Atyiah-Macdonald Ex 3.7
Complement of maximal multiplicative set is a prime ideal
Do tag edits count for Archaeologist? I know tag-edits do not count towards the 500 edits required for the Copy Editor badge. Do they count to the 100 for the Archaeologist badge? It seems they should because you are making a 6-month-inactive post active again. php execute a background process I need to execute a directory copy upon a user action, but the directories are quite large, so I would like to be able to perform such an action without the user being aware of the time it takes for the copy to complete.  Any suggestions would be much appreciated. Counter ions and charge of a complex how can I add the charge and the counter ions to my complex? I'm using chemfig.  \schemestart \chemname{\chemleft[\chemfig{Ni (&lt;:[1]OH_2) (-[2]OH_2) (&lt;:[3]OH_2) (&lt;[5]OH_2) (-[6]OH_2) (&lt;[7]OH_2) }\chemright]}{\ce{[Ni(OH2)6]Cl2}} \schemestop     Thanks! Choosing numbers without consecutive numbers. In how many ways can we choose $r$ numbers from $\{1,2,3,...,n\}$,   In a way where we have no consecutive numbers in the set? (like $1,2$) To prove in a Group Left identity and left inverse implies right identity and right inverse Let $G$ be a nonempty set closed under an associative product, which in addition satisfies :      A. There exists an $e$ in $G$ such that $a \cdot e=a$ for all $a \in G$.      B. Given $a \in G$, there exists an element $y(a) \in G$ such that $a \cdot y(a) =e$.      Prove that $G$ must be a group under this product.   Attempt -Since Associativity is given and Closure also, also the right identity and right inverse is given .So i just have to prove left identity and left inverse.  Now as $ae=a$ post multiplying by a, $aea=aa$. Now pre multiply by a^{-1} I get hence $ea=a$. And doing same process for inverse Is this Right? How to fix "Headers already sent" error in PHP How do I write an Apex unit test? Locking a vehicle bicycle rack We are planning a trip with a bike rack attached to our trailer hitch. What is the best way to lock the bikes while the vehicle is unattended? I see some racks have locks on the knobs but they don't seem very strong.   What do you suggest? How can I separate multiple quote blocks? I have a series of quotes which I want to display in two separate quote blocks.  &gt; Are questions about Microsoft Word Online on-topic here? I don't &gt; really see any posted here, but perhaps I am not searching correctly. &gt; - mweiss &gt;  &gt; Yes, they are. We even have a specific tag: [tag:word-online]. -ale &gt;  &gt; Thanks! I'm sorry I missed that tag. -mweiss   When I try to add these to my post, they display as a single combined quote.     Are questions about Microsoft Word Online on-topic here? I don't   really see any posted here, but perhaps I am not searching correctly.   - mweiss      Yes, they are. We even have a specific tag: . -ale      Thanks! I'm sorry I missed that tag. -mweiss   How can I separate quotes so that they aren't combined into a single quote block? Anyone know a simple method to swap the background color of a webpage using JavaScript? Reoccurring Bent Axles, any reasons or upgrades? Earlier this month I had a issue with a broken axle.  It was the original that was sold with the bike and lasted around 5,000 miles.  I went to my LBS and bought a new axle (unbranded, labeled, etc, paid $13, came with cones, spacers, and hardware).  I did the install myself and had everything adjusted nicely.  The next time I rode it was on my commute to work, but I didn't get 4 miles into it when I found that I bent the axle.    I took the wheel to the shop but they wouldn't do anything about it since I didn't have them do the work.  However, I figured this was a fluke and asked them for a new axle and to true the wheel.  Once again I did the install of the axle and only got 15 miles out of it this time before bending it.  Before I got home I felt the wheel get stiff, but it stayed true.  The question is what will be a cause of bending 2 axles in 20 miles?    I think the guy at the shop said it was a cro-moly axle (while talking to another person today at the shop, these may not be), and he couldn't believe that I bent it.  If so, where would such an axle fall in the quality hierarchy?  Or is my hub toast?  While the wheel was being repaired, I swapped to a new original wheel for about 2 weeks.  I have not had any issues with it, but I wanted to keep it as new as possible.   Reading other questions I saw a comment that said not to use a hub where the bearing was catching.  