Commutative rings exercise
If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative
What is the purpose of /home on a Mac? On Linux, (which I am used to), /home is the place where home folders live.  On a Mac, home is some kind of crazy mutant:  $ mount /dev/disk1 on / (hfs, local, journaled) devfs on /dev (devfs, local, nobrowse) map -hosts on /net (autofs, nosuid, automounted, nobrowse) map auto_home on /home (autofs, automounted, nobrowse)   Other than to be non-conformist, is there some purpose to this auto_home mount point?  Considering how it mucks up things that come right out of the box, there must be some wonderful thing that it accomplishes, but I can't figure out what that would be..... Which version of Ubuntu will work on my PowerPC? I'm using iMac PowerPC G5 1.8 GHz, 1GB RAM, 160GB HDD, and 10.5.4 Leopard OS.  I want to run Ubuntu on my iMac. Please can anyone tell me which version will run in my system? Our 21 month old toddler will use the potty when we leave her bottom bare - she has been successfully doing this for a month. Tired of cleaning stinky diapers, we leave her bottom bare when she is home.   She doesn't particularly enjoy wearing diapers, and when we put one on her, she doesn't take the initiative to tell us that she has to use the potty. However, when we have company over, we put a diaper on her.  We are trying to transition her into wearing panties; however, she is very opposed to them. She equates them with pants and doesn't want to wear them. While she is fully capable of dressing herself in pants, she hasn't mastered taking her pants off yet.   How do you introduce underwear to your child? When do people normally introduce underwear to their child? How do I explain the concept of having to take her panties off before using the potty?  We in no way want to discourage her use of the potty, as she seems to really enjoy pooing/peeing outside (she claps for herself whenever she uses the potty). But if she is interested in potty training herself, I do want to show her the concept of an underwear. Simultaneous collaborative editing of a LaTeX file How should I detect unnecessary #include files in a large C++ project? Write in NTFS using Mavericks I was able to read and write NTFS filesystems normally in Mountain Lion, but after the upgrade this stopped to work.  I installed ntfs-3g and fuse4x using homebrew in Mountain Lion and followed the instructions displayed by homebrew to finish the installation of fuse4x kernel extension.  To try to solve this problem I removed ntfs-3g, fuse4x and fuse4x-kext and reinstalled them. But this didn't solve my problem.  Does anyone know how to solve this? Thanks. Adding Edge Loops Around Airplane Window I would like to add an edge loop around each one of these windows and are not 100% sure on the best way to do it even after watching some videos.  If someone could point me in the right direction that would be great. Thanks. Sci-fi novel, scout crash lands on planet, shaman sees competition and wrecks spherical spaceship I read this years ago. A scout in spherical advanced ship lands on a planet. Natives are friendly, but the shaman perceives competition, and sneaks into the craft and scatters potions on the controls, their acid eating into plastics and metals. He can be rescued at the other side of planet, but he has a deadline. Eventually all make airship and he makes it. Compute $P(Y<3X)$ using joint PDF I'm given a joint pdf  $f_{X,Y}(x,y)=2e^{-x-y}, 0&lt;x&lt;y, 0&lt;y $  and asked to compute $P(Y&lt;3X)$. To do this, I let $Y=3X$ (the boundary) and found that the region of integration is under this line.  To find $P(Y&lt;3X)$, it seems to me that the integral for this region be written as   $P(Y&lt;3X)=\int_0^\infty \int_0^{3x} 2e^{-x-y} dy dx  $  However this isn't right, as the boundaries are actually   $P(Y&lt;3X)=\int_0^\infty \int_x^{3x} 2e^{-x-y} dy dx  $  Why is this? I'm having a hard time seeing how to construct these boundaries (especially the fact that the $y$ boundary goes from $x$ to $3x$ instead of $0$ to $3x$)  Thank you. Are aerosol cans allowed and safe, in checked luggage? What is the name of the black&white movie about a mirror Earth (universe)? Astronauts in a spaceship have an accident and fall back to the Earth. They are tried in court. It turns out to be a mirror Earth. As I remember (though maybe I'm not totally right) only one professor knows something about that and in the final scene he crashes into a mirror with his wheelchair. In another scene an astronaut realizes that his name can be read from the mirror directly.  Does anybody remember the movie? How to apply a macro to each column of a table How to expand primary partition when WUBI is used to install? I have installed Ubunut 12.04 using WUBI. The "Installation size" options I chosen was 18GB i.e the amount of space you want to "give up" for WUBI. That is the space wubi's Ubuntu install is able to use. Wubi creates a disk image (loopmounted device) on your host system in/on which Ubuntu is installed. Now, I am running out of memory in primary partition.   Could anyone help me out to extend/stretch the partition size? Some time ago, I baked a cake (Sachertorte to be exact). The taste and texture was fine, so my recipe is okay.  My biggest issue was the shape. In the middle it was really a lot higher than on the sides. I would like to redo it in the near future, but preferably with a flat(ter) surface. (Especially since I put icing on top and I don't want it to drip off this time.)  Anyone knows the cause of my problem, or even better a solution? I think I put enough butter on the side of my springform pan, in case this is an option. Consider a particle on the ground. This particle is raised by a force of magnitude $mg$ to a height $h$ above the ground. At this point, the work done on the particle by the force is $mgh$, which is equal to the potential energy of the particle. But, during this period, the force of gravity also acts on the particle and is displaced by $h$, and so does a work of $-mgh$ on the particle. Shouldn't  the two cancel and no net work should be done on the particle?   If they don't cancel, then where did the energy that came from the work done by the force of gravity go? Changing threshold in logistic regression Suppose that you have a logistic model and the predicted probabilities of a $1$ all are in $[0.2,0.3]$. So the predictions would be "0" based on a threshold of $0.5$. What would be a good threshold to choose? adjective describing middle distance (neither near nor far) Are there names for consonant-shifts when suffixes are added? Object Positions Reset On Blender Load What is the difference between a deep copy and a shallow copy?