number of regions created by n lines (no 3 are concurrent but parallels are permitted) i have found a formula for the number of regions created by n lines. The only condition is that no three lines are concurrent. Parallels are permitted. The formula is R=1+n+p where n is the number of lines and p is the number of points created by these n lines.  Can someone show me how to prove this formula? Maybe with a proof by induction on l or with a sweep-line argument. Can someone explain me in detail how a proof containing a sweep-line argument works?
Trying to understand formula for counting regions of hyperplane arrangements in $\mathbb{R}^2$ There are up to $\binom{n}{2} + n + 1$ regions created by a hyperplane arrangement in $\mathbb{R}^2$ containing $n$ hyperplanes.  I want to understand this in a demonstrative way. Each hyperplane splits each region it passes in two regions. A newly added hyperplane therefore creates a new region each time it crosses another hyplerplane or the plane. So the number of regions should be $1$ (plane) $+ \binom{n}{2}$ (number of intersections of $n$ lines). But as I know the answer, I know that I am missing $n$ regions.   Do you know where my mistake is? Why do I have to add the number of lines to get the max. number of regions?
How do you send a HEAD HTTP request in Python 2? What I'm trying to do here is get the headers of a given URL so I can determine the MIME type. I want to be able to see if http://somedomain/foo/ will return an HTML document or a JPEG image for example. Thus, I need to figure out how to send a HEAD request so that I can read the MIME type without having to download the content. Does anyone know of an easy way of doing this? Why does Javascript getYear() return 108? multiple regression and multiple comparisons Say I fit a multiple regression of p explanatory variables. The t-test will allow me to check if any single one of those is significant ($H_0: \beta_i = 0$). I can do a partial F-test to check if some subset of them is significant ($H_0: \beta_i=\beta_j=...=\beta_k=0$).  What I often see though is that someone gets 5 p-values from 5 t-tests (assuming they had 5 covariates) and only keeps the ones with a p-value &lt; 0.05. That seems a bit incorrect as there really should be a multiple comparison check no? Is it really fair to say something like $\beta_1$ and $\beta_2$ are significant but $\beta_3$, $\beta_4$ and $\beta_5$ are not?   On a related note, say I run 2 regressions on 2 separate models (different outcome). Does there need to be a multiple comparison check for significant parameters between the two outcomes?  Edit: To differentiate from the similar question, is there any other interpretation to the p-values besides: "B_i is (in)significant, when adjusting for all the other covariates"? It doesn't seem that this interpretation allows me to look at every B_i and drop those less than 0.5 (which is similar to the other post).   It seems to me that a sure fire way to test whether B_i and Y have an a relationship would be to get a correlation coefficient p-value  for each covariate and then do a multcomp (although that would definitely lose signal).  Finally, say I computed the correlation between B1/Y1, B2/Y1 and B3/Y1  (thus three p-values). Unrelatedly, I also did a correlation between T1/Y2, T2/Y2, T3/Y2. I'm assuming the correct Bonferroni adjustment would be 6 for all 6 tests together (rather than 3 for the first group and 3 for the second group - and thus getting 2 "semi"-adjusted p-values). how to hide files from package explorer Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$   Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$.   First, I tried to show that $X \times X \setminus D$ is open using the fact that $X \times X$ is Hausdorff (because $X$ is Hausdorff), but I couldn't find an open set that contains a point outside $D$ and is disjoint to it... Can we have a way to edit bounty custom message? Why is the wild card character * so different between the zip and rm commands? I put together a script to do some file operations for me. I am using the wild card operator * to apply functions to all files of a type, but there is one thing I don't get. I can unzip all files in a folder like this  unzip "*".zip   However, to remove all zip files afterward, I need to do   rm *.zip   That is, it does not want the quotation marks. The unzip, on the other hand, does not work if I just give it the * (gives me a warning that "files were not matched").   Why is this different? To me, this seems like the exact same operation. Or am I using the wild card incorrectly?   in Unix do not really go into this, and I could not locate anything in the rm or zip docs.  I am using the terminal on a Mac (Yosemite). How do I remove the OneDrive folder from my File Explorer folder tree in Windows 10? I don't use or want to use OneDrive, but after upgrading to Windows 10 it's been installed automatically and added to the sidebar of every explorer window:    I can't seem to figure out how to get rid of it. I can't uninstall it like a normal app, and I can't seem to unpin it in Explorer's options.  I tried following  to getting rid of it, but that seems to only remove stored OneDrive files.  I've also tried following another  and , but I can't seem to edit group policies on my computer (maybe because I upgraded from Win7 Home Premium?).  All I would really like is to be able to remove the OneDrive shortcut from my explorer sidebar, though being able to uninstall OneDrive would be a bonus.  I'm not sure what I should try now. Is there a way to remove OneDrive from Windows 10? Can I bring a painted Cow skull to the US with me via plane? I bought this traditionally painted cow skull, a typical decoration item for the day of the dead. Since its bone (completely clean though) can I bring it with me on a plane as carry-on or special luggage, without having to declare it or having it confiscated?    [] Converse of mean value theorem I am wondering if the following converse (or modification) of the mean value theorem holds. Suppose $f(\cdot)$ is continuously differentiable on $[a,b]$. Then for all $c \in (a,b)$ there exists $x$ and $y$ such that $$ f'(c)=\frac{f(y)-f(x)}{y-x} $$ I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical consideration related to these to formalism (like tangent bundle, cotangent bundle, 1-form, 2-form, etc.).  I have , but I would like to go into more details about the [symplectic] geometrical and mathematical foundations of classical mechanics.  Additional note: A chapter about relativistic Hamiltonian dynamics would be a good thing. 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