## fa.functional analysis – Reference request: Is if possible to estimate the local behaviour of the solution of \$nabla cdot a(x) nabla f=g\$ via constant coefficients?

Consider the divergence form uniformly elliptic operator $$nabla cdot a(x) nabla$$
where the coefficient $$a$$ are smooth and bounded and $$D$$ is a bounded
and smooth domain of $$mathbb R^d$$
$$begin{cases} nabla cdot a(x) nabla f (x)=g text{ in } D \ f(x) = 0 text{ in } partial D, end{cases}$$
where $$g$$ for some $$g$$.
Consider now $$x_0in D$$ and $$delta < d(x,partial D)$$ and the function $$f_{x_0}$$ which solves
$$begin{cases} nabla cdot a(x_0) nabla f^delta_{x_0} (x)=g text{ in } B(x_0,delta)\ f^delta_{x_0}(x) = f(x) text{ in } partial B(x_0,delta). end{cases}$$

I was wondering whether it is possible to bound quantities such as
$$M(x_0,delta,r,p):=r^{-d}|f-f^delta_{x_0}|_{L^p(B(x_0,r))}$$
for $$r < delta$$ and for some $$p in (1,infty)$$. In particular, I was wondering about the case asymptotic behaviour for $$r to 0^+$$. That is, can I show that
$$M(p, gamma):= sup_{x_0 in D} sup_{delta < d(x,partial D)wedge c_a} sup_{r le delta} frac{M(x_0,delta,r,p)}{r^gamma},$$
is finite for some $$gamma>0$$ and some constant $$c_a >0$$? If so, does that bound depends on the smoothness of $$g$$?

The idea being that if $$delta$$ is sufficiently small, $$a(x)approx a(x_0)$$ in the ball $$B(x_0,delta)$$ and therefore the two equations should behave similarly. I am not sure if this is indeed enough or if I would need to ask $$delta$$ to vanish as well.

I would appreciate any references or even what are the keywords to find such type of estimates.

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## hreflang – Does Google show the pages we have in the local language by IP address?

hreflang – Does Google show the pages we have in the local language by IP address? – Webmasters Stack Exchange

## differential geometry – Local coordinates of one form on a principal bundle

I am reading Natural and Gauge Natural Formalism for Classical Field Theory by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates.

Let’s say we have a principal bundle $$mathcal{P}=(P, M, pi ; G)$$ and the isomorphism $$T_{e} L_{p}: mathfrak{g} longrightarrow V_{p}(pi): T_{A} mapsto lambda_{A}(p)$$. He fixes a point $$p=(x, g)_{(alpha)}$$. $$theta_{(L)}^{A}=bar{L}_{a}^{A}(g) mathrm{d} g^{a}$$ is the local basis of left invariant 1-forms dual to $$lambda_{A}=L_{A}^{a}(g) partial_{a}$$ where the two matrices are inverse of each other. Then the connection one form can be written as
$$begin{equation} bar{omega}_{p}=left(theta_{(L)}^{A}(p)+mathrm{Ad}_{B}^{A}left(g^{-1}right) omega_{mu}^{B}(x) mathrm{d} x^{mu}right) otimes T_{A} end{equation}$$
Pulling back with a section, he gets
$$begin{equation}sigma^{*} bar{omega}=left(bar{L}_{a}^{A}(g) partial_{mu} g^{a}(x)+mathrm{Ad}_{B}^{A}left(g^{-1}right) omega_{mu}^{B}(x)right) mathrm{d} x^{mu} otimes T_{A} end{equation}$$
However, then he goes on saying “We remark that the local gauge $$sigma$$ also induces a local trivialization of $$mathcal{P}$$. In the induces local trivialization, the section $$sigma$$ has the expression $$sigma: x^{mu} mapstoleft(x^{mu}, eright)$$ and the vector potential is of the form”

$$begin{equation} sigma^{*} bar{omega}=omega_{mu}^{A}(x) mathrm{d} x^{mu} otimes T_{A} end{equation}$$
He also states that the induces connection is of the form
$$begin{equation} omega=mathrm{d} x^{mu} otimesleft(partial_{mu}-omega_{mu}^{A}(x) rho_{A}right) end{equation}$$

