How many closed polygons $n$ lines make on a plane?(not duplicate)
Show that $n$ lines separate the plane into $\frac{n^2+n+2}{2}$ regions
Defining commands/abbreviations that contain numbers I tend to use \newcommand to define macros for symbols I use often, e.g. \newcommand{\hankel}{H^{(1)}_0}.  One inconvenience of this approach is the fact that command names cannot include numbers.  What I ideally want is to map something like \H0 to H^{(1)}_0.  I imagine this could be done with some macro expansion/snippet package, or one might use \newcommand to define a command that takes a parameter, etc.  Before I start experimenting, though -- is there a canonical way of dealing with this problem? How can I view Google Photos that are not in a Google Photos album? Go to  Add a few photos to a photo album Go back to All Photos   See all of the photos there, even ones you copied to the Album. No problem here, per se. At least not yet.  What if you want to delete photos that you did not copy to that album?  Where is the selector or filter that filters those "in an album already" photos from this main view?  Or is there a way to create such a filtered view such that I can then cherry-pick photos for deletion? What is the scientific name of this evergreen plant? This plant is found in India. It is an evergreen plant. Would love to know the scientific name. "The package lists or status file could not be parsed or opened" Everytime I sudo apt-get update it always not complete and will say in the end  E: Read error - read (5: Input/output error) E: The package lists or status file could not be parsed or opened.   Same goes, when I try to install something and do sudo apt-get install &lt;some app&gt;  What should I do ? I admit, that my drive has bad sectors, could that be the reason? Is there anything else I can do? Where does the extra kinetic energy of the rocket come from? How can I install a package on MikTeX? I need to install a LaTeX package on my MikTeX system.  Specifically, I am trying to make biblatex work, but I always get the error message:    ! LaTeX Error: File `biblatex.sty' not found.   I have changed the options to have the "Quick Build" command that includes biblatex too.   Code/commands are correct because it's working under Overleaf. I have a shapefile with date in a string field. I create a new field with date type and use the calculator to make the conversion. But the result only provide the date part, the time part is removed. Even when the preview show the correct conversion.  Here you can see the original string field time and the result date time2. And the use of the conversion function to_datetime() and the preview result at the end    If you go to the layer properties you can change the edit mode to date/time. Then if go to the data and try to edit time2 you get the current date/time. So is possible save date and time in the same field. But the update tool still doesn't work as expected      As J.R said. Even when you can change QGIS edit to allow Date/Time the save to shapefile will discard the time. Distribution of a ratio of uniforms: What is wrong? Suppose that  $X$ and $Y$ are two i.i.d. uniform random variables on the interval $[0,1]$  Let $Z=X/Y$, I am finding the cdf of $Z$, i.e. $ \Pr(Z\leq z) $.  Now, I came up with two ways of doing this. One produces a correct answer consistent with the pdf here: , the other does not. Why is the second method wrong?  First Method  $\newcommand{\rd}{\mathrm{d}} \Pr(Z\leq z) = \Pr(X/Y\leq z) = \Pr(X\leq zY) = \int^{1}_{0}\int^{\min(1,zy)}_{0} \rd x \rd y = \int^{1}_{0}\min(1,zy)\ \rd y$ $  = \left\{      \begin{array}{lr}        \int^{1/z}_{0}zy\ \rd y + \int^{1}_{1/z} \rd y&amp; : z &gt; 1\\        \int^{1}_{0}zy\ \rd y &amp; : z \leq 1      \end{array}    \right. $ $  = \left\{      \begin{array}{lr}        1 - \frac{1}{2z} &amp; : z &gt; 1\\        \frac{z}{2} &amp; : z \leq 1      \end{array}    \right. $  This appears correct.  Second Method  $\Pr(X/Y\leq z) = \Pr(X \leq zY\ |\ zY \geq 1)\Pr(zY \geq 1) + \Pr(X \leq zY\ |\ zY &lt; 1)\Pr(zY &lt; 1) $ by total probability  $ = \Pr(X \leq zY\ |\ zY \geq 1)\Pr(Y \geq 1/z) + \Pr(X \leq zY\ |\ zY &lt; 1)\Pr(Y &lt; 1/z)$  Taking $z&gt;1$ yields $ (1)(1-\frac{1}{z}) + (\int^{1/z}_{0}\int^{zy}_{0} \rd x \rd y)(\frac{1}{z}) = 1-\frac{1}{z} + (\int^{1/z}_{0}zy\ \rd y)(\frac{1}{z}) = 1-\frac{1}{z} + \frac{1}{2z^{2}}$  This is already different. Why is this wrong?  Thanks! Can you turn weather off in Minecraft? Cardinality of the set of all real functions of real variable How does one compute the cardinality of the set of functions $f:\mathbb{R} \to \mathbb{R}$ (not necessarily continuous)? How to send mail from the command line? How to send mail from the command line? Where to get QGIS toolbar icons? Does anybody know where to download the QGIS toolbar icons? Why constructors will always have same name as of class and how they are invoked implicitly? When to comment or answer Prove that if $A$ is a set with $m$ elements, $B$ is a set with $n$ elements, and $A \cap B = \emptyset$, then $A \cup B$ has $m+n$ elements. "Notice: Undefined variable", "Notice: Undefined index", and "Notice: Undefined offset" using PHP When did The Doctor start running so much? Thanks to Netflix Streaming, I've been watching some random classic Doctor Who stories.  One thing that strikes me as different between the modern Who and old Who was that in modern Who, they do an awful lot of running.  It's become a sort of running gag that they've referenced in-universe.  In the classic Who I've seen, he tends to drive around quite a bit instead.  In addition, The Doctor in modern Who is incredibly reliant on his sonic screwdriver, which I often don't see at all in classic Who stories.  Did the Doctor running and the sonic screwdriver get introduced over time in classic Who, or were they both added as part of the modern Who revival? Does $x^{y^z}$ equal $x^{(y^z)}$? If so, why?   Why not simply apply the order of the operation from left to right? Meaning $x^{y^z}$ equals $(x^y)^z$?  I always get confused with this and I don't understand the underlying rule. Any help would be appreciated! Recently I needed to do weighted random selection of elements from a list, both with and without replacement. While there are well known and good algorithms for unweighted selection, and some for weighted selection without replacement (such as modifications of the resevoir algorithm), I couldn't find any good algorithms for weighted selection with replacement. I also wanted to avoid the resevoir method, as I was selecting a significant fraction of the list, which is small enough to hold in memory.  Does anyone have any suggestions on the best approach in this situation? I have my own solutions, but I'm hoping to find something more efficient, simpler, or both. Connectedness of the Hausdorff distance. Does anyone know a proof of connectedness of the Hausdorff distance?  I mean a proof of the following:  Theorem  If $(X, \rho )$ is a connected metric space, then $(F(X), d_h )$ is also connected.  Remark:  $F(X) = \{ A \subseteq X | A \text{ is nonempty, closed and bounded } \} $, and $d_h$ stands for the Hausdorff distance.