I am reading the lecture notes. On page 21, it is said that when $a_{ij}=-1$, we have \begin{align} ad_c(x_i)^{1-a_{ij}}(x_j)=x_i^2 - (q+q^{-1})x_ix_jx_i+x_jx_i^2. \quad (1) \end{align} Here $ad_c(x_i)(x_j)=x_ix_j - q^{a_{ij}}x_jx_x$. I am trying to verify (1). We have \begin{align} & ad_c(x_i)^{...

I am working on the following problem: Suppose $\displaystyle \frac{\partial T}{\partial t} = \kappa \frac{\partial ^{2}T}{\partial x^{2}}, \, 0<x<L,$ where $\kappa > 0$ is constant and $T(x=0,t)=A$, $\displaystyle \frac{\partial T}{\partial x}(x=L, t) = B$ with $A$ and $B$ constan...

Why is rewriting $x^2 -y^2$ as $(x+y)(x-y)$ a way to avoid catastrophic cancellation? We are still doing $(x-y)$. Is it because the last operation in the second form is a multiplication?

For Lie algebra $gl(m)$, the commutator is \begin{align} [E_{ij}, E_{kl}] = \delta_{jk}E_{il} - \delta_{li}E_{kj}. \end{align} What is the commutator $[E_{ij}, E_{kl}]$ in Lie superalgebra $gl(m|n)$? Thank you very much.

It is a very well known result (I guess attributed originally to Zilber and later extended by Cherlin-Harrington-Lachlan) that every totally categorical theory is pseudofinite. However, the papers usually go through several technical lemmas to obtain not only this result but also several others...