Red warning triangle in top bar W:Failed to fetch http://ppa.launchpad.net/kirillshkrogalev/ffmpeg-next/ubuntu/dists/vivid/main/binary-amd64/Packages  404  Not Found W:Failed to fetch http://ppa.launchpad.net/kirillshkrogalev/ffmpeg-next/ubuntu/dists/vivid/main/binary-i386/Packages  404  Not Found E:Some index files failed to download. They have been ignored, or old ones used instead.   This is the error output. What's reason for this and what is a good solution?
How can I fix a 404 Error when using a PPA or updating my package lists? PPA seems to be constantly offline. Whenever I use sudo apt-get update, this error is shown:  W: Failed to fetch http://ppa.launchpad.net/ 404  Not Found   How do I fix these errors?
Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$ I'm having issues figuring out how to approach this problem:     Conclude that if $ad-bc = \pm 1$, then   $$\gcd(x,y) = \gcd(ax +by, cx + dy)$$   The fact that $\gcd(x,y) = \gcd(x+ky, y)$ is a very special case of this exercise   I believe there is a property that if $a = bq_1 + 0$ where 0 is r, then b divides a, and $b=\gcd(a,b)$  Is that at all relevant in this case? What would happen if two wizards made each other a Horcrux? Would they be invincible? How would you kill one of them? About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$ What are your favorite English language tools? To prevent myself from asking an obvious, silly question multiple times: What are the English language tools you found most useful?  I found  extremely useful for looking up .   Please, one tool per answer. Is it now considered acceptable to follow a proper noun ... or any noun ... with a pronoun? E.g.      The President he issued a new executive order. When should I upgrade my camera body? As the question says, when should I upgrade my camera body? In particular, if I have a low- to mid-range body (A DSLR such as a Nikon D3200 or Nikon D5100, or Canon SL1 or T5, or a mirrorless camera such as a Panasonic GH3 or an Olympus E-PL6). How do I know when I need to upgrade?  I have a complement of "kit" lenses, such as an 18-55mm f3.5-5.6 and a 55-200mm f4-5.6. Maybe I've even added a 50mm 1.8 to the kit. Flash is only for indoors (right?) and I usually shoot outside, so that means I don't need an external flash.  I shoot photos of my family and friends whether out at the park or having a picnic, my kids sports practices and games, vacation photos, as well as some landscapes, flowers, and whatever else catches my eye. Pretty familiar territory. I'm not a professional and don't intend to become one. Align equation left Here is my split equation:  \documentclass{article} \usepackage{amsmath} \begin{document} \begin{equation} \begin{split} a_{ijk} =&amp; \frac {Pr(M_{I} =2  \&amp;  M_J=1 \&amp; M_K =1 | I=i , J=j , K=k)}{Pr (M_I =1 \&amp; M_J =1 \&amp; M_K=1 | I=i , J=j)} \\             =&amp; \frac {\mu_{ijk211}}{\mu_{ijk111}}\\ \end{split} \end{equation} \end{document}   I want to align this equation to the left. Is this possible in an equation environment? I am using LaTeX with the article document class. The Bibtex bibliography style that I am using is agsm, part of the Harvard family of bibliography styles.  I have two questions:  (1) If I cite an entry of type @book, then in the reference list I get publisher, address. I would like to re-arrange this order to address: publisher. Which part of agsm.bst should I modify?  (2) I would like to change single quotation marks ' ' to double quotation marks " " for setting off the title of entries of type @article. Which part of agsm.bst should I focus on? I'm currently passing the pid on the command line to the child, but is there a way to do this in the Win32 API?  Alternatively, can someone alleviate my fear that the pid I'm passing might belong to another process after some time if the parent has died? What triggers the benefit of the bard's "Song of Rest" feature? In D&amp;D 5e, the benefit of the Bard's "Song of Rest" feature triggers "if you or any friendly creatures who can hear your performance regain hit points at the end of the the short rest."  Does Song of Rest's benefit apply just when spending hit dice to regain hit points? Or does it apply to any other way that a player might regain hit points at the end of a short rest (e.g. by casting a spell, using a Second Wind, eating a Goodberry, etc.)? How to split algorithm2e over two pages As stated in the question, I am looking for a Noetherian ring over which the Euclidean algorithm is not valid.  