I did have one in this wheel but I figured since it is a cheap wheel I would just burn it up and move on to a better one.  Besides, it has lasted 5k miles without issue to this point.  The bike in question is a 2008 Schwinn High Timber with Joytech freewheel hubs.  I do use a trailer with this bike using a Burley axle hitch.  While I had the hitch installed on both axles I bent, I did not have either trailer hooked up to them.  Something I would like to note is that when I removed the first bent axle (axle #2) that the cones on it had a wavy bearing track.  I didn't think too much of it, and I put them on axle #3.  Only having a few miles on them shouldn't have made a difference? First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an eigenvalue of $BA$?  If it's not true, then under what conditions is it true or not true?  If it is true, can anyone point me to a citation?  I couldn't find it in a quick perusal of Horn &amp; Johnson.  I have seen a couple proofs that the characteristic polynomial of $AB$ is equal to the characteristic polynomial of $BA$, but none with any citations.  A trivial proof would be OK, but a citation is better. Even/Odd Binomial Coefficients I was wondering if there's a nice general solution for the following problem:  How many numbers in the $n^\text{th}$ row of Pascal's triangle are even? How many are odd? QGIS apply style to shapefile Avoiding NullPointerException in Java I use object != null a lot to avoid . What is an alternative to: if (someobject != null) {     someobject.doCalc(); } How do you tell if a D&D book is 3.0 or 3.5? Can there be true conclusions without assumptions? I was thinking of the sentence  "I think therefore I am",  which I had for a long time considered indisputable because it's self-evident.   Then I considered the hypothetical situation where my thoughts are illogical – and of course I can't be aware that they are – and so to a logical outsider I am not saying something that is actually indisputable and self-evident, but in fact may be total nonsense. Perhaps there is a hypothetical evil demon altering my ability to reason. However convincing it seems to me that I MUST exist since, in whatever since "I" or "exist" is, there is the ability to question whether I do, the conclusion can't actually be trusted without doubt since I cannot know if I am in fact reasoning logically.  Even what I consider to be trivial tautologies require this assumption of sanity, and therefore there is no true conclusion without assumption – an assumption about the nature of my own reasoning. The only absolute truth I have is unspeakable subjective experience, and any articulation (and I realize the apparent circular reasoning I've cornered myself into here...) is necessarily uncertain.  I realize that perhaps this doesn't even make sense, but was just curious if anyone had entertained this line of thought and come to some satisfying resolution. I have often heard it recommended that a user account should be disabled by setting its shell to /bin/false. But, on my existing Linux systems, I see that a great number of existing accounts (all of them service accounts) have a shell of /sbin/nologin instead.  I see from the man page that /sbin/nologin prints a message to the user saying the account is disabled, and then exits. Presumably /bin/false would not print anything.  I also see that /sbin/nologin is listed in /etc/shells, while /bin/false is not.  The man page says that FTP will disable access for users with a shell not listed in /etc/shells and implies that other programs may do the same. Does that mean that somebody could FTP in with an account that has /sbin/nologin as its shell?  What is the difference here? Which one of these should I use to disable a user account, and in what circumstances? What other effects does a listing in /etc/shells have? MikTeX 2.9, install packages on the fly? Simulate a uniform distribution on a disc I was attempting to simulate injection of random points within a circle, such that any part of the circle has the same probability of having a defect.  I expected the count per area of the resulting distribution to follow a Poisson distribution if I break up the circle into equal area rectangles.    Since it requires only placing points within a circular area, I injected two uniform random distributions in polar coordinates: $R$ (radius) and $\theta$ (polar angle).  But after doing this injection, I clearly get more points in the center of the circle compared to the edge.     What would be the correct way to perform this injection across the circle such that the points are randomly distributed across the cirlce?