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I’m studying Vakil’s Foundations of Algebraic Geometry, and working through the exercises in 2.4 about compatible germs as a method for constructing a sheafification of a given presheaf. Can this construction be vacuous? Can a nontrivial presheaf fail to have a nontrivial set of compatible germs? Can there fail to be a universal functor thatRead more

To define group cohomology $ H^n_\sigma(G,A)$ , we need to know group $ G$ , Abelian group $ A$ , and the group action $ \sigma: G \rightarrow Aut(A)$ . For trivial group action, I know there is a math program called GAP to compute. See Program for computing group cohomology I want to knowRead more

Fielding writes in his famous dissertation (p.89) that although the entities referenced by a resource may change, the resource’s semantic meaning must not. So a resource prot://bookstore.com/new-books-this-week is OK, because although it will always show different books every week, the semantic meaning does not change. What I can’t come up with, is a good exampleRead more

Let $ X, V\in\mathbb{R}^{n\times r}$ such that $ X^\top V$ is symmetric. The central quantity I care about is \begin{equation} \|XV^\top\|_{F}^2+\|X^\top V\|_{F}^2 +[\text{Tr}(X^\top V)]^2. \end{equation} An easy lower bound for this quantity is given by $ 2\sigma_{r}(X)^2\|V\|_{F}^2$ , where $ \sigma_{r}(X)$ is the smallest singular value of $ X$ . I’m wondering whether there existsRead more