MathOverflow

Can you given me a example of a quandle $Q$ and knot $K$ such that $K$ only has trival $Q$-colorings but $K\#K$ has nontrival $Q$-colorings?

Consider the polynomial $p(z)=\sum_0^na_iz^k$ where $a_n=1$ and $a_k \sim N(0,1)$, $k=0,1,2,\dotsc,n-1.$What is the probability that 2 will be a bound of the roots of the polynomial? How can we find the asymptotic probability for the same? I did a simulation...

Given real numbers $y_i's$, consider the following convex optimization problem:$$\min_{x_i's} \sum_{i=1}^N(y_i-x_i)^2 + \lambda\sum_{i=1}^{N-1}|x_{i+1}-x_{i}|.$$The paper A Direct Algorithm for 1D Total Variation Denoising by Laurent Condat claims that...

It is well known that the function $f(z)=2\cos(\sqrt {-z})$ (or more accurately the entire function $f(z)=2\sum_{n=0}^\infty \frac{z^n}{(2n)!}$) satisfies such a functional equation, i.e. $f(4z)= f(z)^2-2$ ; it is not hard to show that this is the unique...