Can we write the empty set as a union of open intervals?
Why is the empty set considered an interval?
Why do birds sitting on electric wires not get shocked? What's the difference between:        Have someone do something     Make someone do something   Get someone do something.      To have someone do something means to cause someone to do something. To make someone do something means to force someone to do something. To get someone do something means to persuade someone or trick someone to do something.  This is what the dictionary says but I saw they're sometimes interchangeably used, so dictionary definitions are not enough. I wish to calculate the volumes of a truncated cone whith asymetry over all axes and ellipses as base and top - how do I do that?  I have height and radii of the corresponding ellipses. (I assume correctly I need the geometric means of the 2 radii describing the ellipsis.)  Kind Regards Why is it possible to get significant F statistic (p<.001) but non-significant regressor t-tests? How do I start Chrome using a specified "user profile"? No cycle containing edges $e$ and $g$ implies there is a vertex $u$ so that every path sharing one end with $e$ and another with $g$ contains $u$ There is a proof in my textbook for the following claim, which doesn't make a whole lot of sense to me.  My annotations are in bold.  Could someone perhaps elaborate on what's going on?  Claim.  If there does not exist a cycle containing edges $e$ and $g$ then there exists a vertex $u \in V (G)$ such that every path in $G$ sharing one end with e and another with $g$ contains $u$.  Proof: The claim trivially holds if $e$ or $g$ is a loop OK, so we assume that neither is. Let $P$ with vertex set $v_1, v_2, . . . , v_k,$ in order, be a path with $e$ joining $v_1$ to $v_2$ and $g$ joining $v_{k−1}$ and $v_k$. Let $f_i \in E(P_i)$ be the edge with ends $v_i$ and $v_{i+1}$. Let $j$ be chosen minimum so that no cycle in $G$ contains $e$ and $f_j$ We can do this because we know that at least the edge $g$ will not create a cycle by assumption, right?. We will show that $u = v_j$ satisfies the claim.  Suppose not. Let $C$ be a cycle containing $e$ and $f_{j−1}$ What if $f_j = e$?  then what cycle? a single vertex? and let $P′$ be a path from an end of $e$ to an end of $f$ Not sure what the book meant by $f$ here, any guesses? avoiding $u$. Choose a subpath $Q$ of $P′$ with one end in $V (C)$ and another in ${v_{j+1}, v_{j+2}, . . . , v_k}$ as short as possible. Then $C \cup Q \cup P$ contains a cycle containing both $e$ and $f_j$, a contradiction. (The last statement requires some case checking.) This ending seems a bit abrupt and non-obvious to me  Thanks for the help To date I have had an Apple ID tied to my personal email address. It uses my personal paypal account for the purchases, and currently it has a credit balance thanks to some Christmas iTunes gift cards.  With the App Store recently being released, I'd like to buy an OS X app for business purposes. I don't want to use the credit balance on my personal Apple ID account, so I was thinking about creating a new Apple ID just for when I buy apps that I use for business. This way I could have the business Apple ID tied to my business paypal account.  However, I'm concerned that there might be some gotchas with using two different Apple IDs on the same MacBook by the same user. Before I just try this blindly, does anyone have any experience or know of any gotchas? How does the twitter bot work? My months and days in my File Manager are in German, how do I change that? On the photo you see 'Dez' : It means Dezember (German for December). I also saw one day Mittwoch (German for Wednesday). How do I change that German issue? This is only my File Manager and not completly but partially. Weird. Elementary central binomial coefficient estimates How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}&lt;\binom{2n}{n}&lt;\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ &gt; 1 ?     Does anyone know any better elementary estimates?   Attempt.  We have $$\frac1{2^n}\binom{2n}{n}=\prod_{k=0}^{n-1}\frac{2n-k}{2(n-k)}=\prod_{k=0}^{n-1}\left(1+\frac{k}{2(n-k)}\right).$$ Then we have $$\left(1+\frac{k}{2(n-k)}\right)&gt;\sqrt{1+\frac{k}{n-k}}=\frac{\sqrt{n}}{\sqrt{n-k}}.$$ So maybe, for the lower bound, we have $$\frac{n^{\frac{n}{2}}}{\sqrt{n!}}=\prod_{k=0}^{n-1}\frac{\sqrt{n}}{\sqrt{n-k}}&gt;\frac{2^n}{\sqrt{4n}}.$$ By Stirling, $n!\approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$, so the lhs becomes $$\frac{e^{\frac{n}{2}}}{(2\pi n)^{\frac14}},$$ but this isn't $&gt;\frac{2^n}{\sqrt{4n}}$. I have some troble understanding the Kernel density estimation.  If a consider the next example:  s&lt;-0 z&lt;-density(rnorm(25)) f&lt;-function(i)approx(z$x,z$y,xout=i)$y for(i in seq(min(z$x),max(z$x),length.out = 10000)) s&lt;-s+f(i) s   I expected getting that s is approximately equals to one since I am considering an approximation for the integral of the Kernel density function (that I suppose is also a density). However, I don't get the expected result. Why it doesn's sum one?  