I am trying to construct non-trivial examples of Noetherian rings.  Thank you. Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups? I have a graph. This is my code.    \documentclass[10pt]{article}     \usepackage{pgf,tikz}     \usetikzlibrary{arrows,patterns}     \pagestyle{empty}      \definecolor{ttqqcc}{rgb}{0.2,0,0.8}     \definecolor{ffqqtt}{rgb}{1,0,0.2}     \definecolor{ttqqff}{rgb}{0.2,0,1}     \definecolor{uququq}{rgb}{0.25,0.25,0.25}      \begin{document}      \begin{tikzpicture}[line cap=round,line join=round,&gt;=triangle 45,x=1.0cm,y=1.0cm]      \draw[-&gt;] (-3.63,0) -- (3.67,0);     \foreach \x in {-3,-2,-1,1,2,3}       \draw[shift={(\x,0)}] (0pt,2pt) -- (0pt,-2pt)          node[below] {\footnotesize $\x$};     \draw[color=black] (3.48,0.07) node [anchor=south west] { $x$};      \draw[-&gt;] (0,-1.93) -- (0,5.57);     \foreach \y in {-1,1,2,3,4,5}       \draw[shift={(0,\y)}] (2pt,0pt) -- (-2pt,0pt)          node[left] {\footnotesize $\y$};     \draw[color=black] (0.06,5.19) node [anchor=west] { $y$};      \draw (0pt,-10pt) node[right] {\footnotesize $0$};      \clip(-3.63,-1.93) rectangle (3.67,5.57);     \draw[pattern color=ffqqtt,pattern=north east lines,fill opacity=0.1, smooth,samples=50,domain=0:1.0]      plot(\x,{\x^4-4*\x^2+3}) -- (1,0) -- (0,0) -- cycle;     \draw[pattern color=ttqqcc,pattern=crosshatch,fill opacity=0.1, smooth,samples=50,domain=1.0:1.7320508075688772]      plot(\x,{\x^4-4*\x^2+3}) -- (1.73,0) -- (1,0) -- cycle;      \begin{scope}[xscale=-1]     \draw[pattern color=ffqqtt,pattern=north east lines,fill opacity=0.1, smooth,samples=50,domain=0:1.0]        plot(\x,{\x^4-4*\x^2+3}) -- (1,0) -- (0,0) -- cycle;     \draw[pattern color=ttqqcc,pattern=crosshatch,fill opacity=0.1, smooth,samples=50,domain=1.0:1.7320508075688772]        plot(\x,{\x^4-4*\x^2+3}) -- (1.73,0) -- (1,0) -- cycle;     \end{scope}      \draw[smooth,samples=100,domain=-3.6342419080068153:3.665758091993186] plot(\x,{(\x)^4-4*(\x)^2+3});      \fill [color=uququq] (-1.73,0) circle (1.5pt);      \fill [color=uququq] (-1,0) circle (1.5pt);      \fill [color=uququq] (1,0) circle (1.5pt);      \fill [color=uququq] (1.73,0) circle (1.5pt);      \end{tikzpicture}     \end{document}   When I see the DVI output DVI, I have    And if I look at the PDF file, I have Is this defined by the language? Is there a defined maximum? Is it different in different browsers? Does upvoting a duplicate comment do anything? When a question is initially flagged as a duplicate, there is a comment. If the duplicate flag gets enough votes, the question is closed as a duplicate, and the comment is deleted.  Does upvoting the duplicate comment count as a close vote? Does it do anything at all? Just beginning to learn quantum mechanics and my textbook says that all pure eigenstates of the Schrodinger equation ( with a specified potential function) have a zero standard deviation for the expectation values of energy.So every measurement of the energy yields a fixed value. I've also read that pure eigenstates are physically realisable solutions in some situations ( like the infinite potential well). But wouldn't that violate the uncertainty principle? the standard deviation in expectation of energy is zero and hence $$ΔE~Δt=0.$$ Where am I going wrong? Also could someone also give me an insight as to what uncertainty in time even means? I haven't been able to grasp that concept clearly yet. Estimate on the derivative at fixed point Let $D$ be a bounded domain and let $f$ be analytic function from $D$ into $D$. Show that if $z_{0}$ $\in D$ is a fixed point for $f$ , then $|f' (z_{0} )| \leq 1$.  WHAT I WAS THINKING:   To have a conformal mapping from $D$ to the unit disc &amp; after composing it with $f$ ; I was thinking to use Pick's Lemma. So, can anyone tell me any conformal map from any bounded domain to the unit disc?? OR Is there any other way out to solve this problem?? From a previous question, we have that $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[4]{3})$, and I am assuming that we would need to use this to justify the answer to the question.   Would it be right to use some idea of a splitting field here? Or is it something to do with $\mathbb{Q}(\sqrt[3]{3}, \sqrt[4]{3})$ not containing a primitive 3rd root of unity?   This is a past exam paper question worth 8 marks, so it doesn't seem like it's a few lined answer, but I am a little stuck on where to go. Can I use film that's 20+ years old?