a_1, a_2, \dots a_n$ are ...

If the question was stated to appeal to the general public, it would be something like this. For a number such as $\pi$ or $\sqrt{2}$, the digits in base $b$ appear to be randomly distributed. We are ...

Let $\mathbf{x}$ be a random Gaussian vector in $\mathbb{R}^n$, i.e. $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_n)$. Then for any fixed unit vector $\mathbf{u}$, one has $\mathbf{u}\mathbf{u}^\...

Given a vector of $n$ cells and $k$ elements in it, we can define a cluster of elements as a contiguous sequence of elements inside the vector. My goal is to calculate the average cluster size for all ...