How to draw change-of-coordinates notation in LaTeX? For a homework assignment, I'd like to include something like the following image in my document as a single term in an equation:   I know I could simply use another variable name, but my professor prefers this notation. My main problem is placing the P above the B \leftarrow C; is there a way to do this?
Arrow with plus/minus sign on top Below you can see an arrow with a plus and a minus sign on top of it. How is this achievable in latex? Further possibilities: arrow with only plus and arrow with only minus sign on top. How can I do this?
tar files only, no directories Prove that $\int_0^\infty \frac{x\,\log x}{(1+x^2)^2} = 0$ I have been asked to prove that $$\int_0^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx = 0$$  I tried proceeding as follows: \begin{align} \int_0^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx &amp;= \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx + \int_1^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx \\ &amp; = \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx + \int_{u=1}^0 \frac{u\,\log u}{(1+u^2)^2}\,du\quad (\text{where }u=1/x) \\ &amp;= \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx - \int_0^1 \frac{u\,\log u}{(1+u^2)^2}\,du \\ &amp;=^? \int_0^1 0 \,dx \\ &amp;= 0 \end{align}  But I think that I should first show that the improper integral $\int_0^1 {x\,\log x}/{(1+x^2)^2}\,dx$ is convergent.  Do I need to show it? If yes, how to do so?  (I have been taught basic methods such as comparison, limit comparison and Dirichlet test.) How should you interview for QA *Automation* positions? Can we know the fundamental nature of space and time? How can I change a field cardinality? I have two content types: Person and Office with existing content. The Person content type has an entity reference field to Office nodes, which only allows a single reference. I need to change this to allow an unlimited number of Office nodes to be selected.  Drupal 8 prevents changing the cardinality of the field, and it gives the following error message.     There is data for this field in the database. The field settings can no longer be changed.   At first, I thought I needed to write a migration, but they seem to be aimed at bringing data into a Drupal instance. Then, I read about writing  (also ).  I discovered a migrate-plus plugin that adds a couple pages me to export the field storage settings, change the cardinality, and re-import the settings. Given that I'm going from cardinality of 1 to unlimited, I could imagine this change wouldn't destroy data. And my empirical data suggests this is the case. I've now made this change via the migrate-plus screens as well as via configuration export and import and the data is still in the field.  Is this a safe operation, or should I write an update function? When documenting location, should current or historical place names be used? When I am documenting records, do I use the name of a city or country as it is called today or as it was in the year an ancestor was there?  Related to: ERROR: cuvid requested, but not all dependencies are satisfied: cuda/ffnvcodec I am trying to compile FFMPEG with Nvidia Cuda support, on Debian 9.3. Parameters, what I am using:  --enable-cuda --enable-cuvid --enable-nvenc --extra-cflags=-I/usr/local/cuda/include --extra-ldflags=-L/usr/local/cuda/lib64 --enable-gpl --enable-libx264 --disable-x86asm --enable-libx265 --enable-libfdk-aac --enable-nonfree   Nvidia Cuda with drivers are installed. When I try to configure ffmpeg, it says:  ERROR: cuvid requested, but not all dependencies are satisfied: cuda   Newer ffmpeg will show a similar, re-worded message:  ERROR: cuda requested, but not all dependencies are satisfied: ffnvcodec   I absolutely don't know why I'm having this issue, because I am compiling ffmpeg on all of our trans-coding servers. How to disable Macbook Pro internal display when using external monitors I have a Late 2011 Macbook Pro 13" (running Lion - OS 10.7.4) which I use with a dual external monitor setup. I have two 21" LCDs that I use as external monitors. One uses the built in Mini Display Port on the Macbook Pro and the other is connected via a USB to DVI adapter (Diamond DVILink).  The problem I am having is with the 13" built in display of the Macbook Pro. I would like it to remain closed and unused (I prefer the dual monitors at same resolution), however I cannot seem to "disable" the display in the display preferences applet. So even though the Macbook Pro lid is closed the operating system still makes use of the display, even though the apple logo on the back of the display is unlit. I have tried to move the laptop screen off out of the way in display preferences but I find when I "sleep" my setup that any applications that remain open move to the laptop screen. This makes it difficult to locate the application windows when resuming from sleep.    