Is it possible to stack symbols on top of another and how do I make letters vary in size?
Stacking symbols
There is by now a captcha-problem How to find out the terminal command of an application? I want to use some applications shown on the Unity Launcher in a terminal, but I need to know the appropriate command to run them. How can I achieve that? Linux genisoimage Batch Ring Inside an Algebraic Field Extension How to do "Files to store without compression" in 7-zip? Is there any way to selectively prevent certain file types from compression in 7-zip, like in WinRAR?  I'm trying to compress a huge folder (many gigabytes), and I don't want the already-compressed files (e.g. ZIP files) to be re-compressed -- they should be stored without compression. bases and binary I can not figure this out.   I started off by making a table and putting the base(2) on one side, the number in the middle and the remainder in the third row and for 2010 I got 11111011010 but I cant find the next two numbers please help and please explain in detail.  Thank you =)   to express 2010 in base 2, we use exactly 8 ones and 3 zeros. find the next two integers in base 2 that use the same number of ones and zeros Notebook computer battery says: plugged in but not charging with no further information. Is the battery bad already? My 1.5 year old Dell Studio 15 notebook computer has a battery and I keep it attached to the notebook but 98% of the time I just plug in the power adapter.  Lately when I unplugged the power adapter, the battery was only good for 10 minutes, and a few days ago, when I unplugged the power adapter, the computer simply turned off.  When clicking on the battery icon on Win 7, it says "Plugged in, not charging":    is the battery dead?  Or can Windows 7 have any option to not charge the battery?  It is a Dell Studio 15, I wonder if in such case, it looks like it can be a manufacture's defect, would Dell replace the battery? (if there was a recall, then they probably would, i think).  The notebook is 1.5 year old. Check if the application is already installed Is there any way of modifying the preamble, so instead of typing   $\mathtt{x}$ ... \begin{eqnarray}     \mathtt{y} &amp; \mathtt{=} &amp; \mathtt{z^\pi} \end{eqnarray}   I can omit the \mathtt{...} commands? Storing third-party libraries in source control Should libraries that the application relies on be stored in source control? One part of me says it should and another part say's no. It feels wrong to add a 20mb library that dwarfs the entire app just because you rely on a couple of functions from it (albeit rather heavily). Should you just store the jar/dll or maybe even the distributed zip/tar of the project?  What do other people do? If a warlock's invocation states that they can use a spell without a slot does that mean they can cast the spell as often as they want? On page 111 of the 5E hand book the invocations Misty Visions, Mask of Many Faces, Master of Myriad Forms, and Otherworldly Leap all they that you can cast them without expending a spell slot. Does this mean the spells essentially become cantrips and I can cast them as often as I want? Do Legendary Pokemon Respawn After The Elite Four? Why is a vector space equal to its tangent space for any point? I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$.  If $x\in V$, then since $V$ is a manifold, there is a local parametrization $\phi: U\to V$ where $U$ is open in $\mathbb{R}^k$. Without loss of generality, we can require $\phi(0)=x$. Then $T_x(V)$ is defined to be the image of $d\phi_0$ on $\mathbb{R}^k$.   