Source for if $L_A:(\mathbb{R}^n, \left\Vert \cdot \right\Vert_p) \to (\mathbb{R}^n, \left\Vert \cdot \right\Vert_q)$ is an isometry, then $p = q$.
$(\mathbb{R}^n,\|\cdot\|_{p})$ is isometrically isomorphic to $(\mathbb{R}^n,\|\cdot\|_{q})$ iff $p=q,$
How can I get multiple stacked plots (ybar stacked, stack plots=y) in one diagram, such that there are multiple stacked bars for the same x-coordinate, each slightly shifted? Is there a way to reset pgfplot's zero levels after an \addplot command, such that the next \addplot[xhshift=...] doesn't add something to the current stack, but rather starts a fresh stack?  FWIW, overlaying two plots with scale only axis won't help; I need this to work with groupplot Supernatural sites on Stack Exchange Is there any Q&amp;A site under the Stack Exchange network where we can discuss about supernatural incidents? If not, then whom should I petition to create it? We all know you can't do the following because of ConcurrentModificationException:  for (Object i : l) {     if (condition(i)) {         l.remove(i);     } }   But this apparently works sometimes, but not always. Here's some specific code:  public static void main(String[] args) {     Collection&lt;Integer&gt; l = new ArrayList&lt;&gt;();      for (int i = 0; i &lt; 10; ++i) {         l.add(4);         l.add(5);         l.add(6);     }      for (int i : l) {         if (i == 5) {             l.remove(i);         }     }      System.out.println(l); }   This, of course, results in:  Exception in thread "main" java.util.ConcurrentModificationException   Even though multiple threads aren't doing it. Anyway.  What's the best solution to this problem? How can I remove an item from the collection in a loop without throwing this exception?  I'm also using an arbitrary Collection here, not necessarily an ArrayList, so you can't rely on get. how to fill intersection of 2 shapes in latex tikz package I have to fill a specific region as given in the following figure.  The code I have written is   \begin{tikzpicture}  \draw (0,0) rectangle (10,10);  \draw (3.5,3.5) circle (5) node {A};  \end{tikzpicture}   Now, how to fill the intersecting area (area of circle overlapping the square) How to find the kth largest element in an unsorted array of length n in O(n)? I believe there's a way to find the kth largest element in an unsorted array of length n in O(n).  Or perhaps it's "expected" O(n) or something.  How can we do this? I have seen many programs consisting of structures like the one below  typedef struct  {     int i;     char k; } elem;  elem user;   Why is it needed so often? Any specific reason or applicable area? In an hydrogen-like atoms the orbitals are solutions to the Schrodinger equation suitable for the problem.  They describe the regions where an electron can be found.  So, why don't they have spherical symmetric?  As in this picture:    Why, in a certain distance from the nucleus, are there points where an electron can be found, and points where it can't?  EDIT: Is there any intuation about the symmetry breaking? When is it better to use String.Format vs string concatenation? Floor a date in SQL server In SQL Server, how do I "floor" a DATETIME to the second/minute/hour/day/year?  Let's say that I have a date of 2008-09-17 12:56:53.430, then the output of flooring should be:   Year:  2008-01-01 00:00:00.000 Month: 2008-09-01 00:00:00.000 Day: 2008-09-17 00:00:00.000 Hour: 2008-09-17 12:00:00.000 Minute: 2008-09-17 12:56:00.000 Second: 2008-09-17 12:56:53.000 Closed-form of $\int_0^1 \operatorname{Li}_p(x) \, dx$ Did other meta-humans enter a coma? In the first episode of The Flash (2014), Barry's struck by lightning, some chemicals fall on him, and he enters a coma for 9 months.  This happens when the S.T.A.R. Labs particle accelerator is turned on for the first time.  9 months later, he wakes up and has powers.  He then starts fighting crime, especially other meta-humans who were created by the same particle accelerator accident.  However, this is all 9 months after the accident, where were the other meta-humans during those 9 months?  The S.T.A.R. Labs people make it sound like they hadn't seen any meta-humans active during the 9 months Barry was out.  Did all people granted powers by the particle accelerator accident also fall into comas?  Why weren't they active heroes/villains during the 9 month gap? Recording claimable expenses in GnuCash Why is it that a sequence in a closed, bounded interval [a, b] is a bounded sequence? What is the formal proof of this proposition from the definition of a bounded sequence? Since reading the answers to my question about  I've been good about giving my meat time to defrost in the refrigerator. However, my plans for tonight have changed and I'd like to defrost some ground beef that is completely frozen for dinner this evening. What are my options to do so quickly and safely? We are a team with three orgs. Development, UAT and Production.  We are trying to move away from changesets and manual copying and use various deploy mechanisms or scratch orgs but we cannot deploy org wide email addresses since it is not supported in the metadata api.   The problem is that since addresses cannot be deployed, workflows referencing them cannot either. And if we dont deploy all workflows some tests will fail and the environment will be wrong.  I would appreciate any help with this issue, thanks Finding the closest Geometry in PostGIS How do I remove "permission denied" printout statements from the find program? Code   find / -name netcdf   Output  find: `/root/.dbus': Permission denied find: `/root/.gconf': Permission denied find: `/root/.gconfd': Permission denied find: `/root/.gnome': Permission denied find: `/root/.gnome2': Permission denied find: `/root/.gnome2_private': Permission denied Let $f(z)=\sum\limits_{n=0}^{\infty}a_n z^n$ be an analytic function in $|z|\leq R$ such that $|f(z)|\leq M$. Prove an inequality about $f$'s zero What are some typical LuaTeX gotchas? I'm using using algorithm2e package with the following settings:  \usepackage[ruled,vlined,linesnumbered,noresetcount]{algorithm2e}   But for some subset of algorithms in the same document, I want to use this setting:  \usepackage[boxed]{algorithm2e}   Is there a way to override the default setting just for a subset?  I found a similar question:  Here newenvironment is used to override the default setting of an option. But I want to override the option itself.  I wish there was a way to load a package several times with different options for each instance.