Image of a self-normalizing subgroup under a surjective homomorphism
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
I am having a serious trouble finding an agreement between the two facts in Special Relativity:  (1) given two events $A$ and $B$ in spacetime separeted by a spacelike interval $s$, the distance in space between the two events will be the shortest in the reference frame which the two events occur simultaneously:   The invariance of the interval says $s^2=(\Delta x)^2-(c\Delta t)^2$ and $\Delta x$ is minimum when $\Delta t=0$  (2) the length contraction says $L=\dfrac{L_0}{\gamma}$ where $\gamma=(1-(v/c)^2)^{-1/2}$ , $L_0$ is the proper length of an object in a reference frame which the object is at rest and $L$ is the length of the object in a reference frame which the object moves with speed $v$ in the $x$ direction.  This says that the lenght of the object in the $x$ direction is the largest in the comoving reference frame of this object.  The question is: According to (1), the reference frame in which two events $A$ and $B$ occur simultaneously is the reference frame that gives the shortest distance in space between the two events. Let $A$ and $B$ be the two end points of a meter stick in a reference frame in which the stick is at rest and at time $t=0$. This two events occur simultaneously, so the distance in space between the two is the shortest acording to (1). But according to (2) shouldn't this distance be the largest?  What is wrong? Why Re-throw Exceptions? I've seen the following code many times:  try {     ... // some code } catch (Exception ex) {     ... // Do something     throw new CustomException(ex);      // or     // throw;      // or     // throw ex; }   Can you please explain the purpose of re-throwing an exception? Is it following a pattern/best practice in exception handling? (I've read somewhere that it's called "Caller Inform" pattern?) Can decimal numbers be considered "even" or "odd"? Proving by induction $4^n  0$ How to delete a Test Class WITHOUT using eclipse I cannot install Eclipse, even if I wanted to.  It is not approved software for my org.  How can test classes be deleted from production, using deployments? There no 'active' checkbox when I am editing the classes in sandbox mode, so I can't just deactivate the test class, and i cant deploy a deleted class to prod...  how the heck is this done?  I dont want to use ANT or special software.  SF must have a native way to achieve this.  And if not...  then I am very disappointed in humanity. Random test data: How to do it right? Can I legally downgrade from Win 7 64 bits to Win 7 32 bits? Possible Duplicate:          Tired of the unstable Win 7 64bit I got when I purchased my laptop (it was impossible to find a powerful laptop with Win 7 32 bits).  Can I legally downgrade from Win 7 64 bits to Win 7 32 bits? The key/activation will work?    Details: Everybody comments about the stability of Win 64.  Well, I made a mistake. Is not that Win 64 is unstable (it never crashed). It just works very poor. It feels choppy. Compiling a large program (on an old AMD desktop PC) takes 1.5 seconds on Win 32. The same program compiles on Win 64 in 4-5 seconds in this new and "powerful" laptop. All programs are less responsive. How to start Ubuntu in Console mode I want to know if there is a way to switch to console mode from the boot menu.  I have an NVIDIA 1070 and every time there is an update, the GUI stops working. All I need to do is re-install the drivers; however, to do that I need to be in console mode. EDIT: I should have added more information to this question but I'll comment on all answers. In short: I've tried text instead of quiet splash I do see booting activity in text mode; however, I don't get a login screen and ctrl + Alt + F1 doesn't help because GDM keeps trying to restart. Previously I managed to kill GDM through multiple tries but is not working since the latest update. EDIT: After searching for runlevel, I've found  but I think it required me to have a running system to make changes -- please correct me if I'm wrong. Google translate exclude words Let $V$ be spherically symmetric and $W$ i.i.d. Gaussian. Then $E[V|V+W=t]=g(\|t\|)$ A random vector $V \in \mathbb{R}^n$ is said to be spherically symmetric. If for every orthogonal matrix  $A$   the distribution of $AV$ is the same as the distribution of $V$.   My question: Let $W$ be  i.i.d. zero mean Gaussian random variable.     Can we show that  for a spherically symmetric $V$ the conditional expectation  \begin{align} E[V|V+W=t]=   g(\|t\|).  \end{align}  That is the conditional expectation is only a function of the magnitude.   My attempt:  \begin{align} E[V|V+W=t]= \int  V  \frac{f_W(t-v)}{f_{V+W}(t)} dP_V(v)= \frac{\int  v  f_W(t-v) dP_V(v)}{f_{V+W}(t)}= \frac{E[ V  f_W(t-V) ]}{f_{V+W}(t)} \end{align}  Therefore, we can show that the two functions in the above ratio only depend on $\|t\|$.   Observe that we can write $t= \|t\| u_t$  where $u_t$ is a unit vector. Then, for any orthogonal matrix $A$ \begin{align} f_{V+W}(t)=f_{V+W}(\|t\|, u_t)&amp;= \frac{1}{(2\pi)^{n/2}} E[ e^{-\frac{\|t-V  \|^2}{2}} ]\\ &amp;= \frac{1}{(2\pi)^{n/2}} E[ e^{-\frac{\|t-AV  \|^2}{2}} ]\\ &amp;= \frac{1}{(2\pi)^{n/2}} E[ e^{-\frac{\|tA^{-1}-V  \|^2}{2}} ]\\ &amp;= \frac{1}{(2\pi)^{n/2}} E[ e^{-\frac{\|t A^T-V  \|^2}{2}} ]\\ &amp;= f_{V+W}(tA^T)\\ &amp;= f_{V+W}(\|t\|,  u_t A^T) \end{align}  Since, $A$ was arbitrary  we have that $f_{V+W}(t)=h(\|t\|)$(i.e., only depends on the magnitute).  