Let $CDGA$ be the category of commutative differential graded algebras over a field $k$ of characteristic zero. Denote by $\Omega\left(\Delta^n\right)$ the cdga of algebraic differential forms on the $n$-simplex. There is a natural simplicial enrichment of CDGA given by $Map_n\left(A,B\right)=Hom...

Let $0<x < y < 1$ be given. Prove $$4x^{2}+4y^{2}-4xy-4y+1 + \frac{4}{\pi^2}\Big[ \sin^{2}(\pi x)+ \sin^{2}(\pi y) + \sin^{2}[\pi(y-x)] \Big] \geq \frac{1}{2}$$ I have been working on this problem for a while now and I have hit a wall. I have plotted this and it seems to be true. I also tried to...

I came across the following conjecture, reading a recent paper in the Monthly, an orthogonal matrix of order $n\neq 0 \pmod 4$ has a nonnegative (up to a scalar) row vector. It should be straight in dimensions $2$ and $3$.

Given a probability measure $\mu$ on the interval $[0,1]$, the linear operator $$ T_\mu \! f(y) := \int_0^1 f(yx) \, d\mu(x) $$ takes the space of continuous functions $f: [0, \infty) \rightarrow \mathbb{R}$ such that $f(0) = 0$ to itself. Assuming that $\mu$ is not the delta function at the or...