I'm writing some formulas inside a tabular. My formulas have summation but the subscript is showing next to the summation not under it! How can I make it under it?  Example (j is next to summation!):
I have a fraction with sums above and under the line. How can I convince LaTeX to write the indices of the sums under the sigma instead of next to it?  \begin{displaymath}   \frac{\sum_{s \in S} s^2}{\sum_{p \in P} p^2} \end{displaymath}
Can all circles of radius $1/n$ be packed in a unit disk, excluding the circle of radius $1/1$? This problem occurred to me when I came across a similar problem where the radii were taken over only the primes. That question was unanswered, but it seems to me infinitely many circles of radius $1/2, 1/3, 1/4...$ can fit into a unit disk. The area of all those circles would be $\pi \sum_2^\infty 1/n^2 = \pi^3/6 -\pi$, which is less than the area of the unit disk $\pi$. But can the circles actually be packed with no overlaps? In Oblivion, I had two favorite money-makers:   Dump all my stuff except some basic armor and potions, and raid an oblivion tower. The armor and weapons was worth lots, and selling it in Anvil made a lot of cash fast. Harvest an entire farm's worth of crops, create 'restore fatigue' potions and sell them.   Skyrim seems to spoil this:   Merchants have a max-amount per day/QOD/... instead of per item You can't carry as much (or maybe I have yet to discover the strength/feather spells) Alchemy is a lot harder (most common items go into food recipes now, which don't sell for those prices) On top of that, once a dungeon is cleared, it doesn't respawn - so no 'farming' familiar dungeons?   What are some easy/repeatable money making methods in Skyrim? Which are most efficient effortwise/real-life-time wise? Brackets in if condition: why am I getting syntax errors without whitespace? I am using the below script to move two days back when script runs at starting two days of the year and also check first and second days of every month and move two days back. if [$month=&quot;01&quot;] &amp;&amp; [$day=&quot;01&quot;]; then         date=&quot;$last_month/$yes_day/$last_year&quot;         fulldate=&quot;$last_month/$yes_day/$last_year&quot; else         if [$month=&quot;01&quot;] &amp;&amp; [$day=&quot;02&quot;]; then                 date=&quot;$last_month/$yes_day/$last_year&quot;                 fulldate=&quot;$last_month/$yes_day/$last_year&quot;         else                 if [ $day = &quot;01&quot; ]; then                         date=&quot;$last_month/$yes_day/$year&quot;                         fulldate=&quot;$year$last_month$yes_day&quot;                 else                         if [ $day = &quot;02&quot; ]; then                                 date=&quot;$last_month/$yes_day/$year&quot;                                 fulldate=&quot;$year$last_month$yes_day&quot;                         else                                 date=&quot;$month/$yes_day/$year&quot;                                 fulldate=&quot;$year$month$yes_day&quot;                         fi                 fi         fi fi  But my bad am getting the below error message Etime_script.sh: line 19: [06=01]: command not found Etime_script.sh: line 24: [06=01]: command not found Small snap loop devices visible in gnome-disk-utility OR what is the function of snap ubuntu-core I just realized using gnome-disk-utility (3.18.3.1) that my 16.04 installation is featuring several small loop devices.  In detail:   78 MB Loop Device (/var/lib/snapd/sn_ntu-core_352.snap -> /dev/loop0) 76 MB Loop Device (/var/lib/snapd/sn_ntu-core_216.snap -> /dev/loop1) 79 MB Loop Device (/var/lib/snapd/sn_ntu-core_423.snap -> /dev/loop2) 705 KB Loop Device (/var/lib/snapd/snaps/tpad_12.snap -> /dev/loop3) 684 KB Loop Device (/var/lib/snapd/snaps/tpad_11.snap -> /dev/loop5) 705 KB Loop Device (/var/lib/snapd/snaps/tpad_13.snap -> /dev/loop6)   Image showing gnome-disk-utility    My questions are as follows:   What are those loop devices used for? Is that an expected behaviour in 16.04? Can  I remove them? And if so - how (just by unmounting)? Is there any risk in removing the 2 installed snaps mentioned below (tpad and ubuntu-core)   UPDATE  Installed snaps  According to snap list I do have   tpad (1.8.1) ubuntu-core (16.04.1)   installed. I remember I did tinker with snaps for a short time when  I realized this option - but am pretty unsure if  I  installed those 2 snaps - or if they are pre-/auto-installed.  tpad is a Terminal text editor with GUI-like user interface  I  did install manually, as  I don't use it  I removed it now via sudo snap remove tpad. As a result the 2 loop devices pointing to tpad are gone in gnome-disk-utility.  So - at the current point  I am still having 1 snap installed called ubuntu-core and I don't know what it is used for. The summary of this particular snap is   The ubuntu-core OS snap How do "pet" Walkers help ward off other Walkers? In both the comics and the show, Michonne first appears with two walkers that she essentially keeps as pets.    The comics sort of half-explain this - as per :     She notices walkers don't attack one another and hacks off the arms and lower jaws of Mike and Terry. She chains them up by their necks to use them as escorts for an escape that was successful.   In , we see that even if they still sort of smell like Walkers, the Walkers can smell/detect the living and will attack.  In ,      We see Andrea effectively making a new "pet" so that she can use to deter the Walkers that she comes across on her trip to the prison. Along the trip, she is shown to come within close range of Walkers, but they give her space when she points the "pets" in the right direction.   