Prove $\langle a^s \rangle=\langle a^t \rangle \iff \gcd(n,s)=\gcd(n,t)$
Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{lcm (n,k)} \rangle$
Why is a ConcurrentModificationException thrown and how to debug it I am using a Collection (a HashMap used indirectly by the JPA, it so happens), but apparently randomly the code throws a ConcurrentModificationException. What is causing it and how do I fix this problem? By using some synchronization, perhaps?  Here is the full stack-trace:  Exception in thread "pool-1-thread-1" java.util.ConcurrentModificationException         at java.util.HashMap$HashIterator.nextEntry(Unknown Source)         at java.util.HashMap$ValueIterator.next(Unknown Source)         at org.hibernate.collection.AbstractPersistentCollection$IteratorProxy.next(AbstractPersistentCollection.java:555)         at org.hibernate.engine.Cascade.cascadeCollectionElements(Cascade.java:296)         at org.hibernate.engine.Cascade.cascadeCollection(Cascade.java:242)         at org.hibernate.engine.Cascade.cascadeAssociation(Cascade.java:219)         at org.hibernate.engine.Cascade.cascadeProperty(Cascade.java:169)         at org.hibernate.engine.Cascade.cascade(Cascade.java:130) Should I write "X and I", "X and me", "I and X", or "me and X" in a conjoined object? Global JavaScript Variable Scope: Why doesn't this work? My very simple Greasemonkey script is not running? Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$ Let $X,Y,Z$ be subspaces of $V$ so that $X$ is a subspace of $Y$. Prove that $Y\cap (X+Z)=X+(Y\cap Z)$  I know that I need to prove that $Y\cap (X+Z)\subseteq X+(Y\cap Z)$ and $X+(Y\cap Z)\subseteq Y\cap (X+Z)$ but I´m having a hard time.  Can you lend me a hand please? I would really appreciate it I would like to port Ubuntu Touch to a device that is currently not supported. What are the requirements for doing this and where do I begin? Lyircs of Free by Natalia Kills:     I'm free      I just spent all my money       but I rocked that like it don't cost a thing   Shouldn't it be "I've just spent all my money"? As far as I remember, our English told us to use Present Prefect in such case because of the just. ISO image integrity check (new try)? When you open an .iso file with e.g. 7-zip, the tool lets you test the ISO file, just like you can test any RAR or 7Z or ZIP file that the file is still ok.  If that test finishes successfully, does that mean the .iso file data and all the files within the ISO disc image are fine and not corrupted (bitrot)?   I am a bit confused because elsewhere I saw it mentioned that the ISO (raw disc image) format doesn't contain any data integrity checksums, so I am unsure what the 7-zip tool is testing while testing an ISO disc image file. At the end of a letter, is it correct to omit "I am" in this sentence?     I am looking forward to hearing from you   Then it will become:     Looking forward to hearing from you This might be a stupid question, but nonetheless, it has been bothering me.  If you take a photon, make it go through some atoms in a solid, liquid or whatever, then you have the chance of this photon being absorbed by an electron, and thereby exciting the electron. This requires the photon to have enough energy to actually excite the electron to another energy level.  My question is then: How does the photon know if it has enough energy or not? Do they interact very quickly to determine if it's okay or not, or is it just something it "knows?" How to prove that $p-1$ is squarefree infinitely often? How to prove that $p-1$ is squarefree infinitely often, where $p$ is prime?  I had thought of using Dirichlet's theorem on arithmetic progressions.  The number of squareful values of $p-1$ less than or equal to $n$ is bounded by $\sum_{q}{\pi_{q^2,1}(n)}$, each term $\pi_{q^2,1}(n) \approx \frac{\pi(n)}{q \cdot (q-1)}$, and $\sum_{q}{\frac{1}{q \cdot (q-1)}} \lt 1$.  But using for example , the possibility is open that the error in each approximation for $\pi_{q^2,1}(n)$ may continue to increase with $q$, and since the sum runs up to $\sqrt{n}$, this argument is invalid.  Is there a more refined version of Dirichlet's theorem that can be used to circumvent this issue?  Or an entirely different way to prove this statement? How would I have a  action that may have some effects on the current page but would also change the URL in the browser so if the user hits reload or bookmark, then the new URL is used?    It would also be nice if the back button would reload the original URL.  I am trying to record JavaScript state in the URL. I know someone or the other has asked about disabling ⌘Q, but I like the shortcut, and I use it often. However, I use an email client called Airmail that unfortunately does not keep a service (for notifications) running in the background when it is quitted. However, if I press the close button a service runs in the background, allowing me to receive notifications.  But since I have a habit of pressing ⌘Q all the time to close apps, I was looking into a way I could disable it for only Airmail.  Anyone know of a what to do this? If $g$ is the generator of a group $G$, order $n$, when is $g^k$ a generator? Say a cyclic group of order $9$ is denoted by $G = \langle g \rangle$. I have a claim that  $\langle g^k \rangle = G $ whenever $k$ is a unit in $\mathbb{Z}_9$. I'm struggling to understand why this is true. Is this specific to order 9? In general, for a group of order $n$, do we require that $k$'s multiplicative inverse in $\mathbb{Z}_n$ be $n$ itself? Otherwise the generating group will be of order less than $n$, and thus cannot be equal to the group. Infusing olive oil with herbs How to disable Macbook Pro internal display when using external monitors I have a Late 2011 Macbook Pro 13" (running Lion - OS 10.7.4) which I use with a dual external monitor setup. I have two 21" LCDs that I use as external monitors. One uses the built in Mini Display Port on the Macbook Pro and the other is connected via a USB to DVI adapter (Diamond DVILink).  The problem I am having is with the 13" built in display of the Macbook Pro. I would like it to remain closed and unused (I prefer the dual monitors at same resolution), however I cannot seem to "disable" the display in the display preferences applet. So even though the Macbook Pro lid is closed the operating system still makes use of the display, even though the apple logo on the back of the display is unlit. I have tried to move the laptop screen off out of the way in display preferences but I find when I "sleep" my setup that any applications that remain open move to the laptop screen. This makes it difficult to locate the application windows when resuming from sleep.    Does anyone know of a way to disable the laptop built in display so it is not seen by the OS when external monitors are being used? How to modify the page at admin/structure/types? How can I do case insensitive string comparison in Python?  I would like to encapsulate comparison of a regular strings to a repository string using in a very simple and Pythonic way. I also would like to have ability to look up values in a dict hashed by strings using regular python strings. Simple Linq query has duplicated join against same table? How does the twitter bot work? I heard that there was a twitter bot which automatically posted questions and asked for answers for specific questions on every site. How do they work? Is there an algorithm for doing this or is it purely random? I know that it doesn't tweet closed questions, but that's it.