Proof: Determinant of a block matrix
Determinant of a block lower triangular matrix
"Abusing" bounties for unlimited profit Six days ago I put a +50 bounty on  which wasn't asked by me, but to which I'd previously posted a pretty decent answer. Since then I've earned 130 reputation from upvotes on my answer, netting me +80 reputation and a bronze badge from no work.  It seems possible to slowly generate free reputation by "investing" in your answers like this. I expect I could repeat this perpetually and continue to profit from it.  This strikes me as an abuse of the bounty system. Should anything be done to prevent it?  EDIT: For more concrete questions:   If I put another bounty on this question after this one expired, would that be considered abuse? What if I did it several times? ( the answer was  "yes", but the author was doing so to promote his own project.) What if I didn't do it to the same question, but was constantly promoting questions I'd answered, in order to promote my answers? Tempdb Add files require Restart Does adding Tempdb files require a restart? We would like to Add files on the same drive, Not Delete and Not modify. We tested in Dev and QA environment, restart was not required. However for production I would like to be sure everything works properly. Some are stating conditions, not always necessary, sometimes necessary, I am reading resources below:  ALTER DATABASE [tempdb] ADD FILE ( NAME = N'tempdev2', FILENAME = N'G:\tempdb2.ndf' , SIZE = 12288000KB , MAXSIZE = UNLIMITED, FILEGROWTH = 0 ) GO ALTER DATABASE [tempdb] ADD FILE ( NAME = N'tempdev3', FILENAME = N'G:\tempdb3.ndf' , SIZE = 12288000KB , MAXSIZE = UNLIMITED, FILEGROWTH = 0    Do You Need an Outage to Add tempdb files? Brent Ozar  "Technically, not always. But practically speaking, yes. If you have to shrink files, that’s very difficult while SQL Server is in use, and we’ve seen cases where anti-virus/file control tools didn’t act up until SQL Server was restarted after the files were added."   "Once the additional files have been created, it’s not always necessary to perform a service restart.  However, in many production environments, simply modifying the filesize of the existing TempDB datafile (tempdev) is not possible without restarting services first. "  The following resources are saying no restart required: On the one hand, I see very few posts in the review queue when I visit.  On the other hand, I see a lot of pretty awful first posts sitting on the front page.  This leads me to suspect that robo-reviewers are blowing through the queue.  Given how empty the queue is, could you please increase the number of people who have to check in before a post leaves the queue?  Of course, some honey-pot solution to detect and disable the auto-clickers would be even better. I would like to search my table having a column of first names and a column of last names.  I currently accept a search term from a field and compare it against both columns, one at a time with       select * from table where first_name like '%$search_term%' or      last_name like '%$search_term%';   This works fine with single word search terms but the result set includes everyone with the name "Larry". But if someone enters a first name then a space, then a last name, I want a narrower search result. I've tried the following without success.         select * from table where first_name like '%$search_term%' or last_name      like '%$search_term%' or concat_ws(' ',first_name,last_name)      like '%$search_term%';   Any advice?  EDIT:  The name I'm testing with is "Larry Smith".  The db stores "Larry" in the "first_name" column, and "Smith" in the "last_name" column.  The data is clean, no extra spaces and the search term is trimmed left and right.  EDIT 2:  I tried Robert Gamble's answer out this morning.  His is very similar to what I was running last night.  I can't explain it, but this morning it works.  The only difference I can think of is that last night I ran the concat function as the third "or" segment of my search query (after looking through first_name and last_name). This morning I ran it as the last segment after looking through the above as well as addresses and business names.    Does running a mysql function at the end of a query work better than in the middle? I have been developing in the windows space with Visual Studio for a while now with work, but I have also been using Ubuntu for a while and am keen to get into some software development for linux.  I should also note. I am not looking for .NET and I am aware of mono. I am also familiar with c++ development and some python, so the language isn't so much relevant as the "all in one" aspect.  I was interested to know if there is a useful all in one code/debug/design(gui) IDE similar to something like Visual Studio but for linux? I've installed Ubuntu 14.04 and I'm experiencing a fuzzy/blurry screen and bad support for multiple screens.  I realized that Ubuntu installs nouveau as default, but I need to install nvidia drivers, so I tried this:   sudo apt-add-repository ppa:ubuntu-x-swat/x-updates apt-get update apt-get install nvidia-current apt-get upgrade       Now that nvidia is installed, you need to blacklist the nouveau driver   so it don’t pop out when you will reboot. Create or edit the following   file:  nano /etc/modprobe.d/blacklist-nouveau.conf       …and add these lines at the end:  blacklist nouveau blacklist lbm-nouveau options nouveau modeset=0 alias nouveau off alias lbm-nouveau off    I checked via the software &amp; updates if nvidia driver got installed successfully, and it says the driver installed is nvidia 304.117  I then tried to run:  lshw -c video | grep 'configuration'   and the result is that the driver is still nouveau:  configuration: driver=nouveau latency=0   So how do I install and use the nvidia drivers? What is the most efficient way to write the old-school:  StringBuilder sb = new StringBuilder(); if (strings.Count &gt; 0) {     foreach (string s in strings)     {         sb.Append(s + ", ");     }     sb.Remove(sb.Length - 2, 2); } return sb.ToString();   ...in LINQ? General formula for evaluating integrals of the form $\displaystyle\int_0^\infty x^ae^{-bx}dx$ This question has been asked before, at least twice by people who were trying to cheat on the PROMYS Europe 2020 Application, and once in a  because the OP showed no effort. I haven't been able to solve it, so I'm going to to ask it again.  PROMYS applications have closed, and I hope the effort shown below is enough to keep this question from being closed.     