When Stembridge, in the paper Folding by automorphisms, considers folding by automorphism $\sigma$ he considers the root system generated by for each orbit $J$. $$\sum_{i \in J} \alpha_i .$$ Whereas Carter, in Lie Algebras of Finite and Affine Type page 166, he considers generators of the form $$...

Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$: A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive reals. Let $\mathcal{O}(V)$ be the two orientations of $V$ under this definition. A generator of $H...

Are there any good reference to tackle the problem below? Or, are there any know result? Problem Let $f_1...f_n\in \mathbb{Z}[x_1,..,x_n]$ and $V:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a vector field on $\mathbb{R}^n$ defined by each $f_i$. Let's consider a initial value problem for $x(t)\in (\...

Let $T$ be a small triangulated category and let $S\subset T$ be a full triangulated subcategory. We denote this embedding by $I: S\rightarrow T$. Let $F: T\rightarrow S$ be a triangulated functor such that $F\circ I= \mathrm{id}$. Define $L= I\circ F$. Suppose that $L\circ L=L$. If $F(a)=0$ the...

Can $\sf NBG$ class theory prove the foundation scheme: Foundation schema: if $\Phi(X)$ is a formula in which "$X$" occurs free and only free, and in which "$Y$" doesn't occur, whose free variables other than "$X$" are among $\vec{P}$, and if $\Phi(X/Y)$ is the formula resulting from replacing ev...