Does anyone know of a way to disable the laptop built in display so it is not seen by the OS when external monitors are being used? Accelerating electrons via microwaves In Synchrotrons I think they use microwaves to accelerate the electrons bundles that fly through-how does putting a microwave through a cavity accelerate an electron? I know that the Electric and Magnetic fields could and do have an effect on the electron, but if the radiation is moving parallel to the electron, wouldn't the force be perpendicular? Also, how exactly does the microwave radiation look in the vacuum chamber? Are the two ends constant nodes? Compute $\sum_{n=1}^\infty \frac{1}{(n+1)\sqrt[]{n}+n\sqrt[]{n+1}}$ Compute   $\sum_{n=1}^\infty \frac{1}{(n+1)\sqrt[]{n}+n\sqrt[]{n+1}}$ Evaluating the integral $\mathop{\lim}\limits_{n \to \infty} \int_{-1}^{1} f(t)\cos^{2}(nt) \ dt $ Given that $f\colon [-1,1] \to \mathbb{R}$ is a continuous function such that $ \int_{-1}^{1} f(t) \ dt =1$, how do I evaluate the limit of this integral: $$\lim_{n \to \infty} \int_{-1}^{1} f(t) \cos^{2}{nt} \,dt$$   What I did was to write $\cos^{2}{nt} = \frac{1+\cos{2nt}}{2}$ and substitute it in the integral so that I can make use of the given hypothesis of $\int_{-1}^{1} f(t) \ dt =1$. So the integral becomes,   \begin{align*} \int_{-1}^{1} f(t)\cos^{2}{nt} \ dt = \int_{-1}^{1} f(t) \biggl[\frac{1+\cos{2nt}}{2}\biggr] \ dt  &amp;  \\  \hspace{3cm} = \frac{1}{2}\int_{-1}^{1}f(t) \ dt + \int_{-1}^{1} \frac{f(t)\cos{2nt}}{2} \ dt \end{align*}  But I don't really know how I can evaluate the second integral and also I can't realize as to why that integral condition on $f$ has been assumed. Moreover without assuming that condition on $f$ is it possible to evaluate this integral? If yes, then what would the answer be? How do lexical closures work? Is it possible to create bootable linux on USB drive using windows command line? No chatoutput in tellraw onClick Event I'm using a /tellraw command to display some kind of menu to the player. How can I react to the clickEvent without posting something in the chat? When I try using the scoreboard, I get an output in my chat like this Set score of &lt;objective&gt; for player &lt;playername&gt; to &lt;value&gt;. The same thing happens, when trying /setblock, /entitydata or /trigger Is there any way, I can execute a command which commandblocks can detect without writing a success/fail message in the chat?  the only commands without a "real" response message I found were /say, /tell and /tellraw. But all of these are impossible to detect(also, they write something else in the chat). Topologist's sine curve is connected I just came across the example of the topologist's sine curve that is connected but not path-connected. The rigorous proof of the non-path-connectedness can be found .   But how can I prove that the curve is connected? To be honest, even intuitively I am not being able to see that the curve is connected. I am thinking if it is proved that the limit point of $\sin(1/x)$ as $x \to 0=0$, then it would be proved. But, why is this true? IMO, this limit doesn't exist. Intuitively also, it seems that the graph would behave crazily and not approach a particular value as a tends to $0.$  EDIT (Brett Frankel): There are a few different working definitions of the topologist's since curve. For the sake of clarity/consistency, I have copied below the definition used in the linked post: $$ y(x) = \begin{cases} \sin\left(\frac{1}{x}\right) &amp; \mbox{if $0\lt x \lt 1$,}\\\ 0 &amp; \mbox{if $x=0$,}\end{cases}$$ Is the differentiation operator an open mapping? Consider the vector spaces $C([0, 1])$ and $C^1([0, 1])$ with norm, \begin{align*} \displaystyle \|f\|_\infty=\sup_{x\in [0, 1]}|f(x)|, \end{align*} and let $T:C^1([0, 1])\to C([0, 1])$ the operator given by, \begin{align*} \displaystyle Tf=f'. \end{align*} Is $T$ defined this way an open mapping? If it's, how to prove it? Everybody perhaps knows of Jon Favreau's movie  which is going to be released in 2016. But quite recently I learned (and to my utter surprise) that another Jungle book movie is coming in 2017 and it's called .   Now, this movie has big names too with Christian Bale as Baghera, Benedict Cumberbatch as Sher Khan, Cate Blanchett as Kaa and a young Indian boy in the live action role as Mowgli.   How often does this happen, with two big productions with big cast making a movie on the same subject? I want to ask why this is happening and if there are any really significant differences between the two (apart from the cast, of course). Who are the users that work for Stack Overflow? I'm trying to go into an irb session with specific environment variables from a file with this command:  $ env $(cat env.sh) irb   But when I try press Tab after I type env. to complete it, I get this following error:  $ env $(cat env.-bash: unexpected EOF while looking for matching `)' -bash: syntax error: unexpected end of file   Another interesting thing is that if I'm logged in as root, this error does not occur.   Here's the output of find ~ -uid 0:  $ find ~ -uid 0 /home/(redacted)/.rpmdb /home/(redacted)/.rpmdb/Group /home/(redacted)/.rpmdb/Conflictname /home/(redacted)/.rpmdb/Installtid /home/(redacted)/.rpmdb/Sha1header /home/(redacted)/.rpmdb/Providename /home/(redacted)/.rpmdb/__db.002 /home/(redacted)/.rpmdb/Requirename /home/(redacted)/.rpmdb/Sigmd5 /home/(redacted)/.rpmdb/__db.001 /home/(redacted)/.rpmdb/Obsoletename /home/(redacted)/.rpmdb/.dbenv.lock /home/(redacted)/.rpmdb/Name /home/(redacted)/.rpmdb/Basenames /home/(redacted)/.rpmdb/Triggername /home/(redacted)/.rpmdb/Packages /home/(redacted)/.rpmdb/Dirnames /home/(redacted)/.rpmdb/__db.003   Can anyone explain to me why this is happening and if so, how do I fix it in when I'm not a root user? How to calculate information included in R's confusion matrix I want know the formulas for the information highlighted in the confusion matrix in the picture below: