I have a little trouble understanding how to write a mathematically notation for r = x%n. How should I write this in math notation if I want to get the remainder value, after dividing by $n$? $r$ is also not just an integer in this case, just the rest of $x$ (double) divided by $n$ (integer), &quot;$r = \operatorname{mod}(5.4, 3) = 2.4$&quot; in this case. I think $x \equiv r \mod n$ means something different. Edit: After thinking a while, I am still confused by the notation. Why is the notation $x \equiv r \mod n$ used at all? If $\operatorname{mod}$ is a mathematical operator like $\sin$ or $\cos$, why is the notation $r = \operatorname{mod}(x,n)$ not used always?
Is there an equivalent in mathematical language to the modulo (or mod) function in computing?
How can I remap a keyboard key? One of my laptop's keys has fallen off. Is there any way I can remap another key to serve as that key?  I'll 'sacrifice' the other key because I never use it. Maximum sum sublist? Which topics are essential for QGIS beginners? What topics do you think are essential in an introduction for QGIS?  The target audience is those with zero or rudimentary knowledge in mapping and geospatial software. Apply display settings to all user accounts? Spectral radii and norms of similar elements in a C*-algebra: $\|bab^{-1}\|<1$ if $b=(\sum_{n=0}^\infty (a^*)^n a^n)^{1/2}$ Let $A$ be a unital $C^*$-algebra and $a\in A$ such that $r(a) &lt; 1$. Define b  = $(\sum_{n=0}^\infty (a^*)^n a^n)^{1/2}$. We can prove that $b\geq e$ and that $b$ is invertible. I want to show $\| b a b^{-1} \| &lt; 1$.   From the definition of $b$ we see that $a^* b^2 a = b^2-e$ and we know $r(bab^{-1}) = r(a) &lt;1$.   So it suffices to prove $r(b a b^{-1}) = \| b a b^{-1} \|$. It can follow from the fact $c = ba b^{-1}$ is a normal element... I don't know how to prove it (I have tried to compare $c^* c$ and $c c^*$...).    Context: The question appears in Murphy's book, page 74.   I have managed to prove the first part. The second part of the question is to prove $$r(a)= \inf_{c\in Inv(A)}\{\|cac^{-1} \| \}$$ It's easy to see $$r(a) \leq \inf_{c\in Inv(A)}\{\|cac^{-1} \| \}$$ But I can't prove $$r(a) \geq \inf_{c\in Inv(A)}\{\|cac^{-1} \| \}$$ If we have had $r(a) = \|b a b^{-1} \|$ then it was obvious... But this is not true, so how we can prove this inequality? Let $R$ be a simple ring and $T$ be a non-zero ring. Let $f\colon R \rightarrow T$ be a ring homomorphism. Show $f$ is injective.  Proof: $\ker f \lhd R$, so $\ker f=R$ or $\ker f=\{0\}$. If $\ker f=R$ then $R/R \cong \{0\}$, but $T \neq \{0\}$, which is a contradiction. I can't see why does the last part means contradiction.   Thanks for any assistance! Background process of subshell strange behaviour I wonder why some similar bash commands behave the way they do.  I have a bash script foo:  #!/usr/bin/env bash  while true do     echo "reading"     read data     echo $data     echo "stderr msg" &gt;&amp;2     sleep 1 done   It's an infinite loop which reads one line at a time from stdin and outputs that same line. I also have a bash script bar:  #!/usr/bin/env bash  ./foo &amp;   The following commands were run in bash (v. 4.4.19) in a terminal on Ubuntu 18.04 (let's assume the scripts reside in the working directory):   ./foo &amp;   Stops when it tries to read from stdin, in accordance with . Killed when controlling terminal is killed, as expected.  (./foo &amp;)   Appears to be automagically continuously fed with new input from stdin. Shouldn't it too stop when attempting to read from stdin? Whatever it's getting it doesn't show when echoed, thus I'm guessing it's EOF characters.  Continues running even after the controlling terminal is killed. Looking at the output of ps, the controlling terminal changes from pts/x to ?. How did this happen without the use of either disown or nohup? (However, after killing the originally controlling terminal, it yields a "write error: Input/output error" on every write, which I suppose is because its stdout was tied to the now closed terminal. Correct?)  bash -c "./foo &amp;"  Appears to behave exactly like 2). ./bar  Appears to behave exactly like 2) and 3). Adding bash -c "~/foo" (no &amp;) as a startup application  Behaves similarly to 2), 3) and 4), with the differences:   Its controlling terminal is the desktop ttyx. You can't (can you?) kill its controlling terminal.   None of those differences seem odd to me, just the fact that it's continuously fed with new input.   