Let $D=\{z \in \mathbb{C}: |z|&lt;1\}$, then which of the following are true?   There exists a holomorphic function $f:D\to D$ with $f(0)=0$ and $f'(0)=2$. There exists a holomorphic function $f:D\to D$ with $f(3/4)=3/4$ and $f'(2/3)=3/4$. There exists a holomorphic function $f:D\to D$ with $f(3/4)=-3/4$ and $f'(3/4)=-3/4$. There exists a holomorphic function $f:D\to D$ with $f(1/2)=-1/2$ and $f'(1/4)=1$.   Now for option $1$, I see that it is not true by Schwarz lemma. But I don't know how to decide the rest. Schwarz lemma says one more thing that if $|f(z)|=|z|$ then $f(z)=az$ where $|a|=1$. So $|f(3/4)|=|-3/4|=3/4$, is this going to help? How can I proceed further? Help needed please.     Edit: Ok I just found that this is a duplicate of , but I still appreciate some answers as I didn't properly understand the answer given.
Let, $D=\{z\in \mathbb C:|z|&lt;1\}$. Which are correct?   there exists a holomorphic function $f:D \to D$ with $f(0)=0$ &amp; $f'(0)=2$. there exists a holomorphic function $f:D \to D$ with $f\left(\dfrac{3}{4}\right)=\dfrac{3}{4}$ &amp; $f'\left(\dfrac{2}{3}\right)=\dfrac{3}{4}$. there exists a holomorphic function $f:D \to D$ with $f\left(\dfrac{3}{4}\right)=-\dfrac{3}{4}$ &amp; $f'\left(\dfrac{3}{4}\right)=-\dfrac{3}{4}$ there exists a holomorphic function $f:D \to D$ with $f\left(\dfrac{1}{2}\right)=-\dfrac{1}{2}$ &amp; $f'\left(\dfrac{1}{4}\right)=1$.   With the help of Schwarz lemma &amp; its applications, we find that $(1)$ is false &amp; $(3)$ is true.  But, I can not think about options $(2)$ &amp; $(4)$.
What a pluperfect a##hole Geometric solution to find a Set of totally ordered Subsets of $ \mathbb{N} $ is uncountable My question is, does there exist a set of subsets of the naturals, $S$, such that $S$ is uncountable, where for any two subsets $A,B \in S$ we have either $A \subset B $ or $B \subset A $?  Initially my thoughts were that we could construct a chain $A_{1} \subset A_{2} \subset \cdots $ and find a bijection by picking out one element which is 'new' to each set by axiom of choice and so we have bijected to a subset of naturals, however I'm not convinced this fully works.  Furthermore, I was thinking there may be a nice elegant solution by looking for such a set in either $ \mathbb{Q}^2 $ or $ \mathbb{Z}^2 $ for example, but any attempt seems to fail due to the 'ordering' part of the question. How do I use subprocess.Popen to connect multiple processes by pipes? How do I execute the following shell command using the Python  module?  echo "input data" | awk -f script.awk | sort &gt; outfile.txt   The input data will come from a string, so I don't actually need echo.  I've got this far, can anyone explain how I get it to pipe through sort too?  p_awk = subprocess.Popen(["awk","-f","script.awk"],                           stdin=subprocess.PIPE,                           stdout=file("outfile.txt", "w")) p_awk.communicate( "input data" )   UPDATE:  Note that while the accepted answer below doesn't actually answer the question as asked, I believe S.Lott is right and it's better to avoid having to solve that problem in the first place! Handling Calls to Remove a Moderator This discussion informed our process for seven years, but eventually required updates and augmentation. For discussion regarding the successors to this process, please see:    Stack Exchange has some of the best Moderators around — seriously — and that is due in no small part to the communities who scrutinize and vote in our Moderator elections.  We have a rather formal process for electing moderators, but up until now  has been quite informal. On those rare, rare occasion when things just aren't working out, you can see why it becomes such a sticky business to remove someone from a democratically elected position. Our current process is one of &quot;we've got your back.