If $a+b+c+d=0=a^7+b^7+c^7+d^7$, Then prove $a(a+b)(a+c)(a+d)=0$ and all are real numbers.  I am confused and don't even know which to tag for help.
Given $x_1,x_2,x_3,x_4$ real numbers such that $x_1+x_2+x_3+x_4 = 0$ and $x_1^7+x_2^7+x_3^7+x_4^7 = 0,$ how can I use symmetric functions and Newton's sums to prove that $x_1(x_1+x_2)(x_1+x_3)(x_1+x_4)=0$?  This is what I have so far:   Note that $x_1 = -(x_2+x_3+x_4)$ and the relation between Newton's sums, elementary symmetric polynomials, and power sums gives us that $P_7-S_1 P_6+S_2 P_5-S_3 P_4+S_4P_3=0,$ where $P_k$ denotes the $kth$ power sum and $S_k$ denotes the $kth$ elementary symmetric polynomial of $x_1,x_2,x_3,x_4.$   We want to show that $(x_2+x_3+x_4)(x_2+x_3)(x_2+x_4)(x_3+x_4) = 0.$  But how?
How do you make numbered equations in latex? I would like to put in a code that automatically assigns a number to the equations in my documents preferably aligned on the right hand side.   How can I go about this?   \documentclass[a4paper,11pt]{article} %\documentclass[a4paper,11pt]{scrartcl} \usepackage{amsmath}  \usepackage[utf8]{inputenc}  \begin{document}  $$\ddot{\underline{\mathbf{r}}} = \frac{d{^2}\underline{\mathbf{r}}}{dt^2}= 0$$.  \end{document} How to disable autorefresh in snap Is it possible to disable autorefresh in snap? This  says that Snaps are updated automatically in the background to the latest version, every day. Some snaps (e.g. vlc) are built daily, and it makes no sense to install them every day. :     "Food left on plates after eating is   defined as garbage (jutha) because it   has been polluted by the eater’s   saliva. It may be handled in the   family.   Is there an English word for something that is not eatable or usable by others because it has someone's saliva on it? I'm using a flow to set a field to a certain value (01, 02, 03) depending on the value of another field on the same record. I initially wanted to do it with process builder but because of the mapping it seems I would need three 'criteria nodes' or a formula field. Furthermore, it seems Salesforce is recommending using flows instead of process builder for before update actions as it is faster.  The problem with Flows is that I found a way to trigger it when the record is new or updated but not when the specific field is changed so it triggers every time the whole record is updated even when the field stay the same.  On Process Builder, you can use isChanged and then call the flow with specific record but that kinds of defeat the point of using flows for before Update.  So is there a possibility to do the same thing directly in flow ?  Thank you for the help ! I want to load the new entries without reloading the page. I found a great solution for: The ajax script:  &lt;script&gt;     $(document).ready(function() {     var refreshId = setInterval(function()     {         $('#content').fadeOut("fast").load('http://neocsatblog.mblx.hu/new.php').fadeIn("fast");     }, 10000);      }); &lt;/script&gt;   And the php:  &lt;?php require_once("wp-blog-header.php"); ?&gt; &lt;div id="content" &lt;?php cyberchimps_filter_content_class(); ?&gt;&gt;          &lt;?php do_action( 'cyberchimps_before_content'); ?&gt;          &lt;?php if ( have_posts() ) : ?&gt;              &lt;?php while ( have_posts() ) : the_post(); ?&gt;                  &lt;?php get_template_part( 'content', get_post_format() ); ?&gt;              &lt;?php endwhile; ?&gt;          &lt;?php elseif ( current_user_can( 'edit_posts' ) ) : ?&gt;              &lt;?php get_template_part( 'no-results', 'index' ); ?&gt;          &lt;?php endif; ?&gt;          &lt;?php do_action( 'cyberchimps_after_content'); ?&gt;      &lt;/div&gt;   However, this is a special feature that works perfectly here:  However, the main page refuses to load new entries, I do not understand the reason for this, as we use everything the same. Is there a published way to take text that may have accent marks and remove the accent marks?  For example:  System.assertEquals('Hello', transform('Ḧéļḻṏ'));   I believe there is an equivalent method in Java, C# and JavaScript. I can't seem to find the Apex equivalent and I'm disturbed that I might have to do it on my own. How can I optimise a Minecraft-esque voxel world? I found Minecraft's marvelous large worlds extremely slow to navigate, even with a quad core and meaty graphics card.  