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Suppose $a_1, a_2,\dots , a_n$ are $n$ positive real numbers with $a_1a_2 \dots a_n = 1$.   Then what is the minimum value of $(1 + a_1)(1 + a_2). . .(1 + a_n)$ ?    I think $(1 + a_1)(1 + a_2). . .(1 + a_n)$ takes its minimum value when $a_1=a_2=\dots=a_n=1$ and thus the minimum value is $2^n$.  I don't know how to prove it. Please help. I'm still learning, but I'm trying to follow a tutorial to apply texture to a model. I only want to apply the texture to one object in the scene, so I selected it, unwrapped it, and painted over it. Now I selected it in the Edit Mode (I can't select it in the Object mode because all is locked together) and went into Node Editor, but the nodes are greyed out.  Why? Fastest, surest way to erase a hard drive? states:  "In practice, explicit right outer joins are rarely used, since they can always be replaced with left outer joins and provide no additional functionality."  Can anyone provide a situation where they have preferred to use the RIGHT notation, and why? I can't think of a reason to ever use it. To me, it wouldn't ever make things more clear.  Edit: I'm an Oracle veteran making the New Year's Resolution to wean myself from the (+) syntax. I want to do it right EM wave generation from different frames of reference How to pass multiple operator properties via UI layout? Calculate the Factor Group $(\mathbb{Z}_4 \times \mathbb{Z}_6)/\langle(0,2)\rangle$ In column B I have contact number. I want to add a country code to all of them.  For example, if the number is 123456, I want it to become 88123456.    Each number is in 1 row, and there are about a thousand of them, so I want an automatic way to do this that allows me to add the country code to each of them without a space. Are there any drawbacks to legacy BIOS as opposed to UEFI? Are there any drawbacks to installing Ubuntu 13.10 (and soon 14.04) in Legacy BIOS instead of UEFI?  I have installed 13.10 64bit in Legacy by mistake and was just wondering if it would hurt anything to simply leave it there as is?  EDIT  Everything seems to be running just fine in Legacy and all my drivers work. Rep line graph: Say "only 200+ accounts displayed" at all times As discovered/discussed , the cross-site reputation graph doesn't show accounts that have less than 200 reputation. The problem is that, unless you have zero accounts over that bar, it doesn't tell you that. I think the "Accounts must have a minimum of 200 reputation to appear in this graph" notice should appear either (A) at all times or (B) if you have any accounts below 200. Why is the sample distribution the Exponential distribution Gamma distributed? Blender saves a new file every time When I click save or hit Command + S what appears to be a new version of the document is created. So if I save 2 times, I get 2 documents, filename.blend1 and filename.blend2.  What exactly are these files and why is Blender creating them? Non-Metrizable Topological Spaces Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$. In Burton's Elementary Number Theory, when introducing absolute pseudoprimes (also called Carmichael numbers), the author states that they are composite numbers $n$ such that $a^{n-1} \equiv 1 \pmod n$ for all integers $a$ with $\gcd(a,n) = 1$.   When showing that 561 is an absolute pseudoprime, he then concludes that $a^{561} \equiv a \pmod{561}$ for all $a$.   I searched a bit on the internet, and it seems that these two are actually equivalent definitions of a Carmichael number. However I have not been able to find a proof; the fact that the implication in the title of the post is stated so plainly in my textbook makes me think it is quite simple to see, but I can't seem to understand the argument. What is the difference between CentOS "DVD" vs "Everything" ISOs `tail -f` until text is seen I've got a CI server with a command-line interface that allows me to remotely kick-off a job (jenkins CI server and the jenkins-cli.jar tool).  After I kick the job off I tail -f the log (sorry for the messy command):  ssh -t my-jenkins-host.com "tail -f \"/var/lib/jenkins/jobs/$job_name/builds/\`ls -ltr /var/lib/jenkins/jobs/$job_name/builds/ | grep '^l' | tail -n 1|awk '{print \$9}'\`/log\""   After the job successfully completes, usually after at least 5 minutes, I get the following line on the output:  Finished: SUCCESS   Is there a good way to stop tailing the log at this point? i.e. is there like a tail_until 'some line' my-file.log command?  BONUS: extra credit if you can supply an answer that returns 0 when SUCCESS is matched, 1 when FAILURE is matched, and your solution works on mac! (which i believe is bsd based) What exactly is tested when a fire extinguisher is tagged? I am just curious: the obvious is that the  pressure mass   can be inspected /measured to ensure that the device has not been discharged. Is there anything else that is measured \ checked? Let $E=l^p=\left\{x=\left(x_n\right): x_n\in\mathbb{R}, \sum_{1}^{\infty}\left|x_n\right|^p&lt;\infty\right\}$ with $1\le p&lt;\infty$, $\left\|x\right\|_E=\left(\sum_{1}^{\infty}\left|x_n\right|^p\right)^{1/p}.$ Let $\left(\lambda_n\right)$ is a bounded sequence in $\mathbb{R}$ and consider the operator $T$ defined by $$T(x)=\left(\lambda_1x_x,\ldots,\lambda_nx_n,\ldots\right)$$ where $x\in l^p$. Prove that $T$ is a compact operator from $E$ to $E$ iff $\lambda_n\rightarrow 0$.  Can anyone give me some hints to solve this problem? Thank you! Caption issue when placing figure side by side I placed 2 figures side by side using the minipage method from , and the figures look perfectly fine. However, the caption of the right figure stick out into the margin, why is that and how can I fix it?    My code is like this:  \begin{figure}[htbp]   \centering   \begin{minipage}{.5\textwidth}     \centering     \includegraphics[width=6.6cm]{figure1.png}     \caption{Caption for figure 1.}     \label{fig:fig1}   \end{minipage}%   \begin{minipage}{.5\textwidth}     \centering     \includegraphics[width=6.6cm]{figure2.png}     \caption{Caption for figure 2, but longer.}     \label{fig:fig2}   \end{minipage} \end{figure} How do you prove that these four definitions of the operator norm are equivalent? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &amp;= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ &amp;=\sup\{ \lVert Av\rVert\;\colon\; v\in V\text{ with }\lVert v\rVert\leq 1\}\\ &amp;=\sup\{\lVert Av\rVert\;\colon\; v\in V\text{ with }\lVert v\rVert = 1 \}\\ &amp;=\sup\left\{ \frac{\lVert Av\rVert}{\lVert v\rVert}\;\colon\; v\in V\text{ with }v\neq 0\right\}. \end{align*}$$