Recovering the bimodule from the trivial extension

Given a ring $ S$ with an $ S$ -bimodule $ M$ , the trivial extension of $ (S,M)$ is defined as the ring $ R:=T_M(S)$ with $ R= S \oplus M$ with multiplication $ (s,m)(s’,m’)=(s s’, sm’ +m s’)$ . We then have that $ M$ is an $ T_S(M)$ -ideal and $ S \cong T_M(S)/M$ . Thus we can recover $ S$ from $ T_S(M)$ as soon as we know $ M$ as an $ T_S(M)$ -bimodule.

Call $ T_S(M)$ unique in case $ T_S(M) \cong T_{S’}(M’)$ implies $ S \cong S’$ and $ M \cong M’$ .

Question 1: Is there a nice criterion when a trivial extension $ T_S(M)$ is unique?

I am mostly interested in the following special case: Let $ A$ be a finite dimensional algebra over a field $ K$ and $ D(A)=Hom_K(A,K)$ . Let $ T(A):=T_{D(A)}(A)$ , which is a symmetric Frobenius algebra with twice the vector space dimension of $ A$ .

Question 2: Given $ A$ , is there a nice a way that when the trivial extension algebra $ R=T(A)$ is given, one gets all two-sided ideals $ I_i$ (as right modules and/or bimodules) of $ R$ such that $ R/I_i \cong A_i$ for an algebra $ A_i$ such that $ R \cong T(A_i)$ ? Especially: When is $ T(A)$ unique as a trivial extension (meaning $ T(A) \cong T(A’)$ implies $ A \cong A’$ )?

For example for any field $ K$ , the ring $ K[x]/(x^2)$ is unique as a trivial extension. More generally, I wonder whether the trival extensions of local finite dimensional algebra are unique as trivial extensions.

Note that all ideals $ I_i$ would have the same vector space dimension. Are they related in a nice way, which could mean that in case you have one $ I_i$ the other $ I_j$ can be obtained from the one $ I_i$ by a certain operation?

Is there a way to find all such ideals $ I_i$ (as right $ T(A)$ -modules) as in question 2 with the GAP-package QPA for a given algebra $ A$ ? Note that knowledge as right $ T(A)$ -modules would be enough since then one can recover the $ A_i$ as $ A_i \cong End_{T(A)}(T(A)/I_i)$ .

product attribute limit by ACL Magento2 (extension attribute)

I have number of custom of product attributes in my store and I want to limit this attribute access to only few users. so how can do this(in backend)? is it possible by ACL? I read about extension_attribute but not sure it is possible or not.

For example I have one custom attribute factory_price and I want to allow this attribute value to my sales team not from another user .

Any idea on this?

Chrome extension installation process design

Currently I am trying to program a Chrome extension, these are written in javascript. I have only used javascript on webpages and I am in a bit of a situation with having long functions. After installing a Chrome extension I require user input 2 times. Before, in between and after the user inputs I am executing different parts of code. To prevent callback hell I started using but this has ended up in a big mess. This is the current program flow after installing:

Begin installation process

event function function event function require user input event function event function event function event function require user input event function event function 

End installation process

As you can see before the chrome extension is properly installed I have called about 10 functions and raised 10 events. This is a huge mess, but I do not see a way to fix this. The installation process has to finnish before a user can use the chrome extension. Whenever an exception is raised it is handled and the user will start over again. Do you see any way to fix this? I can provide code if needed but it is quite a lot.

Edit I believe my question might not be clear enough. I basically have a waterfall in my code. My chrome extension requires the user to go through a setup process before being able to use the application. This installation/setup process is displayed above and consists of functions emitting events after they are done which call other functions which emit events again. This can be seen as a waterfall. Every function also needs to finnish before the execution of the next

Extension of Valdivia-Vogt isomorphism from $\mathscr{D}(K)$ to $\mathscr{E}'(K)$

