Comma before and
Comma before "and" in compound sentences?
Show that holomorphic $f_1, . . . , f_n $ are constant if $\sum_{k=1}^n \left| f_k(z) \right|$ is constant. While studying for an exam in complex analysis, I came across this problem. Unfortunately I was not able to solve it. Any help would be greatly appreciated.  Let $U ⊂ \mathbb{C}$ be a domain and $f_1, . . . , f_n : U \rightarrow \mathbb{C}$ holomorphic functions, such that $\sum_{k=1}^n \left| f_k(z) \right|$ is constant on $U$. Show that $f_1, . . . , f_n$ are constant. The maximum energy stored by a capacitor is when it’s (theoretically) fully charged, or just before it reaches dielectric breakdown, if you will. It seems, then, that the maximum energy ($E$) $=QV$, where $Q$ is the maximum charge attained and $V$ is the maximum ‘voltage’. However, the correct equation for the maximum energy is $E=\frac{1}{2}QV$. Why? I’m looking for a simple, intuitive explanation because my reasoning that $E=QV$ seems sound to me. I know about , but I’m looking for a reason to disprove my specific argument that $E=QV$ and not the “Oh, it comes from integration.” response. Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ Why is surface charge distribution uniform for a conducting sphere? Can't it be arbitrary? If the charge $q_1$ has to repel the charge $q_2$, the electric field has to go inside the conductor which contradicts the fact that electric field inside conductors is zero. Then why do the charges distribute themselves in a particular manner? Why can't they be distributed over a small place on the conductor? 32-Bit UEFI Boot Support I recently purchased a Dell Venue 8 Pro.  I have read online about what it needs in order to boot into Ubuntu or any other Linux distro.  From what I have gathered, what really needs to happen is 32-bit UEFI support.  I got information from several sites, but where it was really summed up was in this previous post:    "Ubuntu has no 32-Bit Uefi installer, ask them to add support, they may listen."  That thread was specifically about getting the Dell Venue 8/11 Pro to boot Linux.  However, there are a slew of new tablets coming out running 32-bit UEFI, and I think this is a trend that is going to continue to grow:  mjg59.dreamwidth.org/26734.html  "UEFI on IA32 isn't going away. It will show up on more embedded systems as time goes on."  Others have tried this before with mixed success, but haven't gotten things to work quite right:    "It doesn’t really work. But it’s interestingly close."  I'm trying to start a thread dedicated thread on 32-Bit UEFI support, because that is the piece that is missing (from what I understand).  I don't really know how to work with this stuff, but I'm hoping that some people who do will read this and either contact canonical or somehow patch Ubuntu (or any other distro, really) to work this way.  Again, I don't really know how to do any of this...I'm just hoping someone can help!  Dell Venue 8, 11 Pro, Toshiba Encore, Acer w3, w4, Lenovo miix, and the asus transformer a100 are all tablets that will be affected by this...I think these are going to be incredibly popular tablets once their prices go down, and I think people are really going to get interested in putting linux on these.  Any help is appreciated, and thank you all so much!  EDIT 4/1/14:  Just booted into Ubuntu on my DV8Pro!  All you have to do is follow the instructions here:    To summarize, what you need to do is download the latest Ubuntu 14 daily build (AMD 64 is what worked for me) and put that on a USB drive via Rufus.  'For “Partition scheme and target system type”, choose “GPT partition scheme for UEFI computer.”' You can get the latest daily build here:   You will need to put the file found below into your EFI/Boot directory:   Disable Secureboot and such--follow the original instructions link if you don't know how.  Then set the USB to boot first in BIOS.  This should get you to GRUB.  The rest I'll quote the original author, it's real straightforward:  In the GRUB menu, highlight “Try Ubuntu”, and press “e” to edit it. In the editing screen, scroll down to the command line options, where it says “quiet splash”. Delete “splash” and replace it with: video=VGA-1:1368x768e reboot=pci,force [EDIT: For those of us using the Dell Venue 8 Pro, you need to use this line instead: video=VGA-1:800x1280e reboot=pci,force That accounts for the different resolution of screen.]  Then press F10 to boot. You should get all the way to the Desktop.  There you have it!  This got me to the Unity Desktop.  Problems:  Wifi is not working, but I used an Ethernet adapter and that seems to work.  I tried rotating the screen, but it throws your touch input off to one side, so I left it.    Make sure to read the tutorial mentioned earlier, it is more comprehensive, even though it is made for the ASUS T100.  If nothing else, the instructions I have provided should be enough just to get you to the Unity Desktop, which for me was amazing!  Hope you too can have the same success and keep building on it!  