What does "? :" mean?
Question mark and colon in JavaScript
I know that it holds for $n=5$, so the first step is done.  For the second step, my IH is: $4^n &gt; n^4$, and I must show that $4^{n+1} &gt; (n+1)^4$. I did as follows:  $4^{n+1} = 4*4^n &gt; 4n^4$, and if I show that $4n^4 &gt; (n+1)^4$ holds for $n\geq 5$ I'm done, so I expand the right hand side and arrive to:  $$4n^4 &gt; n^4 + 4n^3 + 6n^2 + 4n + 1$$  And this holds iff: $$3n^4 - 4n^3 - 6n^2 - 4n - 1 &gt;0$$ And here I'm stuck, I don't really know how to show that the given inequality holds. Any hints? How can Obito use his Mangekyo so often? Server login attempts NOT blocked after configuring fail2ban How can I boot with an older kernel version? How can I deal with "Low on Space" messages on my Android? I keep getting the "Low on Space: Application data space is low. To free up" [the rest is cut off] message.  I've moved every app I can over to the SD card, cleared the cache, compacted my K9 mail cache.  Advanced Task Killer (ouch!) says I have 98M Available Memory  File Manager says I have 223M/748M (29%) free.  That's not so bad -- but I'm stuck in the no-updates scenario. The market downloads won't really start, gmail won't update, etc. I've rebooted several times, and nothing changes. Aaaargh.  What can be done, other than uninstalling everything ? Does the 2020 Apple M1 MacBook Pro support 2x daisy chained Thunderbolt Displays? The Apple Silicon based  supports Thunderbolt version 3 and I'm wondering if anyone has tried connection one to an ? Of course you'd need to use one of Apple's Thunderbolt 3 to Thunderbolt 2 converters. I also understand that this version is limited to supporting only a single 4K or 6K display, but since the original Apple Thunderbolt Display is only 2K I’d want to try and daisy chain these. Can I run more than one external display from the Apple Silicon M1 CPU/GPU based MacBook Pro (or Air)? How to make `sudo` preserve $PATH? I have a program that is installed in a custom directory under /opt. To make it easier to run it, I edited my bashrc to add said directory to my path: export PATH=$PATH:/opt/godi/bin:/opt/godi/sbin  This works fine if I want to run the program without sudo. However, if I try to run it with sudo it fails with a &quot;command not found&quot; error. $ sudo godi_console sudo: godi_console: command not found  Inspecting the PATH variable after using sudo reveals that its not including the same PATH I have as a normal user: $ sudo sh # echo $PATH                  /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin  Why is the PATH not the same? Am I doing something wrong? I'm on Debian Jessie, if it makes a difference. One thing I tried was to invoke /opt/godi/sbin/godi_console directly, passing the absolute path to the executable. Unfortunatelly, that didn't help in this particular case because godi_console itself depends on the PATH being correctly set. Let $f(z)$ be an entire function such that $$|f(z)|&lt;\frac{1}{|\text{Im}(z)|},\qquad z\in\Bbb C-\Bbb R.$$ The question asked me to prove that $f(z)=0$. At least looking at it, it really seems to have an application of Liouville's theorem lurking around somewhere, but I haven't found it.   My thoughts first led me to think about doing this by contradiction and using Picard's little theorem. So, I've considered a strip containing the real axis (say of width $2$ for simplicity). This will imply the complement maps into the unit disk, which implies that this strip has to map almost everywhere else on the complex plane. Now, on the imaginary axis, I know that $f(z)$ will vanish as $z\to\infty$, and this seems like it might be useful. I'm sort of under the impression $f(z)$ might have an essential singularity at $\infty$, which would mean $f(1/z)$ has an essential singularity at $0$. Either way, it of course can't have a pole at $\infty$ because of $f(z)$ vanishing on the imaginary axis, and if it is a removable singularity, it must be $0$, which still gives the solution. Therefore, I think it must have something to do with $f(z)$ having an essential singularity at $\infty$.   I was hoping to combine this with the above inequality and deduce a contradiction, but I haven't thought of one yet.   Any hints or suggestions? I'm studying for a prelim. exam, so this isn't homework.  Update: So, in the spirit of searching methods using the essential singularity at $z=0$, we have $$\left|f\left(\frac{1}{z}\right)\right|\leq\frac{1}{\left|\text{Im}\left(\frac{1}{z}\right)\right|}=\frac{|z|^2}{|y|}.$$ Thus, for $|y|&gt;1$, we have $$\left|f\left(\frac{1}{z}\right)\right|&lt;|y|\left|f\left(\frac{1}{z}\right)\right|&lt;|z|^2.$$ Thus, if I can show that $|f(1/z)|$ is a polynomial, we'll know it can't have an essential singularity at the origin, concluding my proof. Obviously, you can use the | (pipe?) to represent OR, but is there a way to represent AND as well?  