Let $ M$ be a $ d$ -dimensional (say, Hausdorff, paracompact, connected and oriented) smooth manifold, and $ K\subset M$ compact with $ \mathring{K}\neq\varnothing$ . M. Valdivia has shown (based on previous results by himself and D. Vogt, see e.g. M. Valdivia, A representation of the space $ \mathscr{D}(K)$ , J. reine angew. Math. 320 (1980) 97-98) that the nuclear Fr├ęchet space $ \mathscr{D}(K)$ of smooth functions supported in $ K$ is topologically isomorphic to the space $ s$ of rapidly decreasing sequences: $ $ s=\{(a_n)_{n\in\mathbb{N}}\ |\ ((1+n)^k a_n)_{n\in\mathbb{N}}\text{ is bounded for all }k\in\mathbb{N}\}\ .$ $ Let $ \Phi:\mathscr{D}(K)\cong s$ denote the Valdivia-Vogt isomorphism. It is clear that the transpose $ {}^t\Phi$ of $ \Phi$ yields a topological isomorphism between the dual $ s’$ of $ s$ $ $ s’=\{(a_n)_{n\in\mathbb{N}}\ |\ ((1+n)^{-k}a_n)_{n\in\mathbb{N}}\text{ is bounded for some }k\in\mathbb{N}\}$ $ and the dual $ \mathscr{D}(K)’$ of $ \mathscr{D}(K)$ , which may be identified as a vector space with $ \mathscr{D}'(\wedge^d T^*M\rightarrow M)/\mathscr{D}(K)^\perp$ , where $ $ \mathscr{D}(K)^\perp=\{u\in\mathscr{D}'(\wedge^d T^*M\rightarrow M)\ |\ u(\varphi)=0\text{ for all }\varphi\in\mathscr{D}(K)\}$ $ is the annihilator of $ \mathscr{D}(K)$ . It is clear that $ \mathscr{D}(K)’$ contains $ $ \mathscr{E}'(K)=\{u\in\mathscr{E}'(\wedge^d T^*M\rightarrow M)\ |\ \text{supp }u\subset K\}$ $ as a (closed) subspace (I apologize for the slightly unconventional notation). Since the sequences $ e_j=(e_{j,n})_{n\in\mathbb{N}}$ given by $ $ e_{j,n}=\begin{cases} 0 & (n\neq j) \ 1 & (n=j) \end{cases}$ $ form a Schauder basis of both $ s$ and $ s’$ , it is clear that $ s$ is dense in $ s’$ .

Question: Does $ \Phi$ extend to a topological isomorphism between $ \mathscr{E}'(K)$ and $ s’$ ? Likewise, does the restriction of $ {}^t\Phi$ to $ s$ yield another topological isomorphism between $ s$ and $ \mathscr{D}(K)$ ?

My question is inspired by the known characterization of $ \mathscr{D}([0,1])$ and $ \mathscr{E}'([0,1])$ through the decay / growth of their Fourier coefficients in $ [0,1]$ .

How to play video files with .DM extension?

I had some files I transferred from my computer to my iPhone via Zapya and deleted actual files later on

I wanted to recover later and discovered that actual files are nowhere but I can see same files with same size but with .DM extension in my C:\Users\USERNAME\Documents\Zapya\Video folder

I tried removing .DM extension and played on MPC-HC player and also on VLC Player but it does not play

Any idea on how to recover those files?

Central extension gives a gerbe over stack

Consider a central extension of Lie groups $ 1\rightarrow S^1\rightarrow \hat{G}\xrightarrow{\pi} G\rightarrow 1$ .

I understand that this mean $ \pi:\hat{G}\rightarrow G$ is a surjective homomorphism of Lie groups (not sure if this has to be submersion) and that $ S^1\subseteq Z(\hat{G})$ . There is a local section $ \sigma:U\rightarrow \hat{G}$ such that $ \pi\circ \sigma=1_U$ where $ U$ is an open nbd of $ 1\in G$ . Correct me if I am missing some conditions.

Let $ X$ be a manifold with an action of $ G$ on it. Then we have the notion of quotient stack $ [X/G]$ .

There is an action of $ \hat{G}$ on $ X$ given by $ (\hat{g},x)\mapsto \pi(\hat{g})\cdot x$ .

We can then consider the quotient stack $ [X/\hat{G}]$ .

Given a manifold $ Y$ , objects of $ [X/G](Y)$ are pairs $ (P\rightarrow Y,P\rightarrow X)$ where $ P\rightarrow Y$ is a principal $ G$ bundle and $ P\rightarrow X$ is a $ G$ -equivariant space (see that $ G$ acts on $ P$ and $ X$ ).

“As locally any map $ T\rightarrow G$ can be lifter to $ \tilde{G}$ “, the map of stacks $ [X/\hat{G}]\rightarrow [X/G]$ is a gerbe over stack.

I see that, locally any map $ \theta: T\rightarrow G$ can be lifted to $ \hat{G}$ . As there is a section $ \sigma:U\rightarrow \hat{G}$ , we can consider $ \theta^{-1}(U)\xrightarrow{\theta} U\xrightarrow{\sigma} \hat{G}$ and $ \pi\circ (\sigma\circ \theta)=\theta$ . Thus, any map $ \theta:T\rightarrow G$ can be locally lifted to $ \hat{G}$ . But, I am not able to see why this imply $ [X/\tilde{G}]\rightarrow [X/G]$ is a gerbe over stack.

Any comments are welcome.

extension of a regular path

Given a smooth path in $ \gamma: I \to \mathbb R^n$ . such that $ \gamma(0) = x$ , $ \gamma(1)=y$ , $ \gamma'(t) \neq 0, \forall t$ , $ x \neq y$ . Let $ z \neq x, z \neq y$ , Is it always possible to extend $ \gamma$ to a smooth path connecting $ y$ to $ z$ , such that $ \gamma'(t)\neq 0$ everywhere?

What is the right tool to deal with this question? Maybe should I take piecewise path first then try to smooth it using convolution fixing endpoints?