I have the following question:let $f(z)$ be continuous on the closed unit disk, $\{z: |z|\leq 1\}$, and analytic on the open unit disk, $\{z: |z|&lt;1 \}$, with $f(e^{it})=0$ for $0\leq t \leq \pi/4$. Prove that $f(z)=0$ on the open unit disk.  I try to apply maximum modules principle but I couldnt get the result. Any help would be great.
Suppose that $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic on the open unit disc and continuous on the closed unit disc. Assume that $f(z)=0$ on an arc of the circle $\{z\in\mathbb{C}:|z|=1\}$. Show that $f(z)=0$.
Assume $\{a_n\}$is a sequence which  satisfies the following recursion formula: $$a_{n+1}=a_n+\frac{2a_{n-1}}{n+1}~(n\geqslant 1).$$ and $a_0=\pi,a_1=\pi^2 $. How to compute $\displaystyle\lim_{n\to \infty}\frac{a_n}{n^2}$? Does the canonical commutation relation fix the form of the momentum operator? For one dimensional quantum mechanics $$[\hat{x},\hat{p}]=i\hbar. $$  Does this fix univocally the form of the $\hat{p}$ operator? My bet is no because $\hat{p}$ actually depends if we are on coordinate or momentum representation, but I don't know if that statement constitutes a proof. Moreover if we choose $\hat{x}\psi=x\psi$ is the answer of the following question yes?  For the second one   $$(\hat{x}\hat{p}-\hat{p}\hat{x})\psi=x\hat{p}\psi-\hat{p}x\psi=i\hbar\psi, $$  but I don't see how can I say that $\hat{p}$ must be proportional to $\frac{\partial}{\partial x}$. I don't know if trying to see that $\hat{p}$ must satisfy the Leibniz rule and thus it should be proportional to the $x$ derivative could help. Or using the fact that $\hat{x}$ and $\hat{p}$ must be hermitian  Any hint will be appreciated. Let $F$ be a Galois extension over $\mathbb{Q}$ with $[F:\mathbb{Q}]=2^n$, then all elements in $F$ are constructible Let $F\subseteq\mathbb{C}$ be a Galois extension of $\mathbb{Q}$ such that $[F:\mathbb{Q}]=2^n$; then all elements in $F$ are constructible.    Added.  Here is what I have so far.  Since there exist the finite normal extension $F\subseteq\mathbb{C}$ over $\mathbb{Q}$ which contains an element say, s, then s has a degree of power of 2. From group theory we have 2 propositons we could use. 'A group $G$ is solvable iff it has a normal series whose factors are abelian', and also 'If $G$ is solvable and $N\triangleleft G$, then $G/N$ is solvable'. Then we can say that the galois group F has a normal series $\{e\}=G_0\triangleleft G_1\triangleleft\cdots\triangleleft G_n=\mathrm{Gal}(F/\mathbb{Q})$, whose factors have order 2. Thus there must exist intermediate fields $\mathbb{Q}=F_0 \subseteq F_1 \subseteq F_2 \subseteq\cdots\subseteq F_n=F$, such that $[F_{j+1}:F_j]=2$ for all $j$. Now this is where I got stuck. Next I need to somehow show that $F_{j+1}=F_j(s_j)$, where $s^2_j$ is an element in $F_j$. Then I think I can conclude that if $s_j$ is constructible than every element in $F_{j+1}$ is constructible... then with $F_n$ is constructible... How to detect if JavaScript is disabled? There was a post this morning asking about how many people disable JavaScript. Then I began to wonder what techniques might be used to determine if the user has it disabled.   Does anyone know of some short/simple ways to detect if JavaScript is disabled? My intention is to give a warning that the site is not able to function properly without the browser having JS enabled.   Eventually I would want to redirect them to content that is able to work in the absence of JS, but I need this detection as a placeholder to start. How do I switch back to python2 after Anaconda set python3 as the default? Recently, I installed  on my 15.04 and ended up with this!  :~$ python --version Python 3.5.1 :: Anaconda 2.5.0 (64-bit)  :~$ python2 --version Python 2.7.9  :~$ python3 --version Python 3.5.1 :: Anaconda 2.5.0 (64-bit)   During the last moments of installation, I did enter something as yes in hurry, and I suppose it had to do something with this. (Ok, my fault, I should have handled that carefully, but I need help now, not criticism).    AFAIK this is definitely going to break other programs. What do I do now?    What I think could work.  Can this be done using aliases?  alias python=python2   But I ain't sure. Count days between two dates, excluding weekends (MySQL only) every planar graph can be expressed as a union of five edge-disjoint forests How can I prove that every planar graph can be expressed as a union of (maximum) five edge-disjoint forests?  Only thing I can think about is going somehow like the $5$ color theorem but I get stuck in the middle. Or with induction, but still something doesn't fit. Can you help? When are "if" and "whether" equivalent? I'm trying to set up a way for me to just type newjournal, and a new journal entry named year-month-date-time.txt is created in a year subdirectory in ~/documents/journals.   I've got it set up as an alias using the following:  alias newjournal='nano /home/username/documents/journals/'$(date +%Y)'/'$(date +%F-%k%M).txt''   So the year subfolder resolves correctly, and seemingly so does the first half of the date command. Tried appending the alias with sudo, didn't help.  Issue: For some reason, when I do this command the time value (%k) resolves to a few minutes ago... or maybe the last time nano ran this filename, maybe it's called an old buffer?   