&quot; The Community Team routinely monitors moderator activities, so if a moderator violates the  or fails to uphold the , we can and do intervene. If the disputes and problems persist without resolution, eventually we can ask someone to step down. That's not good enough. We need a much more transparent and prescribed process for when — and more importantly how — to intervene when on-going problems are going unresolved. In urgent situations, we'll step in to protect a site, but ideally the process should be impartial and not subject to accusations of Stack Exchange simply applying undue influence over a community when the whim strikes. The Question: Before we start bouncing around ideas internally, I'd like to hear what you think. What process should be in place to add some checks and balances to deal with higher issues like possible impeachment of a moderator? When and how would such a process come into play? How should such situations be resolved? How do we keep it from being abused? Have at it. This is a round table vetting of ideas. Gmail App in my Android device shows correctly all the labels that I created using Gmail through web interface.  Can I create new labels in Gmail on Android device using Gmail App?   Also, ideally, once I create these new labels in Gmail App, I obviously should be seeing the same in Web Interface of Gmail too (Sync!!).  Any ideas? calculator's angle answer for trig ratios that can work in more than 1 quadrant on the unit circle Why does the calculator do a cc (counterclockwise) rotation for positive trig ratios instead of clockwise,  and a clockwise rotation for negative sine &amp; tan instead of cc and a counterclockwise rotation for negative cos ratios instead of a clockwise   ie. in degree mode  $\cos^{-1}(-5/12)=114.62$  $\sin^{-1}(-5/12)=-24.62$  $\tan^{-1}(-5/12)=-22.61$  Is it maybe picking the value that involves the least amount of computing power? or is it a matter of convention? or am I overlooking something? Do citizens migrate closer to their workplace within the town? By using the Path tool (F2, 7) I have found some laborers walking very long to eat at home or to warm up in winter.  If I build a house close to a Quarry or a Mine, will Stonecutters or Miners move automatically to this house to avoid long walking distances between the workplace and home?  Or can I assign some citizens to houses closer to their workplace? How was Will aware of the lights? In  (2016) Will Byers gets taken to the "Upside Down". But he manages to communicate with his mother, e.g through the christmas lights.  &nbsp;   When Joyce is in the cupboard and says "Once for YES, twice for NO.", how would Will make that happen? He can't see his mother, but does he see the lights she is holding?  &nbsp;   Does the alphabet Joyce draws just appear on the wall in the Upside Down, together with the lights?  In the last episode Joyce and Hopper enter the house in the Upside Down, and when they go through the hallway the kids on the other side see the lights go on (suggesting that just standing close to those lights is enough to make them turn on).    But those lights don't seem to be there in the Upside Down.    My Question:   Is there a reasonable explanation on how Will managed to communicate with his mother through the christmas lights? pgAdmin 4: How to establish a connection through an SSH tunnel/interface in Windows I installed the latest pgAdmin 4 (v2.1) but I found out that (), for the moment, the only way to establish a connection through an SSH tunneling is using another outside process, an  says:  I setup a tunnel as follows: ssh -f dpage(at)hostname(dot)enterprisedb(dot)com -L 5999: hostname.enterprisedb.com:5432 -N  (...) I was running on Mac, but the same procedure should work on Linux. On Windows, there's a tunnelling app in Putty iirc, but I don't know how to use it offhand.  