I have been trying for several days to prove the equality of the title, but without any positive results. I know I have to first prove that if m divides n (it has to be multiple of m) then $p^{m}-1$ divide $p^{n}-1$.   That is the one that I have been able to demonstrate knowing that there is a quotient for this division that has zero rest and is the following: $\sum_{i=1}^{\alpha}(p^{(\alpha-i)\times m})$.  Someone could help me relate it to the title and get the proof.  Thanks.
How to prove $x^a-1\mid x^b-1 \iff a \mid b$, where $x \ge 2$ and $a,b,x \in \Bbb Z$.  I've tried the following in attempting to solve this:  $$a\mid b \Rightarrow aq=b \Rightarrow x^{aq}=x^b \Rightarrow x^ax^q=x^b$$  Because $x^q \in \Bbb Z$, it follows that $x^a\mid x^b$.  This is as far as I have gotten; any help getting further is appreciated.  Note: It may be that this identity is not true at all?
Tiny white very hard cell bug! Please help to identify! I found these bugs walking on the floor/ wall and on my laptop.  And unfortunately I think they could be in my bed, too, since I bring my laptop to bed sometimes.   I live in a big city in the Netherlands without pets. My house has humidity issues.  I have noticed that the bugs are very tough to exterminate; meaning that when trying to squeeze them it really needs a lot of effort for their size. Their size is about 1 - 2 mm and their color is whitish.    Please note that my lightbulb is having a yellow color.so it makes it look darker than normal. It's more white than looks in the picture          This is a pic of it walking on my screen... When Dirac found his equation for the electron $(-i\gamma^\mu\partial_\mu+m)\psi=0$ he famously discovered that it had negative energy solutions. In order to solve the problem of the stability of the ground state of the electron he invoked Pauli's exclusion principle and postulated that negative energy states well already filled by a "sea" of electrons. This allowed him to predict the positron, viewed as a hole in the sea.  This interpretation was ultimately discarded owing to its inapplicability to bosons and difficulties with explaining the invisibility of the infinite charge of the sea.  According to my understanding, the modern argument goes something like this. There is a discrete symmetry of the lagrangian called charge conjugation $\psi \rightarrow \psi^c$ which allows the negative energy solutions to be interpreted as positive energy solutions for a second mode of excitation of the electron field with opposite charge, called positrons. The decay of electrons to positrons is then suppressed by the $U(1)$ gauge symmetry of the lagrangian forcing conservation of electrical charge.  According to this interpretation What Dirac would have missed was the lagrangian formalism. Is this historically and physically correct? Give a deduction of $\forall y \exists x (x=y)$  My thinking is I can prove $\exists x(x=y)$ then use generalization theorem， which is equivalent to prove $\neg \forall x \neg (x=y)$ but I got stuck proving this... thanks in advance  Sorry that I did not specify what deduction system that I want to use here, it is the deduction sequence  where ai is either Axiom or original hypothesis or obtained from MP Why does the Community user sometimes approve or reject edits? How does a machine know if my edit is valid? What does it mean if my edit was rejected by Community?     See also: What are good books for graduates/undergraduates in Astrophysics? There are no book recommendations for Astrophysics here. I will write my own answer, but I am also interested in what are others' views on the question (I will NOT mark my own answer as the best one). Is the Moonbeam spell amazing, or are we doing it wrong? The description of the moonbeam spell (PHB pg. 261) says:  When a creature enters the spell's area for the first time on a turn or starts its turn there, it is engulfed in ghostly flames that cause searing pain... On each of your turns after you cast this spell, you can use an action to move the beam 60 feet in any direction.  We read this to mean a creature takes damage from moonbeam when any of the following conditions are met:  It begins its turn inside the moonbeam It enters the moonbeam on its turn The creature finds itself inside the beam on another character's turn, no matter how it got there  The creature will not take damage in the following instances:  A creature moves from one part of the beam to another (obviously it had to take damage, but it won't take more damage) A creature is in the beam after having been inside the beam already on that turn.  