taking convex hull does not add extreme points
Extreme Points of a Convex Hull
Does this constitute a valid proof that $\frac{x^2}{1+x^4} \leq \frac{1}{2}$? Why doesn't bash expand this variable when I prefix a command with a "one time variable assignment" If I run this bash command and prefix the statement, so that the variable fruit should exist, but only for the duration of this command:  $ fruit=apple echo $fruit  $   The result is an empty line. why?  To quote a comment from wildcard on :     parameter expansion is done by the shell, and the "fruit" variable is   not a shell variable; it's only an environment variable within the   environment of the "echo" command   An environment variable is still a variable, so surely this should still be available to the echo command? "Washroom", "restroom", "bathroom", "lavatory", "toilet" or "toilet room" I've always been confused by the terms washroom, restroom, bathroom, lavatory, toilet and toilet room. My impression is that Canadians would rather say washroom while Americans would probably say bathroom or Saint John's in the same situation.  I guess the difference here is not only in different kinds of English, but also in whether one is referring to a room in their house or in some public place.  Which do you usually use? Please specify the difference if you use more than two from those six with different meanings, and also where you are from (i.e. what type of English you speak). Raise bounty on question I have this question here:  I chose 500 then before accepting decided that was too much and halfed it, but now I just want the answer so I'd like to raise it to 500 but I can't find how to it. Also it's just such an irritating problem that I'd give all my rep away if I could to the clever person who has the answer. It would be nice to have a few minutes after creating the bounty to be able to raise it. But for now could a mod or something please raise it for me to 500. It is an usual practice that any quantum field theory starts with a suitable Lagrangian density. It has been proved enormously successful. I understand, it automatically ensures valuable symmetries of physics to be preserved. But nevertheless the question about the generality of this approach keeps coming to my mind. My question is how one can be sure that this approach has to be always right and fruitful. Isn't it possible, at least from the mathematical point of view that a future theory of physics does not subscribe to this approach? Is there any performance reason to declare method parameters final in Java?  As in:  public void foo(int bar) { ... }   Versus:  public void foo(final int bar) { ... }   Assuming that bar is only read and never modified in foo(). Why does cancellation of dots $\frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$ work? Let $A= a(i,j)$ and $B= b(i,j)$ be ($n\times n$) matrices that are positive definite such that $a(i,j) &lt; b(i,j)$ .Let $C= c(i,j)= a(i,j) - b(i,j)$, then $C$ is also positive definite. Why or why not?  what i know is $x^T(C)x= x^T (A-B) x = x^T(A)x - x^T(B)x$. Now both terms on right-hand side are greater than $0$ as $A$ and $B$ are positive definite. Also  $a(i,j) &lt; b(i,j)$. But in such case the sign of $x^T(C)x$ being positive or negative will also depend on the value of elements of vector $x$. So, we can't assure that $x^T(C)x &lt; 0$ always or that $C$ is not positive definite.  Am i right about this? Implementing i18n string translation in a custom module? I have a custom module I've built which essentially acts as a repository of custom CTools keywords that users can arbitrarily define. The issue I'm facing is I need to be able to translate these user-defined strings, so I think I need to implement string translation using the i18n_string module.  I have never worked with translation or localization before, so I'm a bit lost with regards to going about how to implement this. I think I'm on the right path with i18n_string though. Can someone experienced with i18n provide examples that I can look at so I can learn the proper way to implement this? Graphics crop in guest welcome window I don't expect that any user should have noticed this, since only guests will see them.  Well, it seems that the Stack Overflow welcome splash, containing the basic info of the site, is cropped from the bottom.    Please don't mind the sides, I cropped them to fit on the screen. Just on the bottom is my concern. Spoilers for the end of the second film      Towards the end of the fight against Ego, we see that the bomb Groot set off blows up Ego's core. Shortly thereafter, it cuts to a scene where Quill is shown apparently losing his powers.   