I think the following problem is not difficult, but I cannot solve it.  Prove the following fact.     Let $A, B$ be square matrices.   If $A B = A + B$, then $A B = B A$.
I was doing the problem $$ A+B=AB\implies AB=BA. $$  $AB=BA$ means they're invertible, but I can't figure out how to show that $A+B=AB$ implies invertibility.
Generalizing the case $p=2$ we would like to know if the statement below is true.  Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal. Shape of confidence interval for predicted values in linear regression I have noticed that the confidence interval for predicted values in an linear regression tends to be narrow around the mean of the predictor and fat around the minimum and maximum values of the predictor. This can be seen in plots of these 4 linear regressions:    I initially thought this was because most values of the predictors were concentrated around the mean of the predictor. However, I then noticed that the narrow middle of the confidence interval would occur even if many values of were concentrated around the extremes of the predictor, as in the bottom left linear regression, which lots of values of the predictor are concentrated around the minimum of the predictor.  is anyone able to explain why confidence intervals for the predicted values in an linear regression tend to be narrow in the middle and fat at the extremes? How to set an image as a background for Frame in Swing GUI of java? What is a simple way of downloading a file from a URL path? I've compiled a java project into a Jar file, and am having issues running it.  When I run:  java -jar myJar.jar   I get the following error  Could not find the main class: myClass   The class file is not in the root directory of the jar so I've tried changing the path of the main class to match the path to the class file and I get the same issue.  Should I be flattening the file structure? if so how do I do this. I'm using Ant to build the Jar file if thats of any use.  UPDATE   Here is the contents of the jar and the relevant Ant sections, I've changed the name of the firm I work for to "org":  META-INF/ META-INF/MANIFEST.MF dataAccessLayer/ dataAccessLayer/databaseTest.class org/ org/eventService/ org/eventService/DatabaseObject.class org/eventService/DatabaseObjectFactory.class org/eventService/DbEventClientImpl$HearBeatMonitor.class org/eventService/DbEventClientImpl.class org/eventService/EmptyQueryListException.class org/eventService/EventHandlerWorkItem.class org/eventService/EventProcessor.class org/eventService/EventTypeEnum.class org/eventService/EventWorkQueue$MonitorThread.class org/eventService/EventWorkQueue$PoolWorker.class org/eventService/EventWorkQueue.class org/eventService/FailedToLoadDriverException.class org/eventService/IConnectionFailureListener.class org/eventService/InvalidEventTypeException.class org/eventService/JdbcInterfaceConnection.class org/eventService/NullArgumentException.class org/eventService/OracleDatabaseObject.class org/eventService/ProactiveClientEventLogger.class org/eventService/ProactiveClientEventLoggerException.class org/eventService/PropertyMap.class org/eventService/SQLServerDatabaseObject.class org/eventService/TestHarness.class org/eventService/Utilities.class   And the ant target:  &lt;target name="compile" depends="init" description="compile the source "&gt;     &lt;javac srcdir="src" destdir="bin" classpathref="project.class.path"/&gt; &lt;/target&gt; &lt;target name="buildjar" description="build jar file" depends="compile"&gt;     &lt;mkdir dir="dist"/&gt;     &lt;jar destfile="dist/myJar.jar" basedir="bin" includes="**/*.class" &gt;         &lt;manifest&gt;             &lt;attribute name="Main-Class" value="org.eventService.ProactiveClientEventLogger"/&gt;         &lt;/manifest&gt;      &lt;/jar&gt; &lt;/target&gt; Union of a finite set and a countably infinite set is countably infinite How can I convert a std::string to int? Can I preemptively pay my credit card bill before a charge is posted? Showing that the sample mean and variance are independent by showing two distribution are the same Let $x$ and $y$ be positive integers such that $xy \mid  x^2+y^2+1$. Show that $$ \frac{x^2+y^2+1}{xy}= 3 \;.$$ I have been solving this for a week and I do not know how to prove the statement. I saw this in a book and I am greatly challenged. Can anyone give me a hint on how to attack the problem? thanks Has the Magical Community breached the Final Frontier? (aka Wizards in Space?) Have they? Given that Muggles put the first man in space in 1961, even by 1991 a lot of Muggleborns perfectly aware of the space-travel would have passed through Hogwarts and into the Ministry, even the Department of Mysteries (which seems to be a bit like their research division/Area 51 combo) And even leaving Muggle technology (which Wizards don't care for or about) aside, Humans have wondered about air travel to &quot;other worlds&quot; and meeting aliens from the time of the Ancient Greeks, surely some Witch or Wizard somewhere might have wondered if it really was possible to fly to the moon? So, is there any canon (aka Books + Pottermore + Rowling interviews) evidence of the Wizarding community having ever travelled or trying to travel to space / other planets or satellites? ETA: Note, I am also totally okay (and in fact would be eager for) historical &quot;Famous Wizard&quot; card-like tidbits from canon-sources.  