How does the electron know to "release" the photon at the same angle at which it got absorbed? :     In classical electrodynamics, light is considered as an electromagnetic wave, which is described by Maxwell's equations. Light waves incident on a material induce small oscillations of polarisation in the individual atoms (or oscillation of electrons, in metals), causing each particle to radiate a small secondary wave in all directions, like a dipole antenna. All these waves add up to give specular reflection and refraction, according to the Huygens–Fresnel principle.   My question is, how do all the waves add up to give the direction of angle-in equal to angle-out if they are going in all directions? Also, in reflection does the photon get absorbed and re-emitted or does it act like a dipole antenna?
Explain reflection laws at the atomic level The "equal angles" law of refection on a flat mirror is a macroscopic phenomenon. To put it in anthropomorphic terms, how do individual photons know the orientation of the mirror so as to bounce off in the correct direction?
Number of elements in the quotient ring $\mathbb{Z}[X]/(X^2-3, 2X+4)$ How does bash interpret the equal operator with no surrounding spaces in a conditional? The following script does not behave as I would have expected.  Adding spaces around the '=' in the conditional made it perform how I wanted, but it got me thinking, what is it actually doing inside the conditional?  #!/bin/bash S1='foo' S2='bar' if [ $S1=$S2 ]; then     echo "S1('$S1') is equal to S2('$S2') fi echo $S1 echo $S2   The output is:  S1('foo') is equal to S2('bar') foo bar   The contents of S1 and S2 don't change from what they are assigned, so the = doesn't perform an assignment. Replace environment variables in a file with their actual values? Is there an easy way to substitute/evaluate environment variables in a file? Like let's say I have a file config.xml that contains:  &lt;property&gt;     &lt;name&gt;instanceId&lt;/name&gt;     &lt;value&gt;$INSTANCE_ID&lt;/value&gt; &lt;/property&gt; &lt;property&gt;     &lt;name&gt;rootPath&lt;/name&gt;     &lt;value&gt;/services/$SERVICE_NAME&lt;/value&gt; &lt;/property&gt;   ...etc. I want to replace $INSTANCE_ID in the file with the value of the INSTANCE_ID environment variable, $SERVICE_NAME with the value of the SERVICE_NAME env var. I won't know a priori which environment vars are needed (or rather, I don't want to have to update the script if someone adds a new environment variable to the config file). Thanks! Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$. Let $B$ be a compact subset of a metric space $X$. Let $x \in X \setminus B$. Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.   Is this an existence proof? If yes, do I get to define $U,V$? Why does Earth behave like Natural Bar Magnet? What is the reason for the Earth to behave like a bar magnet and have poles (North and South poles)? Union of uncountable cardinals This is a comic book I read no later than 1993 (so is pre-1993).  And probably after 1988 though I can't be certain.  The protagonist is a huge barbarian with an axe, who seems like an old man, who is defending a series of gigantic walls against an invading army, apparently single-handedly. They keep having to fall back to inner walls. He has to get a massage at one point (not that kind of comic btw).  It's not a super-hero or funny comic, it's more of a serious graphic novel.  That's about it, sorry it's so vague. Showing that localization is an exact functor load a file and read the information I want to open an Excel file through an Apex dialog box, then browse the information in my file line and display it with visualforce. I show you my code  public class ImportExcel {       public String fileName{get; set;}     public Blob contentFile{get;set;}     public String[] csvFileLines{get;set;}     public List&lt;String&gt; acclist{get;set;}     public String[] data{get;set;}     public ImportExcel(){         csvFileLines = new String[]{};         acclist = New List&lt;String&gt;();      } public void readFile(){         try         {             fileName=contentFile.toString();         }catch(Exception e){         ApexPages.Message myMsg = new ApexPages.Message(ApexPages.Severity.FATAL,'Please upload a valid CSV document');         ApexPages.addMessage(myMsg);              return ;          }         csvFileLines=fileName.split('\n');          system.debug(csvFileLines);         for(Integer i=0;i&lt;csvFileLines.size();i++){             data=csvFileLines[i].split(',');             acclist.add(data[i]);         }     } }   Once I execute my code, it puts me this error:     13:52:13:012 VARIABLE_ASSIGNMENT   [15]|e|"common.apex.runtime.impl.ExecutionException: BLOB is not a   valid UTF-8 string"|0x4e4728 How to clear current “read-only” state on a USB flash drive I setup with Rufus? Three weeks ago I used  2.2 to install Windows 10 on my machine, and it has write-protected my flash drive. I’ve tried diskpart:  diskpart att disk clear readonly  clean create partition primary format fs=fat32 quick   And it kinda worked, it removed the image off the USB, but now it’s showing:  Current Read-only State : Yes Read-only  : No Boot Disk  : No Pagefile Disk  : No Hibernation File Disk  : No Crashdump Disk  : No Clustered Disk  : No   And I can’t format it or write on it.    