Multiple root and common factor
GCD in polynomial rings with coefficients in a field extension
Chrome's default search engine is .  How we can force chrome to use  without https?  I need a solution that allows proxies to rewrite my URLs, but https currently makes this impossible... Which of the following is a correct sentence:   Does anyone know where's Linda?  Does anyone know where Linda is?    Does anyone know why and how to use above usage if #2 is correct. Also please let me know this kind of specific structures. Prove that if the identity is written as the product of $r$ transpositions, then $r$ is an even number Theorem. If the identity is written as the product of $r$ transpositions, $id=τ_1τ_2\dots τ_r$, then $r$ is an even number. Proof. We will employ induction on $r$. A transposition cannot be the identity; hence, $r &gt; 1$. If $r = 2$, then we are done. Suppose that $r &gt; 2$. In this case the product of the last two transpositions, $τ_{r−1}τ_r$, must be one of the following cases: $(ab)(ab) = id\\ (bc)(ab) = (ac)(bc)\\ (cd)(ab) = (ab)(cd)\\ (ac)(ab) = (ab)(bc),$ where $a$, $b$, $c$, and $d$ are distinct. The first equation simply says that a transposition is its own inverse. If this case occurs, delete $τ_{r−1} τ_r$ from the product to obtain $id=τ_1τ_2···τ_{r−3}τ_{r−2}$. By induction $r − 2$ is even; hence, $r$ must be even. In each of the other three cases, we can replace $τ_{r−1} τ_r$ with the right-hand side of the corresponding equation to obtain a new product of $r$ transpositions for the identity. In this new product the last occurrence of $a$ will be in the next-to-the-last transposition. We can continue this process with $τ_{r−2} τ_{r−1}$ to obtain either a product of $r − 2$ transpositions or a new product of $r$ transpositions where the last occurrence of $a$ is in $τ_{r−2}$. If the identity is the product of $r − 2$ transpositions, then again we are done, by our induction hypothesis; otherwise, we will repeat the procedure with $τ_{r−3}τ_{r−2}$. At some point either we will have two adjacent, identical transpositions canceling each other out or $a$ will be shuffled so that it will appear only in the first transposition. However, the latter case cannot occur, because the identity would not fix $a$ in this instance. Therefore, the identity permutation must be the product of $r − 2$ transpositions and, again by our induction hypothesis, we are done. Question: That is the proof written in the book, but unfortunately after more than $10$ times reading I don't understand the last two paragraph (i.e., from &quot;In each of the other three cases&quot;). Is there another simpler proof or could someone please clarify what is going on? Thank you. Difference between "while" and "whilst" Possible Duplicate:          In terms of construction of sentences, can the two words while and whilst be used interchangeably? I am currently reading about the interactions between light and matter, but I keep coming across conflicting explanations.  My initial understanding (using classical electrodynamics) was that light (EM) waves do not actually travel through glass. Instead, they are absorbed by the atoms and/or electrons within the glass, which then emit new EM waves. The new EM waves emitted by the energised particles make up the forward-radiation, which is the light observed from the opposite side of the glass. So it appears as if light is travelling through the glass, but it's actually being absorbed and then re-emitted with (mostly) the same properties.  The article  by Chad Orzel, based on a related SE question, confirms the above:      To understand the propagation of a wave through a medium, you can   think of each component of the medium– atoms, in the case of a glass   block– as being set into motion by the incoming wave, and then acting   as a point source of its own waves.   But an answer to a different but related question,  conflicts with the above:     the electrons in glass are tightly bound to atoms so they are not free   to move like the electrons in a metal and therefore they do not absorb   the photons.   I know the above is referring to the quantum electrodynamics understanding, but I think it still conflicts as it claims that EM waves are not absorbed by the electrons in glass because they are too tightly bound to their atoms.  Also, the  provides a small explanation of the mechanics behind refraction within glass, which kind of conflicts with both of the above:     In the case of dielectrics such as glass, the electric field of the   light acts on the electrons in the material, and the moving electrons   generate fields and become new radiators. The refracted light in the   glass is the combination of the forward radiation of the electrons and   the incident light. The reflected light is the combination of the   backward radiation of all of the electrons.   From my understanding, the above claims that some of the light is absorbed and re-emitted by the electrons (which conflicts with the linked answer), and that the rest of the incident light travels through the glass (makes up the rest of the refracted light) (which I think conflicts with the linked article, claiming the light is absorbed (although the article doesn't technically state whether all of the light is absorbed and re-emitted, so maybe that's where I'm going wrong(?))  I'm sure this is my misunderstanding, so I would be very grateful if you could clear it up for me:  When a light wave is incident on a sheet of glass, do the particles that make up the glass absorb the waves and re-emit them in the forward direction? Or do the light waves manage to travel through the glass, without being absorbed and then exit the glass? Since many users are facing the NullReferenceException: Object reference not set to an instance of an object error in Unity, I thought that it would be a good idea to gather from multiple source some explanation and ways to fix this error.  Symptoms I am getting the error below appearing in my console, what does it mean and how do I fix it?  NullReferenceException: Object reference not set to an instance of an object Current in wire + moving charge next to wire creates magnetic force in the stationary reference frame OR electric force in the moving reference frame from special relativity due to change in charge density etc.... I think I understand this and I think it's super cool. Now here's the rub...  Current in wire + stationary charge next to wire creates no net charge. This is how nature behaves. I get it. My question is why can't I use the same special relativity logic as above, that is, current in wire cause electrons in wire to contract in accordance with special relativity so there has to be a net charge on the wire, which then acts on stationary charge next to wire.  