When I logoff it makes the correct file, then I save, wait a few minutes and try again, and it edits the old file instead of making a new one. Even if I remove the old file, it creates a new file with the old filename (incorrect time).  Here's a screenshot of what I'm trying to explain. Here I create a newjournal, save in nano within a second or two of executing the command, print the correct date/time, list the file that was created. Note the incorrect filename time (1249, as apposed to 13:11):    Anyways, it might be something simple, but I've tried different methods in the alias, such as putting double quotes around the whole thing and escaping properly, single quoting, escaping and spacing, etc.  Thanks for the read/help. Ubuntu 16.04 server.  EDIT: Here's the final alias that worked:  alias newjournal='nano /home/user/journals/"$(date +%Y)"/"$(date +%F-%H%M).txt"'   Needed to double quote the date commands inside the overall single-quoted alias. File l3backend-pdfmode.def not found when loading expl3 I updated my TeXlive distribution a couple times in the past week, and now I get this error when I include package expl3 (or any other package that includes it, such as fontspec). Before the updates it was working fine. Am I missing any package that should be installed? I tried searching for l3backend-pdfmode.def, but I found no results.  \documentclass{article} \usepackage{expl3} \begin{document} Hi \end{document}   λ lualatex a This is LuaTeX, Version 1.10.0 (TeX Live 2019)   restricted system commands enabled. (./a.tex LaTeX2e &lt;2018-12-01&gt;  luaotfload | main : initialization completed in 0.042 seconds (/opt/texlive/2019/texmf-dist/tex/latex/base/article.cls Document Class: article 2018/09/03 v1.4i Standard LaTeX document class (/opt/texlive/2019/texmf-dist/tex/latex/base/size10.clo)) (/opt/texlive/2019/texmf-dist/tex/latex/l3kernel/expl3.sty (/opt/texlive/2019/texmf-dist/tex/latex/l3kernel/expl3-code.tex)  ! LaTeX Error: File `l3backend-pdfmode.def' not found.  Type X to quit or &lt;RETURN&gt; to proceed, or enter new name. (Default extension: def)  Enter file name:  ! Emergency stop. &lt;read *&gt;   l.282   }   379 words of node memory still in use:    2 hlist, 1 rule, 1 dir, 3 kern, 1 glyph, 4 attribute, 48 glue_spec, 4 attrib ute_list, 1 write nodes    avail lists: 2:9,3:3,4:1,5:2,7:2,9:2 !  ==&gt; Fatal error occurred, no output PDF file produced! Transcript written on a.log. Is it correct? It's the message I see right before the metro/subway train shows up.  Shouldn't it be "train is approaching" or "train approaches"? Why was Sirius ever convicted? I want to run an app in 'kiosk' mode for public unattended access via touch screen (no mouse or keyboard attached). I can already automate a network user logon and then the app launches (but only in last state - not necessarily full screen - which I want) I know with Workgroup Manager you can hide the dock, use simple Finder etc...but need something more powerful. Ideally it needs to monitor if the app crashes out and restart it / log limited user off and back on again.  Thanks  Phil Can a purely biological account of man show a superiority over other animals other than that of belonging to a later stage of evolution or of greater possibilities of action, a superiority “in his very being”? Is there some special case where a fermion can mediate a force? How many and which are the uses of "const" in C++? As a novice C++ programmer there are some constructs that look still very obscure to me, one of these is const. You can use it in so many places and with so many different effects that is nearly impossible for a beginner to come out alive. Will some C++ guru explain once forever the various uses and whether and/or why not to use them? Display date/time in user's locale format and time offset I want the server to always serve dates in UTC in the HTML, and have JavaScript on the client site convert it to the user's local timezone.  Bonus if I can output in the user's locale date format. How to make a bootable ISO image from a bootable DVD? I want to make a .iso image of a bootable OS DVD, to use in VirutalBox.    Nautilus can make a .iso, but only of data files, it seems. The .iso it created was not bootable.  What is available to make a bootable .iso from a bootable DVD? Meaning of the chemical potential for a boson gas My lecturer told me that $\mu$, the , is zero or negative, and in the following example, mathematically it acts as a normalization constant. But is there any physical insight about why boson gas can be zero or negative? I think it is due to the fact the photon gas can pop up from nowhere (i.e. vacuum fluctuation). $$ f_{BE}(\varepsilon)=\dfrac{1}{e^{(\varepsilon-\mu)/(k_B T)}-1}$$ how to set static ip from bash scriptlinux I want to set static IP using bash script what I would to achieve I want to set the same IP to static which is come from DHCP, for example, my eth0 set to DHCP and it picks IP 51.89.84.82 for eth1 it pick 10.0.0.1 I want to set the same IP to static how can I achieve this in centos7. 1 ) find interface name and its ip , subnet mask , gateway. 2 ) loop each interface and set the same ip , subnet and gateway.