I have found a simple . The connection is established but it seems, when I try to connect through pgAdmin it fails or is refused (using 127.0.0.1:3306 or 127.0.0.1:5432 as the tunnel's destination). Why? What is the difference between a relative phase and a global phase? In particular, what is a phase? I know that $re^{i\theta} = x + iy$ for any complex number $x + iy$ by Euler's formula. How do you calculate relative and global phase? In my preamble, you'll find:  \renewcommand{\baselinestretch}{1} \renewcommand{\familydefault}{\sfdefault} \usepackage[sfmath]{kpfonts} \usepackage[mathrmOrig]{sfmath} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{animate} \usepackage{anysize} \usepackage{arcs} \usepackage{array} \usepackage[scale=0]{background} \usepackage{blindtext} \usepackage{calc} \usepackage{cancel} \usepackage{chngcntr} \usepackage{CJK} \usepackage{CJKspace} \usepackage{CJKpunct} \usepackage{colortbl} \usepackage{enumitem} \usepackage{epstopdf} \usepackage{etoolbox} \usepackage{fancybox} \usepackage{fancyhdr} \usepackage{float} \usepackage{framed} \usepackage{graphicx} \usepackage{hyphenat} \usepackage{ifthen} \usepackage{latexsym} \usepackage{lmodern} \usepackage{multicol} \usepackage{multirow} \usepackage{mwe} \usepackage{pgffor} \usepackage{rotating} \usepackage{tabularx} \usepackage{textcomp} \usepackage{tgheros,tgtermes,tgcursor} \usepackage{tikz} \usepackage{times} \usepackage{tipa} \usepackage{varwidth} \usepackage{wasysym} \usepackage{wboiboites} \usepackage[abs]{overpic} \usepackage[breaklinks,colorlinks=true,linkcolor=red,citecolor=red, urlcolor=blue]{hyperref} \usepackage[framemethod=TikZ]{mdframed} \usepackage[breakable, theorems, skins]{tcolorbox,empheq} \usepackage[default]{gillius} \usepackage[document]{ragged2e} \usepackage[latin1]{inputenc} \usepackage[spanish]{babel} \usepackage[T1]{fontenc} \usepackage[total={18.59cm,26.5cm},left=1.5cm,top=2.5cm]{geometry} \usetikzlibrary{matrix,arrows, positioning,shadows,shadings,backgrounds,calc, shapes, tikzmark} \usetikzlibrary{shadows.blur} \usetikzlibrary{shapes.symbols} \usetikzlibrary{patterns,fadings} \tcbuselibrary{skins,breakable,listings,theorems}    I'm trying to draw an arrow for X axis and Y axis.   \begin{tikzpicture} \draw[yellow!10, fill=blue!5](-1, -1) rectangle (7, 5); \draw[dotted] (-0.75,-0.75) grid (6.75,4.75); \node[below] at (-0.25,0){$O$}; \draw[-&gt;,black, thick] (-0.75,0)--(6.75,0); \end{tikzpicture}   But I get this error:    Additionally, if I remove -> from the options of \draw the tikz picture compiles normally. Due to my reputation on Stack Overflow, I have access to different review queues. I want to know if it will change my score/reputation or if it will only provide me some badges?  Maybe this part seems a bit odd, but I want to know why some users spend so much time in the review queues? Is it something about their passion about Stack Overflow or their passion to help others inside the community or does it have some other reasons which I've missed?  P.S. I have read some information like  and  but I will be grateful to have your answer about this. How to access Winter 13's Push Major functionality We are ISV Partners and all of our packaging orgs have been upgraded to Winter 13. One of the features of Winter 13 was supposed to allow us to push major release versions to subscriber orgs. I would have expected it to be in the previously existing "Push Upgrade" section of our packaging orgs, but that still only allows pushing patch versions.  