Resolved: manually removed from all questions. The tag bourbaki is entirely useless. It gives nothing that searching "Bourbaki" doesn't already do.

Let $\mathfrak {g}$ be a semisimple Lie algebra with a non-degenerate invariant bilinear form $B$ (e.g. Killing form). Then what is meant by saying that $C$ is a Casimir tensor with respect to the bilinear form $B\ $? The wikipedia article on Casimir element says the following $:$ Given a basis $...

$\def\sO{\mathcal{O}} \def\sA{\mathcal{A}} \def\sB{\mathcal{B}} \def\sF{\mathcal{F}} \def\sG{\mathcal{G}} \def\et{\operatorname{\acute Et}} \def\shtop{\mathsf{ShTop}} \def\etl{\mathsf{\acute Etale}} $A morphism of ringed spaces $(X,\sO_X)\to(Y,\sO_Y)$ can be defined to be a continuous map $\varph...

$\def\et{\operatorname{\acute Et}} \def\sh{\operatorname{Sh}} \def\set{\mathsf{Set}} \def\top{\mathsf{Top}} \def\psh{\operatorname{PSh}} $From a google search, it appears to be a well-known fact that there is an equivalence of categories $\et(X)\simeq\sh_\set(X)$, where $X$ is a topological space...

$\def\VB{\mathsf{VB}} \def\sO{\mathcal{O}} \def\Mod{\mathsf{Mod}} \def\LFMod{\mathsf{LFMod}}$For a ringed space $(X,\sO_X)$ and a ring $R$, denote $\Mod(\sO_X)$ and $\Mod(R)$ to the categories of $\sO_X$-modules and of $R$-modules. Let $M$ be a smooth manifold and denote $\sO_M$ to the sheaf of s...

I am taking my first stats course and struggling quite a bit as my professor does not explain things well. I wanted to double check that I am on the right track. My question is: Show that $f(x)≥0$ for all $x ∈ R$ , where $x$ is the exponential density function:$$f(x) = \begin{cases}λ\operatornam...