Why would the Walkers not just attempt to go around the "pets"? What about them creates an effective protective barrier? What is the most accurate way to retrieve a user's correct IP address in PHP? Is there a way to tell Nautilus not to generate previews of PDF files? I tend to download lots of articles as PDFs, and I find that when I browse a directory of these Nautilus takes 1 minute to load the directory. I assume that's because it's generating previews of their front pages. That feature is useless to me, alas.  I'd like to tell it not to do that. In the Avengers movie, was Thor going to kill Captain America during his woods battle with Iron Man? Do Legendary Pokemon Respawn After The Elite Four? What ensures big holes in my bread? Create an interface which is similar to the material list box Mismatch Reputation in the thumbnail and the actual Reputation Actually this was quite surprising. A casual troll in the Users Page revealed few users with negative vote. One example is here    Then trying to see what could have causes such a low reputation, I peeked inside the Profile Page of this particular User and was surprised to see a decent positive score.    Can someone explain to me how this can be possible? If $a^\phi = 1$ then the order of $a$ divides $\phi$ Finding center and radius of circumcircle Find the center and radius of the circle which circumscribes the triangle with (complex) vertices $a_1,a_2,a_3$. Express the result in symmetric form.   I'm not sure where to start in this question. The circumcenter is the intersection of the perpendicular bisectors, but I don't know how to calculate the line equation of the perpendicular bisectors yet. As for the radius, after we find the circumcenter $c$ we can calculate it using $|c-a_1|$ (or $|c-a_2|$ or $|c-a_3|$; all three should be equal.) So how can I calculate the circumcenter? What is the rule of thumb for mixing doughs? Without delving too deeply into specialty doughs, but addressing more than just  and , what concerns can I evaluate when calculating the technique for approaching a dough?  For instance, some recipes say not to mix too much, others would imply that beating every bit of air out of the dough is the best thing for it. What I am looking for is a set of criteria, maybe a flowchart, that outlines how to deal with having a well-reasoned technique (as opposed to just following a recipe).  Crumble, cream, or melt the fat first? Paddle or hook? Beat, fold or whisk? Knead once and rest or knead three times and throw? I would assume that to some extent getting air in or out, activating and mixing ingredients, distributing fats more or less evenly, and so forth play a factor in any decision. However, I have no idea what those different things do, or the food science behind them. What general principles would help me understand what each factor does, so that I can have more confidence when staring down the ingredients, whether for some cookies or pizza crusts or muffins? I have installed some apps through the Ubuntu Software center. How can I find the corresponding terminal command? For many apps, this is not the same as the displayed name. For example, the app called Text Editor can also be launched by typing "gedit" in the terminal, but how can I find this for other apps? bash if not multiple conditions without subshell? I want to combine multiple conditions in a shell if statement, and negate the combination.  I have the following working code for a simple combination of conditions:  if [ -f file1 ] &amp;&amp; [ -f file2 ] &amp;&amp; [ -f file3 ] ; then   # do stuff with the files fi   This works fine.  If I want to negate it, I can use the following working code:  if ! ( [ -f file1 ] &amp;&amp; [ -f file2 ] &amp;&amp; [ -f file3 ] ) ; then   echo "Error: You done goofed."   exit 1 fi # do stuff with the files   This also works as expected.  However, it occurs to me that I don't know what the parentheses are actually doing there.  I want to use them just for grouping, but is it actually spawning a subshell?  (How can I tell?)  If so, is there a way to group the conditions without spawning a subshell? Can you cast a spell at a higher level, even if it doesn't have an "At higher levels" entry? Evaluating $\int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x$ I was generalizing the following integral: $$\int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x \hspace{40pt} n\geq 0$$ We can start by noting that $\sin x =\dfrac{e^{ix}-e^{-ix}}{2i}$ and thus $\displaystyle \sin^{2n+1}x=\frac{(-1)^n}{2^{2n}}\sum_{r=0}^n (-1)^r \binom{2n+1}{r}\sin(2r+1)x$. This implies $$\begin{aligned}\displaystyle \int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x &amp;=\frac{(-1)^n}{2^{2n}}\sum_{r=0}^n (-1)^r \binom{2n+1}{r}\int_{0}^{\infty}\frac{\sin(2r+1)x}{x}\mathrm{d}x \\ &amp;=\frac{(-1)^n\pi}{2^{2n+1}}\sum_{r=0}^n(-1)^r\binom{2n+1}{r} \\ &amp;=\frac{(-1)^n\pi}{2^{2n+1}}\sum_{r=0}^n\left((-1)^r\binom{2n}{r}-(-1)^{r-1}\binom{2n}{r-1}\right)\end{aligned}$$ Where I have used the well known  $\displaystyle \int_{0}^{\infty} \frac{\sin(2r+1)x}{x}\mathrm{d}x=\int_{0}^{\infty}\frac{\sin x}{x}\mathrm{d}x=\frac{\pi}{2}$ and the property of binomial coefficients that $\displaystyle \binom{n}{r}+\binom{n}{r-1}=\binom{n+1}{r}$ Since the above sum telescopes, we have $$\displaystyle \int_{0}^{\infty}\frac{\sin^{2n+1}x}{x}\mathrm{d}x=\frac{(-1)^n\pi}{2^{2n+1}}(-1)^n\binom{2n}{n}=\frac{\pi}{2^{2n+1}}\binom{2n}{n} ~\forall ~ n\in \mathbb{Z^{+}}$$ I would like to know other methods for evaluating this integral. Numerical partial derivative