1. My first question is how one can derive the expression of $$bar{omega}_{p}$$?
2. My section question is why every other book on the subject I know uses $$sigma^{*} bar{omega}=omega_{mu}^{A}(x) mathrm{d} x^{mu} otimes T_{A}$$ as the definition of a form in local coordinates even this seems not to be true for a general section.
3. I am also not sure how one can derive the form of the induced connection.

## local area network – read-only access on Frtizbox FTP

I am trying to access the contents of my hard drive which is connected to a FRITZ!box 7490. The FRITZ! box can be accessed by both SMB and FTP. Sometimes I need to access files from outside of the FRITZ network so I use FTP and access it over the internet with a subdomain from afraid.org My problem is that, even though I have set the webGUI to allow access for me to access the files, I still get a read-only error. See below:

t

## python – Deploy Django em servidor local

Parece ser trivial a minha questão, mas…

Tenho um servidor local no meu trabalho rodando Windows e xampp com algumas aplicações em mysql e php. Ok. Agora eu tenho um novo projeto Django e PostgreSQL. Gostaria de saber se posso aproveitar essa estrutura do xampp para colocar meu projeto Django em produção? Ou qual seria a melhor opção?

Obs: Não se trata de fazer o deploy na própria internet.

## local customs – Is it rude to speak Swedish in Norway?

I speak Swedish quite well, but my active Norwegian is very basic, although I can understand most of it if I concentrate. I’m aware Norwegians can usually/always understand Swedish. I’m told Norwegians and Swedes may have conversations in which each speak their own language. I’ve seen exactly that in films, and I find it weird, for I understand one half of the conversation perfectly well, and the other half only partially and with difficulty.

When I visit a non-English speaking country, I tend to try to speak the local language, however basic. Personally, I find it arrogant or rude to expect the locals to speak a foreign language when I am the visitor, even when I’m in a country where the level of English is generally very high (such as in Norway). Not everybody is at ease speaking English, in particular in off-the-beaten-track rural corners of the country.

But how is it perceived to speak Swedish in Norway? May it be perceived as rude or arrogant to expect that everybody understands what is, after all, a foreign language? Or does the closeness between the languages mean that people likely wouldn’t think about it, and perhaps barely notice it? There may be cultural issues related to history that affect this as well.

In case it matters, consider a rural area of Trøndelag that receives relatively few foreign visitors.

I could either try to speak broken Norwegian with Swedish mixed in, Swedish, or English (or German, but I’ve only ever once come across a Norwegian Sami person where that ended up being our best shared language).

## deployment – Add domain user to local admin group with MDT

I have a working task sequence which install Windows 10 pretty well inside a domain and I would like to add this feature : AdminAccounts.

The rule “SkipAdminAccounts=NO” is set, the page appears correctly during the Wizard, but it does nothing : no user has been add to the local admin group.

I haven’t see any error or warning in any logs (or maybe I didn’t search in the right place).

Maybe I’m missing the right task in my sequence ?

If someone could help me, I’ll appreciate.

## applications – Show remote android emulator window in local

Hi I’ve configured Android development environment in a remote dedicated server running Ubuntu 20.04 with the help of the tutorial https://proandroiddev.com/how-to-setup-android-sdk-without-android-studio-6d60d0f2812a.

