"He was a considerable cut above the leagers" meaning
What is the origin of "a cut above/below the rest"?
I am trying to prove that:  Let $f$ be analytic in the closed unit disc with $f(z)\leq 1$ for $|z|\leq1$.  For a given $0&lt;r&lt;1$ and $|z_1|,|z_2|\leq r$, $z_1\neq z_2$ then,   $$\left|\frac{f(z_2)-f(z_1)}{z_2-z_1}\right|\leq \frac{1}{(1-r)^2}.$$  My attempt is: part a  of the question asks to prove $$\left|\sum_k^\infty a_nz^n\right|\leq \frac{ |z^k|}{1-|z|}.$$   So, \begin{eqnarray*}  \left|\frac{f(z_2)-f(z_1)}{z_2-z_1}\right|&amp;=&amp; \frac{|\sum_1^\infty a_nz_2^n-\sum_1^\infty a_nz_1^n|}{|z_2-z_1|}\\  &amp;\leq&amp; \frac{\frac{ |z_1 |}{1-|z_1|}+\frac{ |z _2|}{1-|z_2|}}{|z_2-z_1|}\\  &amp;\leq&amp;\frac{1}{(1-r)^2} \frac{   |z_2| ({1-|z_1|})+  |z _1| (1-|z_2|) }{|z_2-z_1|}\\ \end{eqnarray*} But could not prove that the last term is controlled by 1. Here's my source:  #!/bin/bash  echo "Running script to free general cached memory!" echo ""; echo "Script must be run as root!"; echo ""; echo "Clearing swap!"; swapoff -a &amp;&amp; swapon -a; echo ""; echo "Clear inodes and page file!"; echo 1 &gt; /proc/sys/vm/drop_caches; echo "";   It clears caches and stuff, and it echoes that it needs to be run as root in the terminal. I basically just want the script to cease running if it detects it's not being executed as root.   Example:  "Running script to free general cached memory!" "Warning: script must be run as root or with elevated privileges!" "Error: script not running as root or with sudo! Exiting..."   If run with elevated privileges, it just runs as normal. Any ideas? Thanks! Is there a difference between taking a far shot on a 50mm lens and a close shot on a 35mm lens? I am looking for a prime lens to pick up, and I'm wondering if there is any difference between 35mm and 50mm in terms of the end product if I just stand back more with the 50mm. I use the Sony a6000 and am looking at the SEL50F18 and SEL35F18.  I understand that something like a fish eye lens will give a different effect, but is that what is going on in 35 vs 50mm? Or is 50mm effectively just taking a 35mm and cropping the center out (but with increased quality)? Macroscopic quantum phenomena and tunneling I have an area which is divided into several zones the borders of which I have defined as separate layers.  The layers are defined by sql queries on a postgis table of line segments joined with a table that says which segments are part of which zone boundaries.  What I want to do is produce a set of maps where each map is a zoom to layer extent for each zone and which has standard header, legend and scale (which will preferably vary with the zone but could be fixed across all maps).  My first attempt was to get a new composer for each map -- that worked but was rather tedious.  Next I tried producing one map and printing it and then deleting the map frame and going into qgis to zoom to the next layer extent and make the boundary visible then hopping back into the composer to load it in.  This worked more or less.  Map was fine, legend was OK because it was the same for all the maps and the zone name was easy to edit but the scale needed to be deleted and set up again for each map.  Next I looked at Atlas Generation.  The manual seems to assume that you know how it works and all you need is a reminder about the details (like so many software manuals ;)  It seems to assume that you have something that you are iterating over but I can't see how to make it work -- there is nothing to iterate over to get the layers.  Any suggestions welcome!!  :)  Russell I got this problem from my teacher as a optional challenge. I am open about this being a given problem, however it is not homework. The problem is stated as follows. Assume we have an infinite tetration as follows $$x^{x^{x^{.^{.^.}}}} \, = \, a$$ With a given $a$ find $x$. The next part of the problem was to discuss the convergence radius of a. If a is too big or too small the tetration does not converge. Below is my humble stab at the problem.  My friend said you would have to treat the tetration as a infinite series, and therefore could not perform algebraic manipulations on it before it is know whether it converges or diverges. However my attempt is to first do some algebraic steps, then discuss the convergence radius. I) Initial discussion At the start it is obvious that the tetration converges when $a=1$ (just set $x=1$) Now after some computer hardwork it seems that the tetration fails to converge when a is roughly larger than 3. II) Algebraic manipulation $$ x^{x^{x^{.