The idea behind this is that I wanted to use the Kernel as the density of my data points. I was interpolating to get my explicit function and use with other x values. Or do you know another way to get a density function estimated by fitting? I am looking for a convenient way to set the background of tabular cells. I'd like to do two things:  (1) Set the background shaded in a color (2) Set the background hatched  I learned that the TikZ package provides means to do this but I am unable to come up with a solution that does not use matrices (such as here: ) but instead uses the tabular package.  Here's what I'd like to do:  \documentclass{article} \begin{document} \begin{tabular}{|l|l|l|} \hline   1 &amp; \setBGGray 2 &amp; 3 \\\hline   4 &amp; 5 &amp; 6 \\\hline   7 &amp; \setBGHatched 8 &amp; 9 \\\hline \end{tabular} \end{document}   The output should look like this:    Can someone please help me achieving this using tabular (or tabularx) and TikZ? Hi can you help me solve this exercise? Thanks. Let $f: [0;+\infty) \to \mathbb{R}$ be nonnegative and continuous function. Suppose $\forall a &gt; 0$  $\sum_{n=1}^{\infty} f(na)$ is convergent. Prove that $\int_{0}^{\infty}f(x) dx$ is convergent. I tried to solve it by using the Riemann sum, but for fixed a it doesn't work. I have no other ideas. How can I switch a 32-bit installation to a 64-bit one? I've got a 32-bit Ubuntu installation running on 64-bit hardware. Now that , I would like to switch to 64-bit without having to reinstall the OS.  This is one of the :  Shawn installed his system using the 32-bit version of Ubuntu, but his hardware is 64-bit and he wants to switch over. He manually installs the amd64 versions of dpkg and apt, replacing the i386 versions and changing which architecture is used as the default; then he installs the amd64 ubuntu-minimal package; then he installs the amd64 ubuntu-desktop package. Over time the remaining i386 packages are replaced automatically on upgrade.  However, when trying to follow the instructions in there, I cannot find any 64-bit version of dpkg or apt.  Did this user story got implemented in a different way in the final spec, or do I need to do something differently?  In short, how can I switch my 32-bit installation to 64-bit? When —the bartender at the Cantina at Mos Eisley spaceport in the first Star Wars—says:     “We don’t serve their kind here.”   In reference to C-3PO and R2-D2 entering the bar, it’s certainly a bit prejudicial, but are these commonly held views in this universe? We don’t see it happen anywhere else in the films but is it implied elsewhere? Restore deleted photos What does "Change" represent in the User Leagues? What does the Change field represent on the  for SO? Refer to the highlighted areas in the image below:   Rank: the user's ranking for the selected time period (week, month, quarter, year) Change: +/- change representing... ??? Total Reputation: the user's total reputation [Time Period] Reputation: the user's reputation for the selected time period (week, month, quarter, year)  So what is the Change field measuring? Solve Euler Project #9 only mathematically - Pythagorean triplet The "Euler Project" problem 9 () asks to solve:  $a^2$ + $b^2$ = $c^2$ a + b + c = 1000  I find answers solving it with brute-force and programmatically, but is there a way to solve the problem ONLY mathematically? Can someone help, please?  Problem as explained in Project Euler website:  A Pythagorean triplet is a set of three natural numbers, a &lt; b &lt; c, for which,  a^2 + b^2 = c^2 For example, 3^2 + 4^2 = 9 + 16 = 25 = 52.  There exists exactly one Pythagorean triplet for which a + b + c = 1000.  Find the product abc. How to use a string column with pgfplotstable to label datapoints in a pgfplot? I have a comma-separated list of data points. They should be plotted as a scatter plot, and should be labeled using data from the .csv file. Having specified the first column of my file as string type, I still keep running into errors saying Could not parse input [...] as a floating point number, sorry.  Given a CSV file of the following format, how can I read Name and Z as labels?  Name,X,Y,Z ODIN,250,45,2001   Here is my MWE:  \documentclass{article}  \usepackage{tikz,pgfplots,pgfplotstable}     \pgfplotsset{compat=1.5.1}  \begin{document}  \pgfplotstabletypeset[columns/Name/.style={string type}]{table.csv};  \begin{tikzpicture}     \begin{axis}[     visualization depends on=\thisrow{Name} \as \labela,     visualization depends on=\thisrow{Z} \as \labelb,     every node near coord/.append style={font={\tiny}},     nodes near coords=\labela/\labelb,     nodes near coords align={horizontal},]      \addplot[only marks] table [x=X,y=Y,col sep=comma] {table.csv};      \end{axis} \end{tikzpicture}  \end{document}   Note that if the data in column Name is just floats it should work fine. How do I set my level, unlocked weapon equip slots and playthrough mode?