I have been reading and reading and reading and I have come up with the following:  (1) When electrons in wire get accelerated to create current the distance between them actually expands in accordance with special relativity - something to do with bells spaceship paradox - which I am not going to pretend to understand  (2) This expansion from (1) above is exactly opposite and equal in magnitude to the contraction special relativity then causes and the expansion and contraction cancel out to keep the charge density in the wire constant and therefore no net charge on the wire  Here are my questions:  Is the explanation above correct? If so, please elaborate because i dont understand  If not correct, what is going on?  This is driving me absolutely nuts. Connecting MySql with Android without using PHP Tool to create text files from a template I have to regularly create 100+ text files from templates. I currently use an overly complicated shell script. I think there is a smarter way to handle this, but I don't know how.  I have a "database" :  # outputfile      template            data1   data2    data3 first.txt         $template_main      $text1  abcd     1234 second.txt        $template_main      $text2  efgh     5678 third.txt         $template_other     $text1  ij       90   And a configuration file :  template_main=main.txt template_other=other.txt text1=whatever text2=blah   The templates are text files with placeholders like %%data2%% (the placeholder form can be changed).  Does someone know a tool to automate this better than with a complicated shell script ? Interior of closure of open set in topological space If $U$ is an open set in a topological space, is it true that $U$ is the interior of the closure of itself?   If $U$ is open, it must be the interior of itself. But is it the interior of the closure of itself? In the closure, we include all points $x$ such that any open set containing $x$ also contains a point in $U$. In the interior of the closure, we take the union of all open sets contained in the closure. How to replace the_post_thumbnail template tag and show the first inside the post image instead What does  signify when using Vim LatexSuite? I have a LaTeX editing environment that does some assistive stuff such as auto-indenting, closing parentheses, etc.  However, one of the things that it does is unknown to me, and is difficult to search for. If I type an empty set of braces, it replaces them with  {}&lt;++&gt;   If I type two empty sets, for example when I'm setting up a fraction \frac{}{} what I actually get is  \frac\left\{ } \right\}&lt;++&gt;}&lt;++&gt;   Is this a problem? What does &lt;++&gt; do?  Edit: This is using I'm trying to prove  the following claim:     If $f$ is continuous  and $\displaystyle\int_a^\infty |f(x)|\;dx$ is finite then $\lim\limits_{ x \to \infty } f(x)=0$.   Here the counter example of all these sine functions won't do, since  their infinite integral does not converge absolutely.   Any hints for a proof? Can a massless rope accelerate? Projectile Motion - Arrow In a 2D game, I simply want to draw the trajectory of an arrow in flight. With the code below, the trajectory (the parabola) looks right, but the angle (or rotation) or the arrow isn't.   float g = -9.8f; float x = (launchVelocity * time); float y = (launchVelocity * time) + (0.5f * g * (float)Math.Pow(time, 2)); float angle = (float)Math.Tanh(y / x);   What am I missing? Thanks. Python: pop from empty list How to deal with "java.lang.OutOfMemoryError: Java heap space" error? Details on how the Yearling badge works Jawa asked me this  and I didn't know the answer. So I present it to the meta SO community:     [To get the Yearling badge] Do you need to earn the 200 rep within that one year or is it enough to get 200 at some point after being a member for a year? And does 'a year' mean the very first 12 months or the last (sliding window of) 12 months?    I would assume that if you hit 200 reputation after you've already been around for a year, then you would get the Yearling badge. But I don't know. Science fiction short story: beam transporting people into the distant future how to install the DDE to the ubuntu 20.04? I have installed the Ubuntu 20.04, and want now I want to install Deepin Desktop Environment (dde)  My steps are:   add the ppa and install using apt  sudo add-apt-repository ppa:leaeasy/dde sudo apt-get update sudo apt install dde    But during the command sudo add-apt-repository ppa:leaeasy/dde I get the following error:  $ sudo add-apt-repository ppa:leaeasy/dde  You can install deepin desktop environment by run  sudo add-apt-repository ppa:leaeasy/dde sudo apt-get update apt-get install dde ------ Ubuntu Bionic: https://drive.google.com/file/d/1xkQPb4tWocknuYeY-t41W--kz-9SSL0V/view?usp=sharing ------ Important Issue: Q: Can dde on Ubuntu 16.04? A: Sorry, the dde depends on qt version above than 5.6.      If you want to install dde, please upgrade to 17.04 or above.  More info: https://launchpad.net/~leaeasy/+archive/ubuntu/dde Press [ENTER] to continue or Ctrl-c to cancel adding it.  Hit:1 http://in.archive.ubuntu.com/ubuntu focal InRelease Hit:2 http://in.archive.ubuntu.com/ubuntu focal-updates InRelease                                      Hit:3 http://in.archive.ubuntu.com/ubuntu focal-backports InRelease                                    Ign:4 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal InRelease                                      Hit:5 http://security.ubuntu.com/ubuntu focal-security InRelease            Err:6 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release             404  Not Found [IP: 91.189.95.83 80] Reading package lists... Done E: The repository 'http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release' does not have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details.   I get the similar error with sudo apt-get update :  $ sudo apt-get update Hit:1 http://in.archive.ubuntu.com/ubuntu focal InRelease Hit:2 http://in.archive.ubuntu.com/ubuntu focal-updates InRelease                                       Hit:3 http://in.archive.ubuntu.com/ubuntu focal-backports InRelease                                     Ign:4 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal InRelease            Hit:5 http://security.ubuntu.com/ubuntu focal-security InRelease Err:6 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release   404  Not Found [IP: 91.189.95.83 80] Reading package lists... Done E: The repository 'http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release' does not have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details.   How can I install dde ?