When i run system update then i got following error:  W:Failed to fetch cdrom://Ubuntu 12.04.3 LTS _Precise Pangolin_ - Release amd64      (20130820.1)/dists/precise/restricted/binary-i386/Packages  Please use apt-cdrom to make this CD-ROM recognized by APT. apt-get update cannot be used to add new CD-ROMs  W:Failed to fetch cdrom://Ubuntu 12.04.3 LTS _Precise Pangolin_ - Release amd64 (20130820.1)/dists/precise/main/binary-amd64/Packages  Please use apt-cdrom to make this CD-ROM recognized by APT. apt-get update cannot be used to add new CD-ROMs  W:Failed to fetch cdrom://Ubuntu 12.04.3 LTS _Precise Pangolin_ - Release amd64 (20130820.1)/dists/precise/restricted/binary-amd64/Packages  Please use apt-cdrom to make this CD-ROM recognized by APT. apt-get update cannot be used to add new CD-ROMs  W:Failed to fetch cdrom://Ubuntu 12.04.3 LTS _Precise Pangolin_ - Release amd64 (20130820.1)/dists/precise/main/binary-i386/Packages  Please use apt-cdrom to make this CD-ROM recognized by APT. apt-get update cannot be used to add new CD-ROMs  E:Some index files failed to download. They have been ignored, or old ones used instead.   please help me on this.
I have recently installed ubuntu 11.10. But when i try to install a software it tries to 'updating cache' but after a while, it stops, showing:                            Failed to download repository information                           Check your internet connection   so, now i am totally unable to install any software in my ubuntu.  Please,let me know how to solve the problem.
Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$ So I am trying to prove the above stated formula. I am completely new to number theory so I thank you in advance for baring with me. I start off by using the definition:  $\frac{\phi(n)}{n} = \sum_{d|n} \frac{\mu(d)}{d}$  Where $\phi$ is Eulers function, and $\mu$ is the Mobius mu function. From here, I use the Mobius inversion formula to give:  $\frac{\mu(n)}{n} = \sum_{d|n}d\frac{\phi(\frac{n}{d})}{n}\mu(d)$  Now summing $n$ up to $x$:  $\sum_{n \leq x}\frac{\mu(n)}{n} = \sum_{n \leq x}\sum_{d|n}d\frac{\phi(\frac{n}{d})}{n}\mu(d)$  Now I'm not fully sure on how I can rearrange things inside the two summations on the right hand side, so leaving things as they are; using that $\sum_{d|n} \mu(d) = 1$, if and only if $n =1$. Hence:  $\sum_{n \leq x}\frac{\mu(n)}{n} = \sum_{d|1}d\phi(\frac{1}{d})\mu(d)$  Thus $d=1$. Subbing this in:  $\sum_{n \leq x}\frac{\mu(n)}{n} = \phi(1) = 1$  I am not sure what I have missed (or done wrong) as to have an equality and not a triangle relation. Any suggestions or ideas would be greatly appreciated Evaluate $\mathop {\lim }\limits_{x \to 0} \left( {{1 \over {{{\sin }^2}x}} - {1 \over {{x^2}}}} \right)$ I tried l'Hospital but that will require a lot (and I mean A LOT!!!) of differentiating  Is there a shortcut? $$\mathop {\lim }\limits_{x \to 0} \left( {{1 \over {{{\sin }^2}x}} - {1 \over {{x^2}}}} \right)$$  Thanks in advance How to choose input method on Kubuntu 18.04? Is it possible to make the default scene fill two monitors at startup? I know how to set up the default scene by pressing Ctrl+U, but whenever I startup Blender it always fills only one of my screens, even if I set it to fill both before setting the default scene. I end up having to resize it every time I start it up. Is there a way to set this to happen by default? How can one change the timestamp of an old commit in Git? The answers to  describe a way to amend previous commit messages that haven't yet been pushed upstream.  The new messages inherit the timestamps of the original commits.  This seems logical, but is there a way to also re-set the times? I have two Ubuntu Software Centers, one can be opened, another one can't How do I set up a Cron job? "Contact us"... and then what? How to add a custom icon or symbol to a flag link? I'm working on a 'sample request' implementation for a commerce site using , , sessions and a little custom code.  I'd like to add an icon or a symbol to let people know that an item they are viewing has been flagged.  I'm not sure exactly how to approach this, I can imagine perhaps adding a conditional field that displays an image or check box that is then changed by  or ajax, or perhaps something more complicated in JavaScript?  It could really be as simple as adding a checkbox that is checked or unchecked as the flag link is clicked or maybe even changing the css/html on click to be identified as marked.   Anyway, grateful for any advice here. Constructing a continuous function whose graph seems 'special' I've been reading through Zorich's "Analysis I" book recently, and I came across this nice little exercise.  