^{.^.}}}} \, = \, a$$ This is the same as $$ x^a \, = \, a$$ $$ \log_a(x^a) \, = \, \log_a(a)$$ $$ \log_a(x) \, = \, \frac{1}{a}$$ $$ x \, = \, a^{\frac{1}{a}}$$ Now, if we let $a=2$ then $x = \sqrt{2}$. After some more computational work, this seems to be correct, which makes me believe this formula is correct. III) Discsussion about convergence By looking at the derivative of $ \displaystyle \large a^{\frac{1}{a}} $ we see that the maxima occurs when $a=e$, which also seems to correspond to the inital computational work. Now I think, that the minima of $\displaystyle \large a^{1/a}$ is zero by looking at its graph, studying its derivative and the endpoints. So that my &quot;guess&quot; or work shows that it converges when $$ a \in [0 \, , \,  1/e] $$ VI) My questions Can my algebraic manipulations be justified? They seem rather sketchy taking the a`th logarithm and so on . (Although they seem to &quot;magically&quot; give out the right answer) By looking at Wikipedia it seems that the tetration converge when $$ a \in \left[ 1/e \, , \, e \right] $$ This is almost what I have, why is my lower bound wrong? How can  I find the correct lower bound? Beamer text and image on the same slide I am new to Beamer and LaTeX itself. Right now, I want to create a simple slide with an image on the right and text and the sidebar (using the Hannover theme) on the left. I cannot figure out how to do that. Any suggestions? What command do I need to unzip/extract a .tar.gz file? I received a huge .tar.gz file from a client that contains about 800 mb of image files (when uncompressed.) Our hosting company's ftp is seriously slow, so extracting all the files locally and sending them up via ftp isn't practical. I was able to ftp the .tar.gz file to our hosting site, but when I ssh into my directory and try using unzip, it gives me this error:  [esthers@clients locations]$ unzip community_images.tar.gz Archive:  community_images.tar.gz   End-of-central-directory signature not found.  Either this file is not a zipfile, or it constitutes one disk of a multi-part archive.  In the latter case the central directory and zipfile comment will be found on the last disk(s) of this archive. note:  community_images.tar.gz may be a plain executable, not an archive unzip:  cannot find zipfile directory in one of community_images.tar.gz or community_images.tar.gz.zip, and cannot find community_images.tar.gz.ZIP, period.   What command do I need to use to extract all the files in a .tar.gz file? Full List of inventory items After beating the game, I still have several empty inventory spots.  I most likely missed several items in my addicted frenzy to finish the game.  What are all the items that can be found in the game? And where can they found? Anthology for children; contains story about friendly Martian on postapocalyptic Earth I'm looking for the title of an anthology of children's science fiction stories from the 1950s. The best of the bunch involved a Martian who visited earth, and realized we'd blown ourselves up in a nuclear war and had fallen into savagery. The Martian is a vegetarian who subsists on biscuits made by a machine he brought with him, and communicates by using one of his tentacles to feed messages direction to/from the brain of the person he's talking to.  Inadvertently left behind by his expedition, he initially befriended some children, and eventually managed to rebuild civilization. Presently he dies, but when another expedition comes a generation later, they find the civilization he rebuilt.  there were other stories, but that was the best one.   If I can't find the name of the book it was in, I'd be happy to settle for the name of the author and the title of the story. Tsunami dampening mechanisms Pass PHP variables to Javascript/jquery There appears to be a poltergeist in my speakers (no, really) I've no idea if there's a suitable Stack Exchange for poltergeists. It appears to be software, hardware or virus related, so I thought I'd try here, seeing as Ghostbusters is just a film, so I can't call them.  I'm not going mad, and this isn't a joke.  There's what appears to be a radio stream coming out of my speakers, except I can only hear it when the volume knob is turned down to the min setting whilst the speakers are on, which makes no sense. Sometimes it's French, which also makes little to no sense.  