The number of regions into which a plane is divided by n lines in generic position
Show that $n$ lines separate the plane into $\frac{n^2+n+2}{2}$ regions
Why does the 'b' in absorb change to a 'p' in absorption? In school, I have always seen sets. I was watching a video the other day about functors, and they started talking about a set being a collection, but not vice-versa. I also heard people talking about classes. What is their relation? Some background would be nice.   It has to do with something called Russell's paradox, but I don't know what that is.   I think that the difference between a family and a set is that the former is a function and the latter is a set. Is this right? Why doesn't my Bash script recognize aliases? Though I never quite understood what those numbers were and they were quite ugly, they were the only reason I ever visited the reviews page.   Can we have similar functionality back on the top bar please? Apply empty style to the entire bibliography How should ambiguous tags be dealt with? How was Hedwig able to find Sirius? In the beginning of Goblet of Fire, Harry sends Snuffles a letter.  Hedwig somehow finds him after the extremely specific command from Harry, "Just find him, okay?"   Theories:   Hedwig has a scanning system, like Baymax in Big Hero 6. Hedwig got super lucky. Hedwig flies really, really fast.   Now I know that owls are intelligent beings in the Harry Potter canon. But.... those ideas are just as crazy as the next.  So how did Hedwig do it? Display Blank when Referencing Blank Cell in Excel 2010 What determines how much experience I get from a hearthstone match in play mode? Sometimes I win handily and get a little experience, sometimes I squeak out a victory and get a lot.  How is experience earned? Is there a function in python to split a word into a list? Why use hex? Hey! I was looking at this code at   I noticed that in some situations they used hex numbers, like in line 134:  for (j = 1; val &amp;&amp; j &lt;= 0x80; j &lt;&lt;= 1, q++)   Now why would they use the 0x80? I am not that good with hex but I found an online hex to decimal and it gave me 128 for 0x80.  Also before line 134, on line 114 they have this:  small_n = (n &amp; 0xffff0000) == 0;   The hex to decimal gave me 4294901760 for that hex number. So here in this line they are making a bit AND and comparing the result to 0??  Why not just use the number? Can anyone please explain and please do give examples of other situations.  Also I have seen large lines of code where it's just hex numbers and never really understood why :( Schrödinger Equation for Two Electrons? I'm having trouble finding a simple answer to this question (maybe because there isn't one), but I'm just confused about how the Schrödinger Equation would look for two electrons. I understand that it would exist in 6 dimensional configuration space, but how does the potential look? It's confusing that the potential would be different for each electron depending on where the other one is. How can I use Bengali script in an English document? How can we find premade adventures for a virtual tabletop? Me and a couple of friends are just getting into P&amp;P rpgs and are really excited to try something new. However, we have a problem that we live quite far apart from each other. Therefore we'd like to use one of excellent virtual tabletops that are available. But none of us have any real experience with creating an adventure and would thus like to use a premade and beginner friendly adventure. But we're having a hard time finding one with premade story, maps etc in an online format. Most seem to be printed books and the Roll20 marketplace hasn't really gotten started yet. So what we're looking for a is online way to play a premade adventure to get us started in the RPG world.  How can we find these adventures for a VTT? Knots in Smoothing Splines In , there's this line under the section describing Smoothing Spline's tuning parameter $\lambda$:       In fitting a smoothing spline, we do not need to select the number or   location of knots - there will be a knot at each training observation,   $x_1,...,x_n$.   But, in my opinion wouldn't that be overfitting?     Reason for such a claim: If there are more points which act as knots in the Spline fit, then the range for each slot for fitting is very small. Thus each change in the data, is recognized and imititated in the fit. Thus fitting the changes, which are random, and do not mean anything. Thus overfitting.    Possibilities: We know that cubic splines and Smoothing splines are different. The basic differences are:     Smoothing splines try to minimize the function $$\sum_{i=1}^n(y_i-g(x_i))^2+\lambda\int g''(t)^2dt$$ Smoothing splines shrink the parameters.   Moreover, Smoothing Splines are basically natural cubic splines, and thus they're smooth too.  So, how can these differences( and properties) save the Smoothing Spline from overfitting? Is there anything else that I'm missing here? Does the command pwd in a shell script output the directory the shell script is in? How can I make schematic of this board? Why should a CTE start with a semi-colon? I was just looking at a post on  where Aaron Bertrand proposes using a CTE instead of a numbers table, which is an elegant way of performing the task at hand.  My question is, why does the first line of the CTE begin with a semi-colon?    ;WITH n AS (SELECT TOP (10000) n FROM    (SELECT n = ROW_NUMBER() OVER     (ORDER BY s1.[object_id])     FROM sys.all_objects AS s1     CROSS JOIN sys.all_objects AS s2   ) AS x ORDER BY n ) SELECT n FROM n ORDER BY n; -- look ma, no gaps!   Is this to ensure the WITH statement does not get parsed into a previous SELECT or something?  I see nothing in SQL Server 2005 BOL about using a semi-colon prior to the WITH. What happens if you call erase() on a map element while iterating from begin to end? How do  I create a desktop wallpaper slideshow with my own set of images?   With the images coming from a pictures folder, or pre-selected With the images displayed in order, or randomly With the images changing every few minutes, or after random periods of time