Let $f: [0,1]\to \mathbb R$ be a continuous function such that $f(0)=f(1)$. Show that    for any $n\in \mathbb N$ there exists a horizontal closed interval of length $\frac 1n$ with endpoints on the graph of this function; if the number $\ell$ is not of the form $\frac 1n$ there exists a function of this form on whose graph one cannot inscribe a horizontal chord of length $\ell$.   The first part can be proven like this: Consider $g: [0,(n-1)/n] \to \mathbb R$ given by $g(x) = f(x) - f(x+1/n)$. Then   $$\sum_{k=0}^{n-1} g(k/n) = 0$$  and therefore either all of these points are zero or there exists both a point where $g$ is positive and a point where $g$ is negative. By continuity, there must then also be a point where $g = 0$. So we are done.  Now, the second statement seemed rather counterintuitive, and I have given it some time now, but don't see a counterexample for $\ell &lt; 1/2$.  (For $\ell &gt; 1/2$ the function $f(x) = \sin(2\pi x)$ will do.)  Can anyone help me out?  Cheers, According to the Third Newton's law of motion:        For every action there is an equal and opposite reaction.   So, I understand that if I hit a brick wall with $50\, \mathrm{lbs}$ of force, the brick wall also hits me with $50\, \mathrm{lbs}$ of force (usually painfully). In this instance, Newton's third law makes sense. What I'm confused about is, if I hit a patch of drywall with $50\, \mathrm{lbs}$ of force, it's probably going to break, and due to the lack of pain in my hand, I can tell it did not hit me back with $50\, \mathrm{lbs}$ of force.   How does Newton's third law apply to situations when one object or the other is destroyed? It certainly seems like at that point it is incapable of delivering the full force of my blow back to me. What happens with the energy? How to correctly enable Desktop Cube in Unity 3D? Why does the dock appear on the left when my screen is locked? The dock doesn't work correctly after I enter my password, either. I'm just trying out Arduino Uno for the first time with 2 blinking LEDs on a breadboard. All the tutorials on the Internet seem to use a resistor. I do know the function of resistors, but does it really matter here? These LEDs are working just fine without a resistor. Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin...(\sin(x)..)$ I tried to evaluate $$\lim_{x\to0}\frac{(1+x)^{1/x}-e}x$$but all my efforts were in vain . L'Hospital rule becomes very complicated. Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as the dual space of $E$.($\dim E=n$)   $\Omega^P(E):=\{ \alpha\colon \overbrace{E\times\cdots\times E}^{p- times}\rightarrow \mathbb R\  \ , \alpha \text{ is alternating multilinear map}\}$ $\Omega^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\  \ , u \text{ is alternating multilinear map}\}$  I wish to prove that there exist an $\textbf{unique}$ bilinear form $B\colon \Omega^p(E^*)\times\Omega^p(E)\rightarrow\mathbb R$   such that for all $ f_i\in E^* \ , u_j\in E^{**}\simeq E \qquad  i,j=1,\cdots,p$, it has the following rule:  $B(u_1\wedge\cdots\wedge u_p,f_1\wedge\cdots\wedge f_p)=\det[f_i(u_j)]$.  $\wedge$ is wedge product between alternating maps i.e. $\alpha\wedge\beta(v_1,\cdots,v_{p+p})=\dfrac{1}{p! p!}\displaystyle\sum_{\sigma\in S_{p+p}}sign(\sigma)\alpha(v_{\sigma(1)},\cdots,v_{\sigma(p)})\beta(v_{\sigma(p+1)},\cdots,v_{\sigma(p+p)})$  How can I do this? If moderators can view , can we see comments we've deleted? We can see our deleted answers, but why not our deleted comments? How to apply custom headers/footers on pages with new chapters? I'm writing my final report in LaTeX for my bachelor's degree. I'm aiming at having a document similar to the .doc template used normally.   I want to have my customs headers and footers applied on pages when I'm starting a new chapter.  I'm using documentclass report with fancyhdr and lastpage packages.  My headers are defined like this:  \lhead{\myTitle} \rhead{\today} \cfoot{\thepage\ of \pageref{LastPage}}   My chapters are created using standard \chapter{title} macro.  On pages with new chapters, the only headers/footers I see is the standard page number at the bottom.  What should I do?  EDIT: I've finally gone with Peter Breitfeld's solution. I'Ve gone a step further and created my own chapter command:  \newcommand{\myChapter}[1]{     \chapter{#1}     \thispagestyle{fancy} }   Thanks for the help! Dollar signs breaks formatting of 'Hot Network Questions' Related to  and .  This :    looks like this in the 'Hot Network Questions' section of e.g. :    I think Latex formatting of question titles should be disabled if the question comes from a site that don't use Latex formatting.