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Let $ L|K$ be a finite separable extension. Denote the algebraic closure of $ K$ by $ \bar K$ . $ \forall x\in L$ , denote $ d=[L:K(x)]$ and $ m=[K(x):K]$ . How to show the set $ \operatorname{Hom}_K(L,\bar{K})$ of all $ K$ -embeddings of $ L$ is partitioned by the equivalence relation $ $Read more

Let $ T = \mathbb{G}_m$ be the torus, and let $ \tilde{T}$ be its étale universal cover (a pro-object in schemes of finite type). Then both $ T$ and $ \tilde{T}$ have a well-defined étale homotopy type. Explicitly, the homotopy types are $ ét(T) = B\hat{\mathbb{Z}}$ and $ ét(\tilde{T}) = *,$ for $ *$ theRead more

One can define a correlation by making convolutions of two radially symmetric functions. \begin{equation*} C_0(r)=B_1*B_2(r) [B_1*B_2(0)]^{-1} \end{equation*} On the other hand, one can also construct a correlation based on a generalised diffusion equation. It is widely adapted in background matrix modelling in data assimilation. Give here an 1D example of equation: \begin{equation*} \frac{\partial \eta}{\partial t}-\kappaRead more

We say that two finite dimensional algebras $ A$ and $ B$ are stably equivalent if there is an equivalence $ F:\underline{mod} A\longrightarrow \underline{mod} B$ between the associated module categories modulo projective modules, where $ mod A$ and $ mod B$ are finitely generated modules categories over $ A$ and $ B$ , respectively. Question:Read more