Following these assumptions, our druid cast a moonbeam and did damage to two enemies, Bob and Frank. On their turn, they took damage again, killing Bob. Frank left the beam. On the druid's next turn, she moved the beam in a circular motion, not exceeding 60 total feet of movement, and in the process the beam came in contact with a dozen enemies and ended on Frank. All dozen enemies received damage, as they were inside the beam at some point during the turn. On Frank's next turn, he again took damage for the fourth time and died. On each of the druid's turns, she moved the moonbeam around like this until the fight ended (it did not exceed the time limit and she made all concentration checks). Is this a legitimate use of moonbeam? It seems awfully powerful for a level 2 spell. Sufficiency to prove the convergence of a sequence using even and odd terms Count total number of files in particular directory with specific extension I want to count the total number of files in particular directory that ends with ".mp4" extension.  I am getting following command:  ls -F |grep -v / | wc -l   It count all the files in particular directory, but I want the count of files that end with .mp4 extension.  Is there any Ubuntu command for that? Linux ACL behaviour: settings the mask changes 'ls' output, but not actual permissions It might be the time of night, but this is puzzling me. Picture the following.  [root@node1 acltest]# getfacl foo/ # file: foo # owner: root # group: testuser user::rwx group::r-x other::---  [root@node1 acltest]# ls -la . total 24 drwxr-xr-x  3 root root     4096 Feb  9 21:53 . drwxr-xr-x 25 root root     4096 Feb  9 21:54 .. drwxr-x---  2 root testuser 4096 Feb  9 21:53 foo [root@node1 acltest]# setfacl -m m::rwx foo [root@node1 acltest]# getfacl foo/ # file: foo # owner: root # group: testuser user::rwx group::r-x mask::rwx other::---  [root@node1 acltest]# ls -la . total 24 drwxr-xr-x   3 root root     4096 Feb  9 21:53 . drwxr-xr-x  25 root root     4096 Feb  9 21:54 .. drwxrwx---+  2 root testuser 4096 Feb  9 21:53 foo [root@node1 acltest]# su - testuser [testuser@node1 ~]$ cd /acltest/foo/ [testuser@node1 foo]$ ls -la . total 16 drwxrwx---+ 2 root testuser 4096 Feb  9 21:53 . drwxr-xr-x  3 root root     4096 Feb  9 21:53 .. [testuser@node1 foo]$ touch bar touch: cannot touch `bar': Permission denied   In words: I create a directory foo, with mode 0750, root as owner and testuser as the group. (testuser is the private group of testuser, but that doesn't matter.)  The getfacl command correctly shows no ACL's on this directory and there is not mask yet. The mask will be set accordingly to the group permissions if I would add a named group or user now.  If I explicitly set the mask to rwx now, the group permissions as shown by ls change too. I know it happens the other way around (the mask changes when the group permissions change), but this seems puzzling.  The more puzzling because the getfacl output does not show the group permissions as rwx but - as said - ls does.  Which one is right? Apparently, the output of getfacl is right, because testuser cannot write to foo. As expected, by the way, because I did not grant the testuser group any permissions to do so.  It goes on. I cannot allow the testuser group write permissions on foo by just using chmod. I have to explicitly set an ACL using setfacl -m g:testuser:rwx foo to allow testuser to finally touch foo/bar.   Can someone explain the rationale behind the difference in the outputs of getfacl and ls? I know it can be tricky to have normal permissions go together with ACL's, but this seems plain wrong. (Though I expect to be missing something glaringly obvious ;))  I have already seen Consider this equation :   $$\sqrt{\left( \frac{dy\cdot u\,dt}{L}\right)^2+(dy)^2}=v\,dt,$$  where $t$ varies from $0$ to $T$ , and $y$ varies from $0$ to $L$.  Now how to proceed ?   This equation arises out of following problem :   A cat sitting in a field suddenly sees a standing dog. To save its life, the cat runs away in a straight line with speed $u$. Without any delay, the dog starts with running with constant speed $v&gt;u$ to catch the cat. Initially, $v$ is perpendicular to $u$ and $L$ is the initial separation between the two. If the dog always changes its direction so that it is always heading directly at the cat, find the time the dog takes to catch the cat in terms of $v, u$ and $L$.  See my solution below :   Let initially dog be at $D$ and cat at $C$ and after time $dt$ they are at $D'$ and $C'$ respectively. Dog velocity is always pointing towards cat.  Let $DA = dy, \;AD' = dx$  Let $CC'=udt,\;DD' = vdt$ as interval is very small so $DD'$ can be taken straight line.  Also we have $\frac{DA}{DC}= \frac{AD'}{ CC'}$ using triangle property.  $\frac{dy}{L}= \frac{dx}{udt}\\ dx = \frac{dy.udt}{L}$  $\sqrt{(dx)^2 + (dy)^2} = DD' = vdt \\ \sqrt{(\frac{dy.udt}{L})^2 + (dy)^2} = vdt $  Here $t$ varies from $0-T$, and $y$ varies from $0-L$. Now how to proceed? I'm going to accessing my ubuntu machine remotely from my windows machine through putty. Are there any suggestions for possible ways to get an X server running on my windows machine?  Thanks. "When people do make changes" - which tense is this? Which tense are sentences like:   "When people do make changes good things happen" "Even if monkeys can play an important role, their actions can pose risk to the general public" Looking for a single word for 'blind worship' Is there a single word for 'blind worship' as in worshiping an actor blindly notwithstanding the bad performance? Assume I have a flight ticket from A->B->C and at A I checked in luggage all the way to C. However after the first flight I don't board the second flight but decide to remain in B.  