The drive is a a Kingston DataTraveler G4 32GB model, without physical switches. How would I restore it back to normal (rw) state? (complex measures) $d\nu=d\lambda +f\,dm \Rightarrow d|\nu|=d|\lambda| +|f|\,dm$ and $f\in L^1(\nu)\Rightarrow f\in L^1(|\nu|)$ Two simple exercises on complex measures I don't know how to solve, from Folland's Real Analysis. I find it difficult to manipulate the definitions in computations. For context, if $\nu$ is a complex measure on $(X,\mathcal{M})$ then $|\nu|$ is defined as $d|\nu|=|g|\,d\mu$ where positive measure $\mu$ and $g\in L^1(\mu)$ are such that $\nu \ll \mu$ and $\nu = g\,d\mu$ (and in the textbook it's shown that this definition does not depend on the choice of $\mu$).     1) Let $\nu$ be a reguar complex Borel measure on $\mathbb{R}^n$, $m$ denote   Lebesgue measure, and   let $d\nu = d\lambda + f\,dm$ be its Lebesgue-Radon-Nikodym   representation. Then ("it is easily verified")   $d|\nu|=d|\lambda|+|f|\,dm$.      2) If $f\in L^1(\nu)$ then $f\in L^1(|\nu|)$. I was able to prove the   reverse implication, but not this one.   The definition of $|\nu|$ in the case of real signed measures (as $|\nu|=\nu^+ + \nu^-$) makes sense to me but I can't see intuitively into the definion of $|\nu|$ for complex measures as given in the textbook. Do you have any advice on this?  Thank you! What does the number next to the review link in the top bar actually mean? I would like to redirect a URL with uppercase characters http://localhost/A/B to a lowercase version http://localhost/a/b using the .htaccess file. Using regex I can capture A and B but is it possible to convert them to lowercase? When I run the following query in query editor I get this error.  Error     Unknown error parsing query   Query  SELECT Id, OwnerId,Account.OwnerId FROM Contact where OwnerId = Account.OwnerId limit 100   Does anybody know what the problem is? pdfTeX error (font expansion): auto expansion is only possible with scalable how to get the original entity on hook_entity_presave? i actually need it for the content_moderation_state entity   function hook_entity_presave(Drupal\Core\Entity\EntityInterface $entity) { if ($entity-&gt;getEntityTypeId() !== 'content_moderation_state' ) {     return; }     $moderation_state = $entity-&gt;get('moderation_state')-&gt;getString();   //code to get the original entity moderation state I know the new SPFx will be released for Sharepoint 2016 sometime in the year 2017. But are there any plans to also bring this for Sharepoint 2013?  Are there any possibilities/hacks to add a Sharepoint app created with SPFx in the Sharepoint 2013 app catalog? anyone tell how is this effect created which change lips color to orange and little orange on face without brush etc tool how to change the only lips color to orange without brush etc tool and how this is created and little orange on face Adding my launcher to the Unity Launcher automatically. How? There is already a similar post, that has an answer to my question, but it is quite short and I don't get it... So I am trying to show that  \begin{align} \{e_K |~ K \subset G~ conjugacy~ class \} \end{align} is a basis for $Z(\mathbb{Z}[G])$. Serre even writes "one immediately checks that the $e_K$ form a basis", but even reading the other post  I don't see that...  My approach to this is:  Be $x \in Z(\mathbb{Z}[G])$, which is equivalent to: $x = gxg^{-1}$ for all $g \in G$. Writing  $x = \sum\limits_{h \in G} \lambda_h h$, we get $x = \sum\limits_{h \in G} \lambda_h ghg^{-1}$. I don't really know how to go on at this point. I mean, I know that when $a,b \in K$ (same conjugacy class) you get $\sum\limits_{s \in K} \mu_s s = \mu\sum\limits_{s \in K} s$, since \begin{align} a = hbh^{-1} \Rightarrow \mu_a = \mu_b ~\forall a,b\in K \end{align} So all elements in the same conjugacy class have the same scalar.  But from here I don't really know how I can generate the $x = \sum\limits_{h \in G} \lambda_h ghg^{-1}$ with the $e_K$ ...  I would be really thankful, if someone could give me quite a basic proof that even I understand...