An arbitrary element of $T_x(V)$ looks like $$ d\phi_0(v)=\lim_{t\to 0}\frac{\phi(0+tv)-\phi(0)}{t}=\lim_{t\to 0}\frac{\phi(tv)-x}{t} $$  but this doesn't seem very useful to show $T_x(V)=V$. What is the right approach? Say i have this PHP code:  $FooBar = "a string";   i then need a function like this:  print_var_name($FooBar);   which prints:  FooBar   Any Ideas how to achieve this? Is this even possible in PHP? Properties of p-adic valuations Let $n$ be a integer not equal to zero, $p$ a prime and $$|n|_p = p^{-\operatorname{ord}_p(n)}; |0|_p=0$$  Proposition: $|xy|_p=|x|_p|y|_p$  We want to show: $\operatorname{ord}_p(ab)=\operatorname{ord}_{p}(a)+\operatorname{ord}_p(b)$ So if we suppose there is a largest integral $v$ such that $p^{v}|(ab)$, then we can split up v such that $p^{v_1}|a,p^{v_2}|b$. If we suppose that there exist $v_1,v_2$ with them both being the largest integers such that $p^{v_1}|a, p^{v_2}|b$,  with $v_1+v_2= : v $ then $p^v|ab$. (I don't believe this to be a formal proof but rather my best own "idea" I can come up with...)  The script also mentions that the padic valuations' sums are bounded: $|x+y|_p\le \sup\{|x|_p,|y|_p\}$, how to show this?   Edit:   According to Robert Israel and Bill Dubuques answers, the first proposition is proven, if I understood correctly:   Given $ord(a)= v$ and $ord(b) = u$ which is the same as $a:= p^{v}m, b:= p^{u}n$ with $gcd(m,p)=gcd(n,p)=1$, because $ab=p^{u+v}mn$ with $gcd(mn,p)=1$ it follows that $ord(ab)=ord(a)+ord(b)$ and so $|xy|_{p}=p^{-ord(xy)}=p^{-(ord(x)+ord(y))}=|x|_{p}|y|_{p}$  Now to show: $|x+y|_{p}\le sup\{|x|_{p},|y|_{p} \}$  and the second can be tackled like this: set $a:=p^{v}m. b:=p^{u}n$, then checking two cases $1:(u=v)$ and $2:(v&lt;u)$: $(a+b)=p^{v}m+p^{u}n= p^{v}(m+p^{u-v}n)$ so in case of (u=v) it follows that  $ord(a+b)= u = sup\{ord(a),ord(b)\}$  and in case of $(v&lt;u)$ because the biggest v is $v:= u-1$: $ord(a+b)\le p^{u-1}(m+pn) \le p^{u}(m+n)= sup\{ord(a),ord(b)\}$  It seems that I did something wrong in both cases... When do I use "can" or "could"? When should I use can? When should I use could? What is right under what context? Why not make  but for questions too?  I love that everything is well though behind SO but I can't see no reason as to why great questions can't be prized with "Badges by subject". Prove :(P → Q) ∨ (Q → P) using natural deduction Allowed inference rules: ∨-I, ∨-E, ∧-I, ∧-E, →-I, →-E, ¬-I, ¬-E  I tried to prove a contradiction by assuming $¬ ((P → Q) ∨ (Q → P))$ but got stuck, or am I doing it in the wrong way?  Edit:  My proof attempt  $1.\qquad¬ ((P → Q) ∨ (Q → P))\qquad \qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad Assum$  $2.\qquad P → Q\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\  Assum$  $3.\qquad(P → Q) ∨ (Q → P)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\; ∨-I(2)$  $4.\qquad(¬ ((P → Q) ∨ (Q → P))) ∧ ((P → Q) ∨ (Q → P))\qquad\qquad\qquad\; ∧-I(1,3)$  $5.\qquad  ¬ (P → Q)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\;\; ¬-I(2,4)$  $6.\qquad  Q → P\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\quad\; Assum$  $7.\qquad  (P → Q) ∨ (Q → P)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\;\; ∨-I(6)$  $8.\qquad  (¬ ((P → Q) ∨ (Q → P))) ∧ ((P → Q) ∨ (Q → P))\qquad\qquad\qquad\;\; ∧-I(1,7)$  $9.\qquad  ¬ (Q → P)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\; ¬-I(6,8)$  $10.\qquad ¬ (P → Q) ∧ ¬ (Q → P)\qquad\qquad\qquad\qquad\qquad\qquad\;\qquad\qquad\quad\;\;\, ∧-I(5,9)$  Stuck at this line, can't spot any contradiction to line 1 Similar to  on the bugs in Minecraft, I would like a complete list of what error codes exist in Minecraft and how they occur, as well as how they can be resolved. 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