My guess is that 2) - 5) all use a subshell of sorts whose stdin is redirected to something like /dev/null, though I'm very unsure and seek a more exact answer. I’m not sure What it is? I’m trying to get a glossy shade to building in my scene as soon as I put roughness to zero I see these trees in glass. I have no trees in my scene anywhere, background is black. Even when I render I see these trees. How can I get rid of them and still keep my glossy building material. Does the order of declaration matter in Java/C#? In C++ I can't use a method if I declare it after the calling method.    Does this order matter in other languages like Java or C#? Show that $\frac{X-\mu}{\sigma}\sim N(0,1)$ using moment functions Let $X\sim N(\mu,\sigma^2)$. Show that $Z=\frac{X-\mu}{\sigma}\sim N(0,1)$ using moment generating functions. \begin{align*} M_Z(t)&amp;=M_{\frac{X-\mu}{\sigma}}(t)\\ &amp;=M_{X-\mu}\left(\frac t\sigma\right)\\ &amp;=e^{-\mu t}M_X\left(\frac t\sigma\right)\\ &amp;=e^{-\mu t}\cdot e^{\frac{t\mu}{\sigma}+\frac{t^2}{2}}\\ &amp;=e^{-\mu t+\frac{t\mu}{\sigma}+\frac{t^2}{2}} \end{align*} I'm not sure how to conclude that $Z\sim N(0,1)$. What information does the trace of a matrix give? YouTube videos reload when I replay On this site I see hundreds if not a thousand questions being asked about issues with a Wireless LAN adapter not being detected/recognized by Ubuntu. Some mainstream adapters appear to be utterly broken in Linux support, while others just require some simple steps to get it to work.  The options to get it to work appear to be infinitely huge and I have no clue where to get started!  What is considered 'best practice'? What order of things to try is most efficient in terms of time, effort and imposes minimal risk to my Ubuntu installation? Please help me save time and guide me what to do, step by step. I use Ubuntu 12.04 64-bit, I did the following: sudo gedit /etc/apt/apt.conf and added  APT::Install-Recommends "false"; APT::Install-Suggests "false";    But it did not work. When I try to install a package, it still wants to install the suggested and recommended packages. How can I solve this? Force MacBook to drain the battery while connected to charger I leave my MacBook Pro on my desk nearly all the time, connected to the charger with the lid closed, connected to an external display.  I've heard this is bad for the battery health, and everyone in my office have to replace swollen batteries every couple of years. I want to regularly discharge the battery, per , but I don't want to physically unplug it because that temporarily turns off the external monitor, and then I have to reconnect the monitor.  So I am looking for a software method of disabling the power input, to force the laptop to discharge its battery to 20% and then re-enable the power input. Find files using ls and wildcards pwd : ../bin under bin I have a directory called datafiles.  Use meta characters and the ls -lL command (with lower and upper case L) to list all filenames under the datafiles directory that contain a dot . with the letter 'f' or 'u' anywhere after the dot.  I tried: ls -lL datafiles *.*f*u  ls -lL datafiles .[f*u]     Thank you for answering. Unfortunately, none of the solutions worked the way I wanted it.  By clarifying the question, /bin/datafiles/file.backup is the root to the directory. From the /bin directory, the command I used is ls -lL datafiles followed by meta characters.  I'm looking for file.backup that has a dot and followed by f or u, not directly followed by f or u (anywhere after the dot), and not ending by f or u. Deleting a channel on slack.com I have just updated MiKTeX and have found that pstricks now appears to cause a problem. The error message I'm getting is     ("C:\Program Files\MiKTeX 2.9\tex\generic\pstricks\pstricks.tex"      ! Undefined control sequence.      l.31 \if@check@engine   Not sure what's happening here. If I stop pstricks from being loaded things work ok.  Is there a known problem with the latest pstricks? Why isn't Obi-Wan's ghost young? One of the changes made to the Special Editions of Star Wars: Episode VI: Return of the Jedi was at the end of the film.  Luke sees 3 ghosts during the celebration on the forest moon of Endor, Yoda, Anakin, and Obi-Wan.  In the Special Editions, the actor who played Anakin earlier in the film, Sebastian Shaw, is replaced with Hayden Christiansen who played young Anakin in Episodes 2 and 3.  My question is, why is Anakin the only one de-aged?  Shouldn't Obi-Wan also be his younger self, along with Yoda? Inaccurate shading on object