AWESOME example (that I'm frankly not expecting): When the Muggles managed to land on the Moon, the Department of Mysteries had to scramble to hide the Silver-backed Moon Dragon conservatory they had been maintaining near the landing site. ALSO AWESOME example (that I am hoping for): Warlock Flibberti and Witch Tigget misused this earth-based spell and amusingly found themselves on the moon with their pet monkey instead! Last upgrade crashes network manager (no internet connection, no applet) While upgrading today, something caused my wireless network manager to disappear. Not even connecting via wired connection.  This error is being shown by files:  **(nm-applet:2716): WARNING **: Could not initialize NMClient /org/freedesktop/NetworkManager: The name org.freedesktop.NetworkManager was not provided by any .service files   (nm-applet:2716): nm-applet-WARNING **: Error connecting to ModemManager: Error calling StartServiceByName for org.freedesktop.ModemManager1: GDBus.Error:org.freedesktop.DBus.Error.Spawn.ExecFailed: Cannot launch daemon, file not found or permissions invalid   (nm-applet:2716): nm-applet-WARNING **: Could not find ShellVersion property on org.gnome.Shell after 5 tries   (nm-applet:2716): nm-applet-WARNING **: Failed to register as an agent: (2) The name org.freedesktop.NetworkManager was not provided by any .service files    Is there a patch I can download through windows and send through to my ubuntu, or is it a case of having to reload everything? I tried running two versions back but still no internet access. How do you run a Python script as a service in Windows? I am sketching the architecture for a set of programs that share various interrelated objects stored in a database. I want one of the programs to act as a service which provides a higher level interface for operations on these objects, and the other programs to access the objects through that service.  I am currently aiming for Python and the Django framework as the technologies to implement that service with. I'm pretty sure I figure how to daemonize the Python program in Linux. However, it is an optional spec item that the system should support Windows. I have little experience with Windows programming and no experience at all with Windows services.  Is it possible to run a Python programs as a Windows service (i. e. run it automatically without user login)? I won't necessarily have to implement this part, but I need a rough idea how it would be done in order to decide whether to design along these lines.  Edit: Thanks for all the answers so far, they are quite comprehensive. I would like to know one more thing: How is Windows aware of my service? Can I manage it with the native Windows utilities? What is the equivalent of putting a start/stop script in /etc/init.d? What is the typographic reasoning here? Does it make the text look less crowded? After reading  and reading the comments on the page, one person was wondering if it would be possible to use the Physical Device Object Names of other hardware such as the microphone to find out the names of processes using that device.  I tried the same approach, but so far it only seems to work for the webcam. Is there any other way I could get this to work in ?      UPDATE:  The Lifehacker post was about finding out which Windows process is currently using your webcam.  is how they went about doing it:        Start Device Manager (WIN+R &rarr; "devmgmt.msc" &rarr; OK)   Find your webcam among the list of devices (check under Imaging Devices)   Open the properties window of the device and switch to the Details tab  (Right click &rarr; Properties &rarr; Details)   In the dropdown menu, select Physical Device Object Name and copy the string(Right click &rarr; Copy)   Download    Make sure you have opened Process Explorer in Administrator Mode(File &rarr; Show Details for All Processes)   Hit CTRL+F and enter the string you copied earlier(it should be something like \Device\000000XX)   Hit the Search button and you should see a list of processes using the webcam(if there are any) Java generics type erasure: when and what happens? What is the best way to increase my reputation and privileges? I'm not sure if this is a stupid question, but here goes anyway... I have only recently started learning PHP and Javascript and I'm reasonably proficient. One of the reasons I'm so proficient is because I LOVE Stack Overflow and find most of the solutions to my dilemmas here.   But I've found that I never get the chance to ask a question because it has normally been asked many times before and when I go through the unanswered questions to try and help someone else, they are generally WAY over my head.  So I'm not sure how to get more privileges. I can't even vote up answers that I like or that helped me. I can't comment to thank someone for their solution. It is most tragic and demoralising... How do I change the situation? Strange folder in my home folder after a failed run of BleachBit Can the preposition 'in' be omitted here? How to tell Google that I have changed my website URLs? I have done major updates to my website, and renamed all my URLs.  How can I tell Google that I have renamed these URLs and let Google refresh its index?  My website URL: http://www.pndmasr.com/  My sitemap: http://www.pndmasr.com/sitemap.xml  I have uploaded my sitemap via Google Webmaster Tools many times.  However, every time I search Google for "pndmasr", I still get results with old pages. I have waited more than 3 days but the problem remains. Any suggestions?  Is there a problem with my sitemap?