This article presents an effort to use Induction in geometry to prove Sylvester Gallois Theorum.
Theorem statement can be viewed here
For the proof, let us rephrase the theorem as follows –
(1) Given a finite set of points in Euclidian space such that any line passing through two of the points passes through at least one more of them, all points have to be collinear.
Let n be the finite number of points in the given set.
Given 3 points, such that a line containing two of them passes through one more of them, they are by definition, collinear.
Thus (1) is true for n=3.
Let (1) be true for a given k points set. So in the given k-point set in space, all the points are collinear.
For an external point in the same space to pass through any line passing through two of the points in the k-points set in P(k), it has to pass through the line passing through all of them.
Thus the only way to expand the given set of k points to a set of k+1 number of points, is to include an external point which is collinear with the existing points.
Hence all k+1 points of the expanded set are collinear.
Thus if (1) is true for n=k, then it is true for n=k+1
Thus by principle of induction, (1) is true for all n>=3.
Problem : Ctrl+Space does not trigger auto-suggest in Eclipse, on Kubuntu 14.04.
Reason: IBus captures ctrl+space keystroke.
1. Right click Ibus Icon from task bar
2. From the preferences, remove the option Ctrl+Space as the short cut for next command. (Or simply quit this program every time you boot up)
Tab selector (tab bar) can be made to appear on the top side of terminal in Konsole as follows -
- Menu Bar => Settings => Configure Konsole
- Select TabBar pane
- Set “Tab bar position” to “Above Terminal Area”
As you might have noticed now, you can make Konsole open new tabs adjacent to current tab by setting “New tab behavior” to “Put New Tab After current tab”
This is how I installed skype on my Kubuntu 13.10 system -
- Open bash/Konsole
- Run: sudo add-apt-repository “deb http://archive.canonical.com/ $(lsb_release -sc) partner”
- Run: sudo apt-get update
- Run: sudo apt-get install skype
Problem: The URLs including development/local ones are forwarded to Yahoo Search page by a Conduit plugin via their search.conduit portal (as seen from status bar message)
Reason: Spyware by Conduit (Anything that changes the settings of your computer without explicitly asking you, and sends out information from your computer without letting you know – is a spyware/malware indeed)
Fix: Available here : “http://www.techsupportall.com/how-to-remove-conduit-search/” (Download and run Adware-Removal-Tool-v3.8.exe from there. Verified working.)
Problem: When running mvn jetty:run to launch jetty, Appfuse throws error “java.util.zip.ZipException: invalid distance too far back”
Reason: One of the jars in maven repository is corrupted.
Fix: (Kind of nuke all, but works)
- Delete repository folder from .m2 folder : rm -r ~/.m2/repository
- Add any local jars you may have added to repository
- From AppFuse project folder, run : mvn:jetty-run
- Find the offending jar from the exception stacktrace.
- Delete the folder containing the jar inside ~/.m2/repository
- Run “mvn install” or mvn:jetty-run.
Problem : The map instance created using util:map does not give expected keys in Java class when obtained using autowiring.
Reason: When autowired, spring searches for the first Map with same name, and the result might not be the Map configured by the developer in the context xml.
Fix: Instead the tag @Autowired, use @Resource(name=”mapName”) for the map instance in the caller class.
All done and dusted, when you are ready to upload your application to the Android Market for the world to use it, you’ll have to follow a few steps below.
- Remember to change the AndroidManifest.xml file to the specifics such as min SDK version etc.
- In your manifest file, switch to the manifest tab. In the Exporting section is an option Use the Export Wizard to export and sign an apk.
- Select the Project that you want to export and click Next.
- Now for the Keystore selection, this provides a certificate to you application. Since you do not have a keystore yet, click on Create New Keystore. Remember to save this certificate in a safe location and not lose it. If you decide to update your application in the future, you’re going to need the same certificate. If you somehow happen to lose this certificate, you’re going to have to upload a whole new application again.
- Select the location and provide and confirm a password.
- In the Key Creation section provide an Alias so as to refer to the key being created.
- Provide a password. For validity in years put in 25.
- Fill in your First and Last Name. You can also go ahead and provide the other details if you want.
- In the next step, browse and select a location for you final apk file.
- When you look for an application on the Android Market, you must have seen a lot of pictures and graphics on the application page. You can choose to create those graphics for your application too. I’m going to skip this step.
- Now got to http://market.android.com/publish.
- Click on Upload Application towards the bottom. Follow the on-screen instructions and upload the apk and other necessary graphics.
You’re done!! Your application is up and running on the Market!!