Installed related tools by `sdkmanager --install platform-tools emulator "build-tools;30.0.3" "platforms;android-30" "extras;android;m2repository" "extras;google;m2repository" "system-images;android-30;google_apis;x86_64" "system-images;android-30;google_apis_playstore;x86_64" "patcher;v4"`

Created the avd by `avdmanager --verbose create avd --force --name "generic_30" --package "system-images;android-30;google_apis;x86_64" --tag "google_apis" --abi "x86_64"`

I had to install `apt-get install libgl1-mesa-dev libglu1-mesa-dev` to run the Android Emulator.

Ran the emulator with `emulator -avd generic_30 -noaudio -no-window -no-boot-anim -gpu off -screen touch` and it ran with desired output.

``````emulator: Android emulator version 30.5.5.0 (build_id 7285888) (CL:N/A)
emulator: ERROR: FeatureControlImpl.cpp:269: Feature control already exists in create() call
handleCpuAcceleration: feature check for hvf
emulator: feeding guest with passive gps data, in headless mode
emulator: WARNING: Your AVD has been configured with an in-guest renderer, but the system image does not support guest rendering.Falling back to 'swiftshader_indirect' mode.
emulator: INFO: GrpcServices.cpp:301: Started GRPC server at 127.0.0.1:8554, security: Local
Your emulator is out of date, please update by launching Android Studio:
- Start Android Studio
- Select menu "Tools > Android > SDK Manager"
- Click "SDK Tools" tab
- Check "Android Emulator" checkbox
- Click "OK"

emulator: INFO: boot completed
emulator: INFO: boot time 24232 ms
emulator: Increasing screen off timeout, logcat buffer size to 2M.
emulator: Revoking microphone permissions for Google App.
``````

From my Windows 10 machine created SSH tunnel with the command `ssh -NL 5554:localhost:5554 -L 5555:localhost:5555 user@xx.xx.xx.xx` and connected adb with `adb connect 127.0.0.1:5555`.

With the command I could see that the device is also connected.

``````Connected devices & emulators
Searching for devices...
┌───┬─────────────────────────┬──────────┬───────────────────┬────────┬───────────┬─────────────────┐
│ # │ Device Name             │ Platform │ Device Identifier │ Type   │ Status    │ Connection Type │
│ 1 │ sdk_gphone_x86_64_arm64 │ Android  │ 127.0.0.1:5555    │ Device │ Connected │ USB             │
└───┴─────────────────────────┴──────────┴───────────────────┴────────┴───────────┴─────────────────┘
``````

Until this everything has worked as expected (or I think so).

Now my question is how to show the emulator window in my local Windows machine? It’s possible I think. At-least this is how SauceLabs or other online mobile test suites may be working. Do you know any solutions as such for this?

## Partial differential with respect to local coordinates

Let $$M^n$$ be a smooth manifold and $$(U,phi)$$ be a chart with local coordinates $$x^1,…,x^n$$, that is $$phi(u)=(x^1(u),…,x^n(u))$$, with $$x^i:Utomathbb{R}$$. For $$fin C^infty(M)$$ we define
$$tfrac{partial}{partial x^i}f:Uto mathbb{R},quad pmapstotfrac{partial}{partial x^i} f(p)=partial_i(fcircphi)(phi(p)),$$
where the partial derivative is w.r.t. the $$i$$-th coordinate is just partial derivative in $$mathbb{R}^n$$. Now if we take a second chart $$(V,psi)$$ with local coordinate functions $$y^1,…,y^n$$, why is it true that for $$pin Ucap V$$ one has
$$tfrac{partial}{partial x^i}|_p=tfrac{partial}{partial x^i}|_p(y^j)tfrac{partial}{partial y^j}|_p=sum_{j=1}^n tfrac{partial}{partial x^i}|_p(y^j)tfrac{partial}{partial y^j}|_p.$$
(In the identity the Einstein notation for sums was used).
I think, I understand why it makes sense to define the partial derivative like this and for the case of $$M^n=mathbb{R}^n$$, we can just take $$x^i$$ and $$y^i$$ to be the standard coordinates, in which case the identity would be trivial. But how to argue for general manifolds?