Triangularise a matrix
Short way for upper triangularization
What is reflection and why is it useful? What is reflection, and why is it useful?  I'm particularly interested in Java, but I assume the principles are the same in any language. Is there a keyboard command for 'show desktop'? Like Windows, I want to hit Super+D to jump to the desktop or minimize all windows. I don't see command like that in Keyboard Shortcuts. Any idea? Is the use of a hyphen between "non" and an adjective strictly necessary? Do I need to put a "-" between "non" and an adjective? As an example in physics we say "a non isolated photon", "non tight photon"... The context is very formal (paper publications and similar). Is there a general rule? Are there some differences between countries? Is a continuous function $ f\colon(0,\infty)\to R$, such that $f(x)\leq f(nx)$ increasing? Coupon Collector Problem with Batched Selections I am trying to solve a variation on the .  In this scenario, someone is selecting coupons at random with replacement from n different possible coupons. However, the person is not selecting coupons one at a time, but instead, in batches.  Here's an example problem formulation: There are 100 distinct coupons. A person makes selections in 10-coupon batches at random (each coupon with replacement). What is the expected number of batches necessary to have selected 80 unique coupons?  I have been able to determine the expected number of selections necessary to have selected k unique coupons when selecting one at a time (much like Henry's ), but I'm a bit stumped as to how to go about solving it with this particular wrinkle.  Any tips/guidance would be greatly appreciated. If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$? If I have a manifold $M$ and $\omega \in \Omega^q(M)$, considered an open subset $U$ of $M$, and choose a point $p \in M$, is it true that $(\omega|_{U})_p=0 \rightarrow \omega_p=0$? With $\omega|_{U}$ I mean the pull-back of $\omega$ by the embedding $F:U\to M$. The converse is obviously true, but this implication? In ubuntu 11.04 there was a workaround using gconf-editor and setting key /apps/gnome_settings_daemon/plugins/xrandr/active to false.  However there is no such key in gconf-manager using ubuntu 11.10 (gnome 3). I am animating a Rubik's cube, and as soon as I get to the second rotation the blocks that have previously been rotated start tripping out even though it should have been reset. I tried making the graph linear which fixed a lagging issue I had, but the blocks still start to spin instead of rotating around a single axis. I believe I have solved a famous open problem. How do I convince people in the field that I am not a crank? 1 Plot, 2 Scale/Axis How to run a program without typing the full path? System utilities such as ls don't need to have the full path typed before executing them. However, to execute Netbeans, I have to type /home/chris/netbeans-7.0.1/bin/netbeans. How can I make it so that I only have to type netbeans? Debugging a custom-bib bst file that crashes on book items I created my own bibliography file with custom-bib (makebst), but when I run it with a reference to a book I get:  $ bibtex PhD [...normal output] Database file #1: PhD.bib You can't pop an empty literal stack for entry oecd2001brc while executing---line 1504 of file wdsphd.bst   The only thing on that line is a high-level ITERATE call.   The style was created with these options:  %% merlin.mbs  (with options: `head,exlang,annote,seq-no,nm-rvvc,dt-beg,yr-par,xmth,tit-qq,qx,bt-qq,vol-bf,volp-sp,num-xser,pub-date,isbn,issn,doi,english,pp,ord,url,url-nl,nfss,,{}') %% english.mbs  (with options: `exlang,annote,seq-no,nm-rvvc,dt-beg,yr-par,xmth,tit-qq,qx,bt-qq,vol-bf,volp-sp,num-xser,pub-date,isbn,issn,doi,english,pp,ord,url,url-nl,nfss,,{}') %% merlin.mbs  (with options: `tail,exlang,annote,seq-no,nm-rvvc,dt-beg,yr-par,xmth,tit-qq,qx,bt-qq,vol-bf,volp-sp,num-xser,pub-date,isbn,issn,doi,english,pp,ord,url,url-nl,nfss,,{}')   The code for book items looks like:  FUNCTION {book} { output.bibitem   author empty$     { format.editors "author and editor" output.check     }     { format.authors output.nonnull       crossref missing$         { "author and editor" editor either.or.check }         'skip$       if$     }   if$   format.btitle "title" output.check   crossref missing$     { format.bvolume output       new.block       format.number.series output       new.sentence       format.publisher.address output     }     {       new.block       format.book.crossref output.nonnull     }   if$   format.edition output   format.isbn output   format.doi output   new.block   format.note output   fin.entry   write.url }   And this is the entry:  @book{oecd2001brc,  author = {OECD},  doi = {10.1787/9789264193550-en},  isbn = {9789264186903},  publisher = {OECD Publishing},  title = {{Biological Resource Centres: Underpinning the future of life sciences and biotechnology}},  url = {http://www.oecd-ilibrary.org/science-and-technology/biological-resource-centres\_9789264193550-en},  year = {2001} }   Is there a way to get a better stacktrace from bibtex? What I have now doesn 't really tell me what is going on. If it can be fixed simply, help would also be appreciated. Limit of the geometric sequence $\{r^n\}$, with $|r| < 1$, is $0$? Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$ As the title says I need to prove the following by induction: $$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$   When trying to prove that P(n+1) is true if P(n) is, then I get that:  $$\sum_{i=1}^{2^{n+1}} \frac{1}{i} = (\sum_{i=1}^{2^n} \frac{1}{i} ) +(\sum_{i=1}^{2^n} \frac{1}{2^n+i} ) \ge 1+\frac{n}{2} + (\sum_{i=1}^{2^n} \frac{1}{2^n+i} )$$ using the inductive hypothesis. And for P(n+1) to be true it would be sufficient to prove that  $$1+\frac{n}{2} + (\sum_{i=1}^{2^n} \frac{1}{2^n+i} ) \ge 1+\frac{n+1}{2}$$  or rewriting $$\sum_{i=1}^{2^n} \frac{1}{2^n+i} \ge \frac{1}{2} $$ and here I'm a bit stuck  Thank you for your help! Do I really need to replace my helmet every 2-3 years if it hasn't been in a crash? Why would a company care about the price of its own shares in the stock market? Because the default 'Profile Graph' tool was not sufficient, I wrote a custom VBA script that creates scaled and exaggerated profile cross-sections in ArcGIS but I need more features:   Visualize more than 1 surface Visualize vertical and directional boreholes Ability to symbolize / edit the output Linked with underlying data - changes in terrain (or profile 3D line) reflected in output   A few years ago I tested an extension by A-Prime software called 'CrossView For ArcGIS' but I am not sure if it is compatible with ArcGIS 10 (Contacted A-Prime but they have not yet responded)   Area there any alternatives? Other 3rd party software? A functioning script? that can do this? When should I use a List vs a LinkedList Should log file streams be opened/closed on each write or kept open during a desktop application's lifetime? Asserting that Atheism and Theism are both faith based positions I am recently testing an assertion that I have concluded namely that atheism is a faith based position just like theism is a faith based position.  The reason I arrived at this conclusion is that theists have no proof that God actually exists. It's purely a matter of faith. Yes, there maybe strong implicit proof that God exists in their opinion (scriptures, history, etc) but there is no concrete proof of God's existence. In my view, atheists also suffer from the same problem of providing a proof of their position. Specifically that they cannot prove that God is non-existent.  The typical rebuttal I get is that the burden of proof is on the theists. But I view this as a cop out and they hide behind the wall of burden of proof which is just a bias in the debate.  Given the above, I claim that both atheism and theism are positions based on faith. Would it be incorrect to claim that?