The speakers are Logitech 5.1s, I forget the model number, and I'm running XP 64bit. If that makes a difference.  I'd suspect it could be some sort of crazy virus, but I've been randomly killing processes whilst the poltergeist is active and it hasn't killed it, so I'm not sure about that.  Once again, I'm perfectly sane and this really isn't a joke. Perhaps other people have had this problem but have been too afraid of the reaction to ask anyone (please don't have me committed to an insane asylum).  Googling for strange noises coming from speakers didn't really get me anywhere. Has anyone heard of a virus that can do this and affect the volume controls on the hardware in that way? I am looking for a word that would describe someone who always goes out of their way to take their time and do a good job. Sort of the opposite of half-assing something.  I am looking for something that is more about a work ethic than a technically proper way of doing things. Something like canonical or protocol would not be a good fit. How to remove duplicate values from a multi-dimensional array in PHP The vertical line at the end of the cell where i used cell merge is missing. What's this? What is it? but not What's it? - Why? Can anyone give a cogent, simply described explanation of why the verb BE in:  What is it?  ... doesn't seem to be able to be contracted with the subject:  What's it? *  Compare the sentences above with:  What's this? What's that?  These are perfectly fine. In fact, the contractions here should be expected in almost all examples of spoken English. Answers with references to authoritative vetted sources would be greatly appreciated. I've just started using TikZ and need to create a simple triangle commutative diagram like this one.    The closest I've been able to get is this.   How do I get the h arrow to go from A2 to A3? Everything I've tried just causes errors.  Here's my current code:  \begin{tikzcd}[column sep=small] A \arrow{r}{i}  \arrow{rd}{g}    &amp; A3 \arrow{d}{h} \\     &amp; A2 \end{tikzcd} I've seen some of the other threads on using strings as x axis labels, but nothing that fits my situation.  I'm trying to create a bar chart from a CSV file. I can do it so long as everything in the data is numeric. However, in my actual data table, the first row is multi-word string descriptions. I need these descriptions to be the x axis labels for each bar.  I've tried symbolic x coords, but I get the "could not parse input as a floating point number" error. I've also tried adding an ID column and using that as the x values, and assigning xtick labels as the string descriptors. I get the same issue that way.  Here's an attempt at an MWE:  \documentclass{article}  \usepackage{lmodern} \usepackage{tikz} \usepackage{pgfplots} \usepackage{pgfplotstable}  \begin{document}  \pgfplotstabletypeset[columns/category/.style={string type},col sep=comma]{Book3.csv} \vspace{1cm}  \begin{tikzpicture} \begin{axis}[ ybar, xlabel=Xstuff, ylabel=Value, symbolic x coords={cat a,cat b,cat c,cat d,cat e,cat f,cat g}, %xtick labels={cat a,cat b,cat c,cat d,cat e,cat f,cat g},% this is from when i tried the xtick label method xtick=data, nodes near coords, nodes near coords align={vertical}] \addplot table[x=category,y=value]{Book3.csv}; %\addplot table[x=category,y=value]{Book3.csv};&amp; again, from trying the xtick label method  \end{axis}  \end{tikzpicture} \end{document}   Since I suspect this might get answered quickly by you chart gurus, I have a follow-up question:  I have to do a bunch of these charts, and I would like to automate the symbolic x coords process. Is there a way to populate the dictionary of accepted x coords by reading a column of the data? (something like symbolic x coords={columns/category/{Book3.csv}} )?  Book3.csv looks like this (ignore the ID column if using the xtick label method):  ID, category, value, value 2 1,  cat a,    1,     7 2,  cat b,    2,     6 3,  cat c,    3,     5 4,  cat d,    4,     4 5,  cat e,    5,     3 6,  cat f,    6,     2 7,  cat g,    7,     1 Can I configure bash to execute "clear" before every command typed in the console? I would like to configure bash to execute clear command every time I type some command in the terminal (before executing my command). How can I do that?  I'm using Debian Linux.