Now I am not sure how to show that $E[ V  f_W(t-V) ]$ only depends on $\|t\|$? Meta Meta Privilege I just took a look at meta today and noticed something weird in the meta privileges. If you look at the meta privilege list, you can see that at . I know that there is no meta for meta, and the link they give you in the privilege page leads you to meta.stackexchange.com, right where you started. Why do they have this privilege? I have a feeling that this is a bad recursive joke. You must be  member of meta to get the privilege of being a member of meta! :) Integral Inequality $|f''(x)/f(x)|$ Let $f$ be a $C^2$ function in $[0,1]$ such that $f(0)=f(1)=0$ and $f(x)\neq 0\,\forall x\in(0,1).$ Prove that  $$\int_0^1 \left|\frac{f{''}(x)}{f(x)}\right|dx\ge4$$ Excel automatic hyperlinks I'm trying to show that if $f$ is a continuous function on $[0,1]$ and $\int_0^{1} f(x)e^{nx}\,{\rm d}x = 0$ for all $n = 0, 1, 2, \dots$, then $f(x) = 0$.  I'd like to use Weierstrass approximation theorem to find a sequence of polynomials $p_m$ that converge uniformly to $f(x)$. Then we could say $\lim\limits_{m\to \infty} \int p_m(x)e^{nx}\,{\rm d}x = \int f(x)e^{nx}\,{\rm d}x = 0$, but I'm struggling to deduce that then all the $p_m$ are zero which would give the result. Are comments safely hidden once the document is compiled? error while loading shared libraries: libssl.so.1.1 (and libcrypto.so.1.1) As you can see I'm missing libcrypto and libssl files. I locked out myself from my server after running apt-update and apt-upgrade, I'm running this from web console, because ssh nor any other program won't work because of missing SSL files.  I tried removing and installing apt install openssl so did libssl-dev and/or compiling own openssl from a distro. None of these create the needed .so files, I ran locate, those files are not present anywhere on the server. There's also libssl1.0.0 which cannot be reinstalled and removing it would cause removal of 700MB of packages that would literally kill server and delete all data. Is it possible to programmatically change connectedApp settings in production? Background: I have a requirement to make it so that an SFDC connected app defaults/is super easy to set to Admin Users are Pre-authorized instead of All users may self-authorize. This setting is important to make this app work and we have historically have had problems with admins not configuring this setting correctly.  Here is a quick history on what I have tried to do to get the isAdminApproved set to "true":   Directly, via xml definition (doesn't get honored) Directly via apex code (doesn't get honored)  (ended up being more pain than what it was worth and was told that it wouldn't work in prod because of security restrictions) Indirectly via a connected-app connected service to make a metadata API call: works in a scratch org but not in prod, which I assume is because it is a managed package. This would occur when an Admin clicks on a button, not from a post-install script.   Details on what I mean by it isn't working in prod: When I try to do a readMetadata call on my connectedApp objects that are in my managed package, the response comes back as [{\n    'fullName': None\n}] instead of with the metadata configuration of my connected app.  Here is a sequence diagram of what I am trying to do in #4 right now:    My question is this: Am I on a fool's errand, where this is impossible to do? ...Or is there a way to set a connected app (in a managed package that I publish) to Admin Users are Pre-authorized without making the admin just do it themselves directly in the settings? How exactly is Arda supposed to be an ancient Earth? As most Tolkien fans know, Arda (Tolkien's world) is supposed to be Earth in its past, and all its stories are supposed to have happened long ago on this very same planet. Of course, this is fiction, so this is not supposed to be actually true.  But, if one wanted to roll with it... how could it possibly make sense from a geographical standpoint? All maps of Middle Earth (before or after the destruction of Beleriand and Numenor) just don't match anything on Earth, even when considering continental drift.  Of course, one could simply say "the Valar reshaped it (again)", but not only this is a bit unsatisfactory, it also doesn't make much sense in context, as Valar are well known for not wanting to meddle with Arda unless absolutely needed, and even then, as little as possible.    Update:  Hey, please calm down, guys. Mine was not a rant at all. And I know perfectly well this is fiction (and I've stated it plainly). The sense of my question was: "If one wanted to follow this line of thought and assume Arda should be related to Earth as Tolkien said, then how could this relationship be explained?". Mods folder missing from .minecraft folder I have Windows 10 and when I go to look in my .minecraft folder in the AppData folder, the mods folder is not there.    How can I find the mods folder? unblock slime presta valve My Son's bike has slime inner tubes.  The bike has been sitting in the garage since last summer and the front tyres valve now seems blocked.  Its has what feels like 15-20 psi in it but I can not get the valve to let any air out or take any more in.  I have no idea at this point how to get it to unblock the valve so I can use it again.  For that matter I don't know how I could get the tire off as it has a reasonable (not ridable) amount of air in it without slashing the tyres (nice kings 2.2 which are pretty new).  I have tried putting the pump on it and putting a bit of air into it to try and clear the blockage but that didn't do anything.  Any ideas please?