I assume Minecraft's slowness comes from:   Java, as spatial partitioning and memory management are faster in native C++. Weak world partitioning.   I could be wrong on both assumptions. However, this got me thinking about the best way to manage large voxel worlds. As it is a true 3D world, where a block can exist in any part of the world, it is basically a big 3D array [x][y][z], where each block in the world has a type (i.e BlockType.Empty = 0, BlockType.Dirt = 1, etc.)  I assume that to make this sort of world perform well, you would need to:   Use a tree of some variety (//) to split all the cubes out; it seems like an oct/kd would be the better option, as you can just partition on a per cube level not a per triangle level. Use some algorithm to work out which blocks can currently be seen, as blocks closer to the user could obfuscate the blocks behind, making it pointless to render them. Keep the block object themselves lightweight, so it is quick to add and remove them from the trees.   I guess there is no right answer to this, but I would be interested to see peoples' opinions on the subject.  How would you improve performance in a large voxel-based world? Counting invertible $0$-$1$ matrices that remain invertible under negating the diagonal There is an invertible $n×n$ matrix with each entry $0$ or $1$. Now we negate ($x\to 1-x$) the diagonal of the matrix, and it is still invertible. How many such matrices exist for a given $n$?  Can we derive a general formula? If $n=2$ there are $2$ such matrices, if $n=3$ there are $48$. But I have no idea whether a formula exists for higher $n$. Castellation metal loop connector How to create a filter in Gmail to identify messages forwarded from Zimbra? We need a profanity filter I searched SE.Meta and was unable to find such question.  Someone just swore at us in a  as an  (warning: profanity, may be NSFW), sure, Arqade (profanity in phrases in games) or some SE (ELU, ELL or Anime.SE maybe?) can sometimes contain profanity as a small parts of question.  I think that a part should be added to the review system that will automatically hide questions/answers with profanity, just like the close vote system, the question/answer will be on hold and will be voted to be shown or be deleted.  Tip, if this gets implemented, u can be typed as \/ or v, and most other can be written as leet (l = 1 etc). Maybe it can be called "suspicious activity", and can be triggered when there is too many number or Unicode characters. (I can write hi as |-| |, |-|I, |-|l, l-li, hî, 1-1i, H1 and many more) Trying to put a footnote in subsection title:  \subsection{Effect of Lateral Confinement on SW Band Structure\footnote{Parts of this sub--section are based on \citet{Venkat2013}}}   produces the following error:  Argument of \@sect has an extra } ...section are based on \citet{Venkat2013}}}   I am using:   \documentclass[paper=A4, fontsize=12, toc=listof, toc=index, toc=bib]{scrreprt}   Please advise. How to test for equality of means for proportions? And two-sided or one-sided? I have the following problem: two groups A and B.  The proportions of married and single are ($p_1$) and ($p_2$) respectively, and the standard deviations ($s_1$) and ($s_2$). The sample consists of  ($n_1$) A and  ($n_2$) B observations.  Now I want to test whether the proportions are equal. I use the following formula to compute the z-statistic:  z = $abs(p_1 - p_2) / \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} }$  and test the hypothesis: h0: $p_1 \neq p_2$  An alternative approach I tried is to estimate the t-value based on the standard deviations (hence just comparing means), that is by the formula (assuming unequal variances):  t = $abs(p_1-p_2)/\sqrt{ s_1^2/n_1 + s_2^2/n_2 }$  My question are:    do you agree with my approach or; should it be two-sided or one-sided?   Thank you. Root partition size remains same after partition resize with gparted I had resized my root partition from 30gb to 70gb using gparted. But Ubuntu still shows that I have the same old disk size while gparted shows that I acually have 70gb and that the other 40gb is somehow used. How can I fix this?  