The Mathematical Forest is grown in a two-dimensional plane, where   trees can only grow on points with integer coordinates. To start with,   there are no trees at all. The foresters plant the first tree at   $(0,0)$. Each year, they carry out tree planting according to the   following rule. If there is a tree on the point (𝑚,𝑛) but there are   no trees on the points $(m+1,n)$ and $(m,n+1)$, then they can choose   to remove the tree on $(m,n)$ and plant new trees on the points   $(m,n+1)$ and $(m+1,n)$. For an integer $k\geq1$, the $k$th diagonal   consists of all points $(m,n)$ with $m+n=k−1$. Is it possible for the   foresters to arrange their planting so that eventually there are no   trees on the first $2$ diagonals? What about the first $3$ diagonals?    $4$ diagonals? Can you generalize?   It's easy to get to a position with no trees on the first two diagonals.  (It only takes $4$ plantings.)  I can prove it's impossible to reach a position with no trees on the first $4$ diagonals, and I believe it's impossible to reach a position with no trees on the first $3$ diagonals, but I can't prove it.  For the $k=4$ case, I used a pagoda function, as in a peg solitaire problem.  For $k=1,2,3,\dots$ define the potential of a tree on the $k$th diagonal as $\frac1{2^{k-1}}$, and the potential of the forest as the sum of the potentials of all the trees in it.  When a tree of potential $\frac1{2^{k-1}}$ is removed, it is replaced by $2$ trees of potential $\frac1{2^k}$ so the potential of the forest never changes.  Initially, the potential of the forest is $1$.  Since there are $k$ trees on diagonal $k$, the potential of all points beyond the fourth diagonal is $$\sum_{k=5}^\infty\frac k{2^{k-1}}=\frac34&lt;1,$$ so it is impossible that there are no trees on the first $4$ diagonals.  The potential of the fourth diagonal is $\frac12$, so this argument doesn't show that it's impossible to have no trees on the first $3$ diagonals.  I've done a lot of experimenting, though, and I believe that the statement is true.  (I even wrote a little computer game so I could experiment quickly.)  The best I've been able to do is to get down to one man on the third diagonal.   The black circles are trees eligible for removal, and the gray circles are other trees.  You can see that there is a "traffic jam" in front of the lone tree on diagonal $3$, and there seems to be no way to clear it.  I've been trying to come up with an argument that reflects this traffic jam, but I haven't been at all successful. We know that after $n$ removals there are $n+1$ trees, and I've been trying to prove somehow that they can't all be sufficiently far from the origin to allow all trees to get past diagonal $3$, but I haven't come close.   In the diagram above, if the tree at $(1,1)$ were moved to $(0,2)$, and the tree at $(1,2)$ were moved to $(2,0)$, then it would be possible to remove the tree at $(0,2)$, emptying the second diagonal.  In the fictitious position, we still have one tree on diagonal $2$ and two on diagonal $3$.   So, I think an argument along the lines I was trying cannot just be concerned with the distance from the origin; it must somehow take into account how the trees interfere with one another.  I haven't been able to do this.    I'd be grateful for solutions, hints, or counterexamples.  I wasn't really sure what tags to attach.  Please add any that seem appropriate. Convergence of $A_nT$ to $AT$ in operator norm for compact $T$ $A_n:Y\rightarrow Z$ are operators that strongly converge to $A$. Also, $\|A_n\|_\text {op}\le c$ for $c&gt;0$. Given a compact operator $T:X\rightarrow Y$, I need to show that $A_nT$ converges to $AT$ in operator norm (all spaces in this question are Banach spaces).     I was unable to prove this and I also do not understand why we need to assume that $T$ is compact. Any ideas? Trichromelibrary was suddenly downloading from Google Play Store From nowhere, my friend got a notification that com.google.android.trichromelibrary was downloading together with a progress bar.  Why? Fantasy book series with non-human races and labyrinth At school, I really struggled to understand the concept of imaginary numbers.  My teacher told us that an imaginary number is a number that has something to do with the square root of $-1$.  When I tried to calculate the square root of $-1$ on my calculator, it gave me an error.  To this day I still do not understand imaginary numbers.  It makes no sense to me at all.  Is there someone here who totally gets it and can explain it?  Why is the concept even useful? "Thus" vs. "so" in formal English Expected value of sums Suppose we draw cards out of a deck without replacement. How many cards do we expect to draw out before we get an ace? How can I move an installed game from one Steam library folder to another? I use (and have it much bigger) drive D: to store all my user data and games, but I have installed the Steam client on drive C:. When I was installing a first game (Warhammer 40000), I've chosen to place my library folder on drive D: and the game went there.  But now as I've got installed the second game (Portal), I can see that it was put on drive C: despite to the choice I've done previously. When I open Steam - Settings - Downloads + Cloud - Steam library folders, I can see both installation places in the list. Can I merge them into one?  I would even agree to use the default drive C: library (as I've replaced it with a symlink already). When I press to remove one library it says it can only remove empty ones. What is the best way to copy text to the clipboard (multi-browser)? I have tried: function copyToClipboard(text) {     if (window.clipboardData) { // Internet Explorer         window.clipboardData.setData(&quot;Text&quot;, text);     } else {         unsafeWindow.netscape.security.PrivilegeManager.enablePrivilege(&quot;UniversalXPConnect&quot;);         const clipboardHelper = Components.classes[&quot;@mozilla.org/widget/clipboardhelper;1&quot;].getService(Components.interfaces.nsIClipboardHelper);         clipboardHelper.copyString(text);     } }  But in Internet Explorer it gives a syntax error. In Firefox, it says unsafeWindow is not defined. A nice trick without using : Creating your own Tinyurl style uid Sometime I see many application such as msn, windows media player etc that are single instance applications (when user executes while application is running a new application instance will not created).  In C#, I use Mutex class for this but I don't know how to do this in Java. Best books for mathematical background?