Let $\mathfrak g$ be a simple Lie algebra. Is there any way to find the $\mathfrak g$-invariant subspace of $\mathfrak g \otimes \mathfrak g\ $? I am familiar with the result for $\mathfrak g = sl_2(\mathbb C)$ in which case the subspace is $\mathbb C \Omega,$ where $\Omega$ is a Casimir element...

I'm developing a binary classification tree and having some touble interpreting my training/validation curves. I used the CART algorithm with information gain as my splitting criterion. Below, i'm increasing the depth of the tree and observing the performance in terms of F1 score (harmonic mean o...

I need to devise a module for next academic year which is an introduction to pure mathematics. They need to use this module as a step stone module such as number theory, group theory, combinatorics, and real analysis. What should I cover to make this interesting and be used as a hook for them to ...

Many statements of mathematics are phrased most naturally in terms of multisets. For example: Every positive integer can be uniquely expressed as the product of a multiset of primes. But this theorem is usually phrased more clumsily, without multisets: Any integer greater than 1 can be writte...

From googling, it seems commonly believed that Euclid did this, but it seems nowhere in Euclid does he even state this property of a tangent line explicitly. Rather Euclid gives 4 other equivalent properties, that the line does not cross the circle, that it is perpendicular to the radius, that ...