As I understand procedures my luggage will need to be unloaded again from the plane in B when they notice I did not board the flight.    How and when can I retrieve my luggage at B? (Or if not, what is the fate of my luggage?) Is there any way I can advise airport/airline staff in B to not load my luggage in the first place? (I understand that asking to do so when checking in in A would be declined / met with a fare recalculation)   This question is different from  in that I am looking for a general answer, whereas in that case B happened to be the port of entry to the US where you have to collect your luggage in any case. Also it is different from   in that I do want to check in  luggage from A to B. Finally I am also aware that possible further legs of my trip would be cancelled.  is asking essentially the same but was marked duplicate to the above two questions that do not answer my question. Let $\varphi:G\rightarrow H$ be a group homomorphism from group $G$ to group $H$. Show that, if $H$ is abelian, all subgroups of $G$ that contain $\mathrm{ker} (\varphi)$ are normal in $G$. Should north, south, east and west be capitalised? Should cardinal points be capitalised? Please give a reason why they should or should not. Git is automatically merging changes from a different branch to master I am trying to use the MailManager class in Drupal 8 to send HTML mail. I got it working, but it isn't parsing the HTML markup. I think something is wrong with the headers of the email.  What I have is a custom module (Custom_forward) that contains a form and that calls Mailmanager::mail() in its submitForm() method, which contains the following code.          // Email body         $header = $sender . ' thought you would like to see this page from the website.';         $personal_message = $form_state-&gt;getValues()['message_body'];         $entity_url_value = '&lt;a href="' . $entity_url-&gt;toString() . '"&gt;Go to article&lt;/a&gt;';         $message = $header . '&lt;br&gt;' . $personal_message . '&lt;br&gt;' . $entity_url_value;          // Email         $langcode = 'en';         $mailManager = \Drupal::service('plugin.manager.mail');         $to = $form_state-&gt;getValues()['recipient'];          $params['title'] = 'Forward';         $params['message'] = $message;         $result = $mailManager-&gt;mail('custom_forward', 'forward', $to, $langcode, $params, NULL, true);          if ($result['result'] !== true) {             drupal_set_message(t('There was a problem sending your message and it was not sent.'), 'error');             \Drupal::logger('mail-log')-&gt;error($message);         }         else {             drupal_set_message(t('Your message has been sent.'));         }   According to Drupal you also need the hook_mail method in mytheme.theme to be able to send message, you can see this method here:  function custom_forward_mail($key, &amp;$message, $params) {   $message['body'] = [];   $message['subject'] = [];   $message['headers'] = array(     'content-type' =&gt; 'text/html',     'charset' =&gt; 'UTF-8',     'format' =&gt; 'flowed',     'delsp' =&gt; 'yes',     'from' =&gt; \Drupal::config('system.site')-&gt;get('mail')   );    switch ($key) {     case 'forward':       $message['subject'] = $params['title'];       $message['body'][] = '&lt;html&gt;&lt;body&gt;' . $params['message'] . '&lt;/body&gt;&lt;/html&gt;';       break;   } }   As you can see, I am setting the content type to text/html and that should tell the mail system to parse the content as an HTML, but this is not working for some reasons.  This is a testing email. All the HTML markup is not shown, but also not working.  Bassem thought you would like to see this page from the website. personal message Go to article [1] [1] http://myhost/node/81 Searching through ODT documents without opening them? Let's say I have 2000 .ODT files. Their names are random numbers. How do I go about searching for, let's say, "pricing list"? How do I find the document I need without opening all of them one by one and checking to see if it's the right one?  Is there some program to search through the words in the documents without opening them? US and Canada border crossing and computer privacy: What do I need to know? I've read stories about border officials searching laptops when crossing the border in one direction or the other. Does this really happen, typically speaking?  As a US citizen, can I be forced to give them my laptop's password for any/no reason? I don't believe they are allowed to deny entry to the US to a US citizen? ....but who knows these days?  Canada can deny me entry for any reason, so that's fine. But in practice, does this even happen in either crossing direction?  EDIT - I've now made the trip. They didn't so much as glance in the direction of my laptop. My biggest issue turned out to be lack of international wireless data service!