A metric has positive sectional curvature if and only if ${\rm Ric}_{ij} < \frac{r}{2}g_{ij}$

This is a cross-post from my question on MSE.

It is well known that

In dimension three a metric has positive sectional curvature if and only if $ {\rm Ric}_{ij} < \frac{r}{2}g_{ij}$ . where $ r$ denotes the scalar curvature.

Is there a higher dimensional analogues of this theorem? Any reference or counterexample?

What are all pairs $(R,M)$ of a ring $R$ and a left $R$-module $M$ such that all endomorphisms of $M$ are scalar multiple of $\text{id}_M$?

I was playing with some endomorphism rings and got curious whether a classification of all (not necessarily unitary) module $ M$ over a (not necessarily unital) ring $ R$ such that for every $ R$ -module homomorphism $ \varphi:M\to M$ , there exists $ r\in R$ such that $ $ \varphi(x)=r\cdot x$ $ for all $ x\in M$ . I know that it means $ $ \text{End}_R(M)\cong R/\text{Ann}_R(M)$ $ but I can’t make any interesting conclusion from this. In the case that $ R$ is a division ring and $ M$ is an $ R$ -vector space, then it is obvious that $ M$ must be at most one-dimensional. Could you please give me some references if there are any studies regarding this question?

“Almost” absolute continuity of derivative of BV function if ${\rm Tr}\,D_Sf = 0$

Let $ f: \mathbb R^N \to \mathbb R^N$ be a $ BV$ function. Suppose that $ \mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure: $ \operatorname{div} f \ll \mathcal L^N$ . This implies, as seen in a related question, that $ $ {\rm Tr}\,D_Sf = 0.$ $

Does it mean that $ D_S f$ is almost absolutely continuous in some sense? What is the correct way to formalize this notion of “almost” absolute continuity?

Here is a more precise question:

  • As mentioned in the question Lusin Lipschitz approximation in BV and Sobolev space, $ f$ is Lipschitz outside a small set (small with respect to the Lebesgue measure). Does $ {\rm Tr} D_S f = 0$ imply that this set is negligible with respect to the singular measure $ D_S f$ ?

Related question are asked in the posts BV function with absolutely continuous divergence and Role of absolute continuity of divergence of BV function in proof of renormalization property

Can’t grant myself permissions to execute `rm` command as sudo

I have a user account on my machine:

$   (whoami) > foo_user 

I’ve updated by /etc/sudoers file to grant myself permissions to use the rm command:

User_Alias      OPERATORS = foo_user Cmnd_Alias      RM = /bin/rm OPERATORS       ALL = RM OPERATORS       ]ALL = (ALL) ALL 

However, when I try to run rm on a directory, I get a permissions denied error:

rm: /root-directory: Permission denied 

What am I doing wrong?

Disallow `rm` to remove `*`

Note: If you googled this by the title of this question, don’t use this script unless you know what it is supposed to do.

This is a script in bash 3+ that I have used for long for preventing rm * and rm -rf * from accidentally invoked and removing important files by mistake. I put it in my ~/.bash_aliases.

alias rm='set -f;rm' rm(){     if [[ "$  -" == *i* ]]     then         if [ "$  1" = "*" ] || [ "$  2" = "*" ] || [ "$  1" = "./*" ] || [ "$  2" = "./*" ]         then             echo "Abort: refusing to remove *, please go to the parent folder and do rm <folder_name>/*" 1>&2             set +f             return 1         fi     fi       set +f     /bin/rm -i $  @ }  set +f 

I would like to know whether there are any vulnerabilities and whether it can be improved.