Consider directed graphs where all nodes have 2 inputs and 2 outputs. If we design a box with N inputs and N outputs, what is the smallest number of nodes it must contain to satis …

I am not very sure if this is a proper question, but I'm trying to investigate what the area of math can offer in researching the differential equation in polar coordinates: $r'^2+ …

The operadic butterfly is a diagram in the category of operads in vector spaces. It extends the short exact sequence relating commutative, associative and Lie operads. $$\begin{ar …

I want to study quasicrystals from mathematical point of view, but I'm having hard time finding materials about it. If you could suggest me some books, articles or papers, I would …

I'm interested in (node/edge-)symmetric 6-regular graphs on 20 vertices and 60 edges, especially ones with a A5/icosahedral/dodecahedral symmetry group and especially their chromat …

Given a differential equation on a Banach space $\mathcal{X}$ of the form $\frac{d u}{d t} = F(u)$, it is often the case that $F$ is equivariant under translations, i.e. that $T_\a …

It is well-known that to maximize the horizontal distance traveled by a projectile fired from the ground at a given speed, one should fire it at a $45^\circ$ angle. What's less-kno …

Suppose, for example, you are simulating samples from a (multivariate) Gaussian with mean zero and covariance $\Gamma=BB^T$. If you had generated a sample $x$, you could generate m …

Let $n \ge 2$ be some positive integer. Given a norm $p : \mathbb{R}^n \to \mathbb{R}$, one can inquire about the structure and properties of its isometry group, i.e. the group of …

There are many optimization problems in which the variables are symmetric in the objective and the constraints; i.e., you can swap any two variables, and the problem remains the sa …

It is well known that 1) if there exists a non-trivial automorphism of a graph $G$ with corresponding permutation matrix $P$ then if $(v,\lambda)$ is an eigenvector-eigenvalue pai …

Hello MO, I've been working on a problem I'm working on in ergodic theory (finding Alpern lemmas for measure-preserving $\mathbb R^d$ actions) and have found some neat tilings, th …

Let $P$ be a polyhedron which satisfies the following three conditions: $P$ is built out of regular hexagons and regular pentagons. Three faces meet at each vertex. $P$ is topolo …

In classical mechanics: If a Lagrangian L is preserved by an infinitesimal change in the state space variables qi -> qi + εKi(q) leads to only second order change in the L …

In non-relativistic quantum mechanics, what are the necessary conditions on the potential (or on the hamiltonian in general) for the ground state to be non-degenrate?