If it didn't make the final Winter 13 release, I would have expected Salesforce to take it off the release page. Is this something we have to ask to be enabled? XeTeX: What happened to \mathcal and \mathscr? Recently I switched from pdflatex to xelatex to be able to use OpenType fonts. I like to use variables highlighted using \mathcal for specific variables, while I used to use \mathscr-typeset variables for matrices. But when I compile with XeLaTeX they look exaclty the same.  Old code  \documentclass{standalone} \usepackage{mathrsfs} \begin{document} $\mathcal{C} = \mathscr{C}$ \end{document}   Output        New code  \documentclass{standalone} \usepackage{unicode-math} \begin{document} $\mathcal{C} = \mathscr{C}$ \end{document}   Output        Broken code  I tried to get back the glyphs from mathrsfs, but it failed this way.  \documentclass{standalone} \usepackage{unicode-math} \usepackage{mathrsfs} \begin{document} $\mathcal{C} = \mathscr{C}$ \end{document}   Output        Question: How can I get back the behaviour? In principle I'd like to have the old \mathcal letters from the pdflatex-version of Computer/Latin Modern.    Alternate solutions (Addendum 2016/05/11) Why is the even root of a number always positive? Let $n \in \mathbb N$ be a natural number and $a \in \mathbb R$ be a real number. The $n$-th root of the number $a$ is defined as follows:  Case I: $n$ is an odd number. In this case the $n^{\text{th}}$ root of $a$ is defined to be that number $b \in \mathbb R$ such that $b^n = a$.  Case II: $n$ is an even number. In this case the $n^{\text{th}}$ root of $a$ is defined to be that number $b \geq 0$ such that $b^n = a$.  Why is it that when $n$ is even, we only consider $b \geq 0$. For example, both $+2$ and $-2$ squared equal $4$, but when we say the square root of $4$ is $2$. Is there a reason for this? How do I center text horizontally and vertically in a TextView? Why "hide" Luke from his father somewhere Anakin knows? Let $a_n$ be a sequence such that $\forall n\in \mathbb n: 0&lt;a\le a_n\le b &lt;\infty.$      Prove:          $\displaystyle\limsup_{n\to\infty}\frac 1 {a_n}=\frac 1 {\displaystyle\liminf_{n\to\infty}a_n}$   $\displaystyle\limsup_{n\to\infty}a_n\cdot \limsup_{n\to\infty}\frac 1 {a_n} \ge 1$ and there's an equality iff $a_n$ is converging.       Suppose there are two subsequences: $a_{n_l}, \ a_{n_k}$ such that $\lim a_{n_k} = k, \ \lim a_{n_l}=l$ and suppose $l\le k$, so $\lim \frac 1 {a_{n_k}}=\frac 1 k , \ \lim \frac 1 {a_{n_l}}=\frac 1 l$ so clearly: $\frac 1 k\le \frac 1 l\le l\le k$ so it's easy to see once the largest limit (supermum) is 'inverted' it has to become the smallest limit (infimum).   I realize this doesn't show equality, I don't know how to do the other way and I'm not even sure if what I did is good. If $a_n$ converges, suppose to $L$ as its limit then we have: $L\cdot \frac 1 L=1$.  If it does not converge then $a_n$ may tend to infinity or won't have a limit. From 1 we can change it to $\displaystyle\limsup_{n\to\infty}a_n\cdot \frac 1 {\displaystyle\liminf_{n\to\infty}a_n} \ge 1$ and from BW, every sequence has a converging subsequence, and since for converging subsequences: $\liminf a_n\le \limsup a_n$ we have $\frac {\limsup a_n} {\liminf a_n}=\limsup a_n\cdot \limsup\frac 1 {a_n} \ge 1$.   This should probably be in absolute value since one of those subsequnce limits can be negative, but it isn't in absolute value in the question. Accessing the web page's HTTP Headers in JavaScript Let $h$ the irreducible polynomial over $\Bbb F_2 = \{0, 1\}$  $$h  =  x^4 +x +1$$  Let $E=\Bbb F_2(\alpha)$, i.e. the field $\Bbb F_2$ extended with $\alpha$, where $\alpha$ is a root of $h$.   How do we determine the numbers of elements in the extension $E$? Can every non-zero element in the extension be expressed in the form $(\alpha)^n$ for some $n\in\Bbb N$ ? How do we find all the roots of $h$ over the extension $E$?