Given that the movie builds up this finale so much, I'm left wondering:      Did Quill lose his powers or merely the means to access them after Ego's apparent death?   Answers from the film or comics are very much welcome. Does Linux have the capability to use "junctioned" symbolic links? I'm not sure if this is an actual term or not, so let me explain the concept.  I have a git repository containing all my configuration dot files in ~/dotfiles. I use symbolic links into this directory in order to "activate" them. For example, by executing ln -s ~/dotfiles/bash/bash_profile ~/.bash_profile to produce the link:  ~/.bash_profile -&gt; ~/dotfiles/bash/bash_profile   However, I find myself in a situation where I want to combine the contents of multiple files. For example, I want the ~/.bash_profile symbolic link to point to two separate files, one for each project. E.g.:  ~/.bash_profile (1) -&gt; ~/dotfiles/bash/bash_profile (2) -&gt; ~/dotfiles/proj/bash_profile   I know I can simply concatenate the two files (e.g., cat ~/dotfiles/{bash,proj}/bash_profile &gt; ~/.bash_profile), but if I can do the same thing with symbolic links, I would prefer to.  I imagine that if such a feature exists (Nix is pretty huge), then under the hood it would have to map the two different files together, hiding all sorts of complexities under the hood (mapping file offsets of all non-first "mapped" files, locking all files when writing to the junctioned symbolic link, etc, etc).  If such a feature doesn't exist, are there are plans to implement it? Short Science Fiction story, told in a series of letters This was a short story told in a series of letters.  The protagonist starts by writing a letter to a utility company about an error in his bill.  By the last letter he is on death row trying to get his sentence commuted.  Each letter is slightly misunderstood.  I think it's from the late 60's and may have been in anthology like "Orbit". If $U$ is unitary operator then spectrum $\sigma(U)$ is contained inside the unit circle In a Hilbert space, let $U$ be a continuous operator which it unitary. Prove $\sigma (U)\subseteq \Bbb{S}^1$.   It is important for me to know how I am doing, and I didn't come by a clear explanation so it would be appreciated to have your correcting and guiding.  Attempt: $U^{*}U=UU^{*}=I$ and therefore $U^{-1}=U$. I have $$ \| U x \|^2 = \langle U x, U x \rangle = \langle U^* U x, x \rangle = \langle x, x \rangle = \|x\|^2 $$ and the same goes for $U^*$ and therefore $\| U \| = \| U^{-1} \| = 1$.   Looking at $U-\lambda$ I can take $$ -\lambda U(U^{-1}-{1\over \lambda}I) =U-\lambda I. $$ Clearly, for $\lambda \ne 0$, $-\lambda U$ is defined and is a linear operator. $(U^{-1}-{1\over \lambda}I)$ is invertible if $|\lambda|\ge 1$ and therefore $\sigma (A^{-1})=\sigma (A)\subseteq \{|z|\le1\}$. But from the RHS, $U-\lambda I$ is invertible if $|\lambda|\le 1$ and therefore $\sigma(A)\subseteq \Bbb{S}^1$. I have noticed my claims tend to have holes in them many times which I cannot spot. If it is substantially true, how can I make sure it is completely formal? How can I make the map::find operation case insensitive? Can Git hook scripts be managed along with the repository? We'd like to make a few basic hook scripts that we can all share -- for things like pre-formatting commit messages.  Git has hook scripts for that that are normally stored under &lt;project&gt;/.git/hooks/.  However, those scripts are not propagated when people do a clone and they are not version controlled.    Is there a good way to help everyone get the right hook scripts?  Can I just make those hook scripts point to version controlled scripts in my repo? Localization of a UFD that is a PID Maximum number of edges in a simple graph? I found that the maximum number of edges in a simple graph is equal to $$\sum\limits_{i=1}^{n-1} i$$  Where $n =$ number of vertices.  For example in a simple graph with $6$ vertices, there can be at most $15$ edges.  If there were any more edges then $2$ edges would connect the same pair of vertices and thus would not be a simple graph.  Is this a known theorem? Hair particles render not same as 3dView(Eevee) I deleted the "/etc/passwd" file and cannot log in I was trying to change my username and also my home directory (/home/username) and my system started to crash. I deleted the passwd file but I had a backup named passwd_bkp. I tried to rename this passwd_bkp as passwd and it did not work. No commands were being executed... I was in a terminal window.   I restarted my system and now I cannot log in. GRUB gives two options: Linux and recovery mode.   I tried to open a sessions as root but it says the filesystem is corrupted. I cannot access my files.  Did I lose all my files?