Specifically, I'd like to match paragraphs of text that contain ALL of a certain phrase, but in no particular order. How to NOT show keyboard layout chooser popup when changing language in GNOME 3 Since I installed 17.10, when switching language with the assigned keyboard shortcut I get this switcher "window" that obstructs me from typing for a couple of seconds. And yes, a couple of seconds is a huge amount of time when switching languages a lot.      This happens both in X11 and Wayland but doesn't happen (the obstructive window will not appear) when I switch language by clicking on the lang indicator on top bar. Is there any way to tell this window "please go away"? I am trying to retrieve list items using date range based on date column. It implicit both 'Fromdate' and 'Todate'.My CAML query is also querying on two other conditions as well. I am using the below query but it is not retrieving the data.when I remove the date range from query data is retrieved fine. But i want date range as well in my query how would i do this. Little help would work for me.   camlQuery.set_viewXml('&lt;View&gt;&lt;Query&gt;&lt;Where&gt;&lt;And&gt;&lt;And&gt;&lt;Neq&gt;&lt;FieldRef Name="ID" /&gt;&lt;Value Type="Number"&gt;0&lt;/Value&gt;&lt;/Neq&gt;&lt;Eq&gt;&lt;FieldRef Name="_ModerationStatus" /&gt;&lt;Value Type="ModStat"&gt;0&lt;/Value&gt;&lt;/Eq&gt;&lt;Geq&gt;&lt;FieldRef Name="Created" /&gt;&lt;Value Type="DateTime"&gt;' + fromDate + '&lt;/Value&gt;&lt;/Geq&gt;&lt;Leq&gt;&lt;FieldRef Name="Created" /&gt;&lt;Value Type="DateTime"&gt;' + toDate + '&lt;/Value&gt;&lt;/Leq&gt;&lt;/And&gt;&lt;/And&gt;&lt;/Where&gt;&lt;OrderBy&gt;&lt;FieldRef Name="_ModerationStatus" Ascending="True" /&gt;&lt;/OrderBy&gt;&lt;/Query&gt;&lt;/View&gt;'); Frequency dependence of electromagnetic reflection I was surprised to see that the Fresnel equations for reflection depends on refractive index and angle of incidence, but they do not depend on frequency. Why is this case? Are they restricted to regions that do not show frequency dependence? Debugging scripts, what is the difference between -x to set -euxo pipefail? The main way I know to debug scripts is adding -x to the shabang (#!/bin/bash -x).  I recently came across a new way, adding set -euxo pipefail right under the shabang, as in:  #!/bin/bash set -euxo pipefail   What is the main difference between the two ways of debugging? Are there times you would prefer one above the other?  As a freshman, , I couldn't extract such conclusion. Get Label value for specified language Process management and high technology weapon. Novel name and author? I am looking for help to point me out at a very old novel (likely from the '60s) about two empires at interstellar war. When it comes to starships, one is technologically very advanced, and the other has very low and inefficient technology.  The highly advanced empire has apparently strong chances to win the war, in particular considering a new technology they were developing. I remember the technology to work like this: consider the space as a flat surface, and on this surface, a disk around the spaceship. When the technology was turned on, this disk was stretched out, as made of rubber, so that it produced a cone, with the tip being the spaceship. The spaceship was therefore teleported 10 parsec away (if I remember correctly), but retained all the connections to the original position. If the spaceship moved, the cone moved as well. This allowed to move in position among enemy ships completely unnoticed, by stretching while afar, then moving in the proper position 10 parsecs away, then destretching.  Despite this technology, the empire lost the war because of delays. It's a perfect example of how having a disruptive technology does not guarantee to win against the competitors.  I am trying to find out the author and name of this novel, hopefully with the date of publication and in which journal/magazine. Can I manually put an external hard disk to sleep? I tried to install Tor on my new version of Ubuntu, but it seems that Tor is not available in the repositories.  How else can I install and use Tor? $\operatorname{gcd}(ab,a+b)=1$ if $a$ and $b$ are relatively prime What's the easiest way to unrar a file? How can I force my browser to search Google in English? I'm too bored of seeing sites like Google and such show up in my native language, I would rather like them to be in English. Yet, I have to explicitly change the URL to .com and en and that kind of parameters in order for them to show up in English. Can I somehow force this?  So, how is Google configured?  However, it is set to English on the site itself so it has to be my browser:    Then, how does my browser land up on non-english pages, like Google?  It usually shows up in non-English when I'm performing a search, which uses:  {google:baseURL}search?{google:RLZ}{google:acceptedSuggestion}{google:originalQueryForSuggestion}{google:searchFieldtrialParameter}{google:instantFieldTrialGroupParameter}sourceid=chrome&amp;ie={inputEncoding}&amp;q=%s   When performing a search, it fills these variables in with non-english values.  How can I tell my browser to fill these in with the English values?  My Google Chrome options give preference to English: