Three stars or asterisks
Bizarre spacing with 4 $\star$s
How do I know if food left at room temperature is still safe to eat? RSA encryption with private key and decryption with a public key I need help!  I need to prove that for any $2 \le n,k$ positive integers $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$  Thanks! Install packages from newer release without building (apt pinning) I need these packages with the latest upstream version backported for ubuntu/natty   libccid_1.4.5-0ubuntu1_amd64.deb libpcsclite1_1.8.1-0ubuntu1_amd64.deb libpcsclite-dev_1.8.1-0ubuntu1_amd64.deb libusb-1.0-0_1.0.9-0ubuntu1_amd64.deb libusb-1.0-0-dev_1.0.9-0ubuntu1_amd64.deb opensc_0.12.2-1ubuntu1ppa1~natty1_amd64.deb pcscd_1.8.1-0ubuntu1_amd64.deb pcsc-tools_1.4.18-0ubuntu1_amd64.deb   I tried to build them from source with update but it failed. I am not an expert at packaging and there are so many tutorials and recipes out there.  Do you know a good and up to date tutorial for backporting Ubuntu packages?  How do I get packages from other releases without having to build them myself? What's wrong with this argument that $[0,1]$ is countable? How to create polygons with straight lines and right angles in QGIS? When creating a polygon, how do I do it so that the lines are straight (90, 0 degrees, etc.) and when I create corners they can be right angles? For instance, how do I create a perfect square or rectangle? How can I backup my RAW photos while travelling without Internet access? Clone hard drive to another, smaller hard drive Is there a good tool/program that allows me to clone a system drive to another, smaller hard drive?  I've bought a new 64GB SSD drive and would like to move my current Windows 7 Professional 64-bit installation to that new drive. My current drive is a 200GB SATA drive with one partition only. Of course there's enough free space to shrink my current partition to less than 64GB. How will the super massive black hole affect our galaxy? Capturing Blue Screen of Death (BSOD) screen Possible Duplicate:          Sometimes I get a BSOD screen for a split second and I have no chance to see the cause of it. It also happens when I am not at the computer. The event log doesn't have enough information. Is there a way or tool that captures the screen when it happens so I can look at the cause later.  I already did a memory diagnostics test so that's not the problem. texstudio: show line numbers in editor How do I show line numbers in the editor? According to the picture in the doc (e.g. about bookmarks) there should appear by default.  I'm using Texstudio version 2.7, win8.1. Last time I tried to make Chocolate eclairs (similar to cream puffs), the choux pastry just sat in the oven and basically fried.  It didn't rise/grow and so I couldn't hollow out the shells.  Any suggestions?  Recipe was from The Australian Women's Weekly Original Cookbook (Golden Press Pty Ltd, 1977) p. 204  It is almost identical to the one published on their website , with the exception that the cookbook recipe used 1C water and 1C plain flour. Change post permalink to external URL from custom field Pigeonhole Principle Question - Group of 6 people, do 3 either know each other or not? Prove that in any group of 6 people there are always at least 3 people who either all know one-another or all are strangers to one-another.  Hint: Use the pigeonhole principle.  I don't see how this applies to the pigeonhole principle because I keep imagining a group of 4 strangers, and then 2 friends. This would be 6 total but against what the proof is asking. Maybe I don't understand the proof in question... Gmail, how to forward email to group using a filter Someone edited my message on StackOverflow, but it really bugs me out.  I'm not sure what's wrong with it:   As you see, the bigger the circle becomes, the more vertices I need for hiding the straight lines. As you see, the bigger the circle becomes, the more vertices I need to hide the straight lines.   The latter (corrected) sentence just bugs me out for some reason. It would sound better if there was a comma after the word "need", perhaps, but I guess that would be even worse in means of grammatical correctness.  Can someone explain why is the first sentence bad and the second sentence correct? And could there even be better solution to fix the first sentence? Assume $T \in \mathbb{R}^{d\times d}$. Let $\rho(T)$ be the spectral radius of $T$ and $\rho \ge 0$. Prove that  $\rho(T) &lt; \rho$ if and only if there exists a $P \succ 0$ satisfying $T^TPT - \rho^2 P \prec 0$.  I can prove the Necessity: If $P = I$, for any $x \in \mathbb{R}^d$,   \begin{equation} x^T T^T P T x - \rho^2 x^T P x = \| Tx \|^2 - \rho^2 \| x\|^2 &lt; 0 \end{equation} so $T^T PT - \rho^2 P \prec 0$  I get stuck in proving the Sufficiency. What does the exclamation mark do before the function? Prove there are no other solutions of functional equation $f(x+y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$ I have the following functional equation. Find all continuous functions $f:(-1,1) \to \mathbb R$ such that $$ f(x+y)=\frac{f(x) + f(y)}{1 - f(x)f(y)} $$ The first obvious solution is $f(x) \equiv 0$. Another one I guessed, it is $f(x) = \pm \tan x$. I suspect, that there are no more solutions. The problem is that I don't know how to prove that.  Since we have a rational equation, I have no idea how to make any substitutions in order to get expression for $\tan x$ (as it is not a rational expression).  P.S. I can also show that it is true that (a) $f(0) = 0$ (take $x=y=0$), and (b) $f(-x) = -f(x)$ (take $y=-x$).  Update: there is a solution of this problem also here: How to intuitively understand eigenvalue and eigenvector? I'm learning multivariate analysis and I have learnt linear algebra for two semester when I was a freshman.  Eigenvalue and eigenvector is easy to calculate and the concept is not difficult to understand.I found that there are many application of eigenvalue and eigenvector in multivariate analysis. For example     In principal components, proportion of total population variance due   to kth principal component equal    $$\frac{\lambda_k}{\lambda_1+\lambda_2+...\lambda_k}$$   I think eigenvalue product corresponding eigenvector has same effect as the matrix product eigenvector geometrically.   I think my former understanding may be too naive so that I cannot find the link between eigenvalue and its application in principal components and others.   I know how to induce almost every step form the assumption to the result mathematically. I'd like to know how to intuitively or geometrically understand eigenvalue and eigenvector in the context of multivariate analysis(in linear algebra is also good).  Thank you!