Output of 'df -h':  Filesystem      Size  Used Avail Use% Mounted on /dev/sda5        30G   16G   13G  55% / none            4.0K     0  4.0K   0% /sys/fs/cgroup udev            1.9G  8.0K  1.9G   1% /dev tmpfs           391M  884K  390M   1% /run none            5.0M     0  5.0M   0% /run/lock none            2.0G  284K  2.0G   1% /run/shm none            100M   48K  100M   1% /run/user /dev/sda6       341G  115G  227G  34% /media/Data /dev/sdb1       932G  459G  474G  50% /media/sidharth/passport   Output of mount:  /dev/sda5 on / type ext4 (rw,errors=remount-ro) proc on /proc type proc (rw,noexec,nosuid,nodev) sysfs on /sys type sysfs (rw,noexec,nosuid,nodev) none on /sys/fs/cgroup type tmpfs (rw) none on /sys/fs/fuse/connections type fusectl (rw) none on /sys/kernel/debug type debugfs (rw) none on /sys/kernel/security type securityfs (rw) udev on /dev type devtmpfs (rw,mode=0755) devpts on /dev/pts type devpts (rw,noexec,nosuid,gid=5,mode=0620) tmpfs on /run type tmpfs (rw,noexec,nosuid,size=10%,mode=0755) none on /run/lock type tmpfs (rw,noexec,nosuid,nodev,size=5242880) none on /run/shm type tmpfs (rw,nosuid,nodev) none on /run/user type tmpfs (rw,noexec,nosuid,nodev,size=104857600,mode=0755) /dev/sda6 on /media/Data type fuseblk (rw,nosuid,nodev,allow_other,default_permissions,blksize=4096) binfmt_misc on /proc/sys/fs/binfmt_misc type binfmt_misc (rw,noexec,nosuid,nodev) gvfsd-fuse on /run/user/sidharth/gvfs type fuse.gvfsd-fuse (rw,nosuid,nodev,user=sidharth) /dev/sdb1 on /media/sidharth/passport type fuseblk (rw,nosuid,nodev,allow_other,blksize=4096) I am trying to find the splitting field of $x^6+x^3+1$ over $\mathbb{Q}$.  Finding the roots of the polynomial is easy (substituting $x^3=t$ , finding the two roots of the polynomial in $t$ and then taking a 3-rd root from each one). The roots can be seen  [if there is a more elegant way of finding the roots it will be nice to hear]  Is is true the that the splitting field is $\mathbb{Q}((-1)^\frac{1}{9})$ ? I think so from the way the roots look, but I am unsure.  Also, I am having trouble finding the minimal polynomial of $(-1)^\frac{1}{9}$, it seems that it would be a polynomial of degree 9, but of course the degree can't be more than 6...can someone please help with this ? Compactness of  Multiplication Operator on $L^2$ Suppose we have an bounded linear operator A that operates from $L^2([a,b]) \mapsto L^2([a,b])$.  Now suppose that $A(f)(t) = tf(t)$.    Is A compact?  Edit: I know $A = A^*$ but I'm not really sure how to start on this.  It's not homework, just summer fun :D I can't understand where the final keyword is really handy when it is used on method parameters.    If we exclude the usage of anonymous classes, readability and intent declaration then it seems almost worthless to me.   Enforcing that some data remains constant is not as strong as it seems.    If the parameter is a primitive then it will have no effect since the parameter is passed to the method as a value and changing it will have no effect outside the scope.  If we are passing a parameter by reference, then the reference itself is a local variable and if the reference is changed from within the method, that would not have any effect from outside of the method scope.    Consider the simple test example below. This test passes although the method changed the value of the reference given to it, it has no effect.  public void testNullify() {     Collection&lt;Integer&gt; c  = new ArrayList&lt;Integer&gt;();           nullify(c);     assertNotNull(c);            final Collection&lt;Integer&gt; c1 = c;     assertTrue(c1.equals(c));     change(c);     assertTrue(c1.equals(c)); }  private void change(Collection&lt;Integer&gt; c) {     c = new ArrayList&lt;Integer&gt;(); }  public void nullify(Collection&lt;?&gt; t) {     t = null; } How do you write (and run) a correct micro-benchmark in Java?  I'm looking for some code samples and comments illustrating various things to think about.  Example: Should the benchmark measure time/iteration or iterations/time, and why?  Related: Subgroup of $\mathbb{Q}$ with finite index Consider the group $\mathbb{Q}$ under addition of rational numbers. If $H$ is a subgroup of $\mathbb{Q}$ with finite index, then $H = \mathbb{Q}$.  I just saw this on our exam earlier and was stumped on how to show this. Any ideas? Find the limit $ \lim_{x \to \infty} \frac{e^{x}}{(1+\frac{1}{x})^{x^{2}}} $ Find the limit $$ \lim_{x \to \infty} \frac{e^{x}}{(1+\frac{1}{x})^{x^{2}}}. $$  I know that the limit is of indeterminate form type $\frac{\infty}{\infty}$, but it seems using L'Hopital's rule directly does not help here.  Do I somehow use the fact that $\lim_{x \to \infty} (1+\frac{1}{x})^{x}=e$?