For what it's worth, someone else did this and posted a youtube video here: Show that any prime ideal from such a ring is maximal. Ubuntu 16.04 multitouch gestures Isometries of the sphere $\mathbb{S}^{n}$ Got this as homework and I don't know how to tackle this. Help please!  Prove that the isometries of $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$, with the induced metric, are restrictions to $\mathbb{S}^{n}$ of the linear orthogonal transformations. How can I confirm that my I-94 form was correctly processed after leaving the US in 2007? Restrictive vs Non-restrictive Relative Clause Can you tell me the difference in the meaning of the two sentences below?  As a defining relative clause.     The location which was called Central Park was a park in New York.    As a non-defining relative clause.     The location, which was called Central Park, was a park in New York. I understand that the Bernoulli effect is a flawed explanation for the cause of lift, and does not cause much at all, but how much?   Is there any experimental data on the force caused by the Bernoulli effect? Maybe implicitly through data of the pressure difference between the top and underside of an aeroplane's wings. After that, I assume I could (crudely approximating the pressure to be acting perpendicularly to the flight direction) use $\Delta P A$ to work out the net force on the plane.    Perhaps there is another way to quantitatively analyse the extent to which the Bernoulli effect causes lift.  Edit: see  short cartoon (content similar to Mike Dunlavey's answer). Repeatedly taking differences on a polynomial yields the factorial of its degree? "Notice: Undefined variable", "Notice: Undefined index", and "Notice: Undefined offset" using PHP How to remove the left part of a string? I have some simple python code that searches files for a string e.g. path=c:\path, where the c:\path part may vary. The current code is:  def find_path(i_file):     lines = open(i_file).readlines()     for line in lines:         if line.startswith("Path="):             return # what to do here in order to get line content after "Path=" ?   What is a simple way to get the text after Path=? Prove that the equation $x^4+(2k^2)x^2+2k^4$ is not a perfect square Flash/SWF Editor For Ubuntu 12.10 How can I vertically align entries in my acronyms' list? I use the Glossaries package to generate it. Here is MWE:  % !TeX program = xelatex % !TeX encoding = utf-8 % !TeX spellcheck = ru_RU  \documentclass[12pt, a4paper]{article}  \usepackage{polyglossia} \setdefaultlanguage{russian} \setmainfont[Ligatures=TeX]{Times New Roman}  \usepackage[xindy={glsnumbers=false}, nonumberlist, nopostdot, nogroupskip]{glossaries} %\setglossarystyle{super} % uncomment for style "super" \makeglossaries  \begin{document}  \newacronym{bam}{ДАП}{Двунаправленная Ассоциативная Память} \newacronym{ai}{ИИ}{Искусственный Интеллект} \newacronym{am}{АП}{Ассоциативная Память} \newacronym{ann}{ИНС}{Искусственная Нейронная Сеть} \newacronym{an}{ИН}{Искусственный Нейрон} \newacronym{it}{ИТ}{Информационные Технологии} \glsaddall  \printglossaries  \end{document}   With the default style I have the following:  As you can see, the long names are not vertically aligned. What I want can be achieved with the    \setglossarystyle{super}   But it this case, short names lose the BOLDness, some small horizontal indent appears and the spacing between lines becomes smaller:  where RED is some extra indent, which I don't need. Thank you for your answers! Is the set of points in the plane whose coordinates are either both irrational, or both rational connected? Execute PowerShell Script from C# with Commandline Arguments I need to execute a PowerShell script from within C#. The script needs commandline arguments.  This is what I have done so far:  RunspaceConfiguration runspaceConfiguration = RunspaceConfiguration.Create();  Runspace runspace = RunspaceFactory.CreateRunspace(runspaceConfiguration); runspace.Open();  RunspaceInvoke scriptInvoker = new RunspaceInvoke(runspace);  Pipeline pipeline = runspace.CreatePipeline(); pipeline.Commands.Add(scriptFile);  // Execute PowerShell script results = pipeline.Invoke();   scriptFile contains something like "C:\Program Files\MyProgram\Whatever.ps1".   The script uses a commandline argument such as "-key Value" whereas Value can be something like a path that also might contain spaces.  I don't get this to work. Does anyone know how to pass commandline arguments to a PowerShell script from within C# and make sure that spaces are no problem? Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$. Given the function   $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} &amp; x \neq 0 \\ 0 &amp; x = 0 \end{array}\right. $$  show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.  So I have to show that nth derivative is always equal to zero $0$. Now I guess that it is about finding some dependencies between the previous and next differential but I have yet to notice one. Could you be so kind to help me with that?  Thanks in advance!