Pearl Dive

I apologize if this is put inelegantly as I'm new to this type of mathematics, but I thought of this experiment and can't stop turning it over in my mind. Take any set of points on a 2d plane and connect every point to every other point using line segments. Now take the midpoints of each of those...

Curators Much of the effort towards improving question quality goes towards sanctioning bad questions-- closure, review, etc. There is a lot of room for greater recognition of good questions. On our site, we have experimented with ways to do this in the form of curators. Curators seek out excepti...

Every one of the functions $\sin(x), \ldots, \sin(x+n)$ solves the ODE $y'' + y = 0$. But the solution space of a linear (homogeneous) second order ODE is only two-dimensional, which tells you that these sine functions must be linearly dependent if you take at least $3$ of them. That is, there ex...

Let $H$ and $N$ be two groups, $\phi$ and $\psi$ be two homomorphisms $H \to \operatorname{Aut}(N)$, and $G_1 = N \rtimes_\phi H$ and $G_2 = N \rtimes_\psi H$ corresponding semidirect products. What are conditions on $\phi$ and $\psi$ for $G_1 \cong G_2$? If for some automorphisms $h \in \operato...

Consider the following family of polynomials $$P_n(X)=\sum_{i=0}^n(n+1-i)X^i,\,n\ge 1$$ Let’s write down the first few $$ \begin{align} &P_1(X)=X+2\\ &P_2(X)=X^2+2X+3\\ &P_3(X)=X^3+2X^2+3X+4\\ &P_4(X)=X^4+2X^3+3X^2+4X+5 \end{align} $$ My claim is that this family is a family of irreducible polyno...