How do you display words, not variables, in equations in LaTeX?
What is the "correct" way of embedding text into math mode?
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. Applying Arzela-Ascoli to show pointwise convergence on $\mathbb{R}$. The statement I am trying to prove:  Let $\{ f_n \}$ be a sequence of equicontinuous, real valued, uniformly bounded   continuous functions on $\mathbb{R}$. Show that $\{ f_n \}$ has a convergent subsequence   which converges uniformly on any bounded subset of $\mathbb{R}$ and pointwise on all   of $\mathbb{R}$ to a continuous function. (Royden pg 210, problem 9).  I believe the first portion follows directly by Arzela Ascoli, once it is noted that if we are given some bounded subset $E$, we can look at $\overline{E}$ a compact set on which the family is equicontinuous and uniformly bounded.  For pointwise convergence on all of $\mathbb{R}$, the question seems to require some sort of diagonalization argument: Take some dense sequence of $\{x_i\}$ (the reals are separable) and write columns of $f_n$ applied to each individual point. There will be a convergent subsequence by Bolzano-Weirstrass and since the outputs are real and bounded by the uniform boundedness of the family. I am a bit fuzzy on this part: Repeating this literally pointwise and moving far enough diagonally, we can correct for different convergence "speeds" pluck out a single convergent subsequence for any given epsilon and have $\sup|f_n(x)-f(x)|&lt;\epsilon$ for any given epsilon. Finally, continuity should follow from an $\epsilon/3$ argument and equicontinuity of the family.  However, in formalizing the above, I am getting caught up in the notation of that sort of argument and am hoping for a more elegant solution. Is there a better way? Am I just generally confused and that's why I'm having trouble writing things down formally?   Thanks! lstlisting directly after section start removes margin before next section If I place an lstlisting directly after the start of a (sub)section, the margin before the following (sub)section (title) disappears.  Here is a an example document showing the issue:  \documentclass[twocolumn]{article} \usepackage{listings} \begin{document}  \section{Lorem ipsum} Consectetur adipiscing elit.  \begin{lstlisting} Lorem \end{lstlisting}  Sed quis ante nec mi volutpat congue eu id leo.  \section{Dolor sit amet} Suspendisse at lorem mattis, semper massa id, euismod risus.  \newpage  \section{Lorem ipsum} \begin{lstlisting} Lorem \end{lstlisting}  Sed quis ante nec mi volutpat congue eu id leo.  \section{Dolor sit amet} Suspendisse at lorem mattis, semper massa id, euismod risus.  \end{document}   And you can see the (annotated) result here:    Why is there no margin before the fourth section (title)? Is this a bug in my LaTeX environment (MacTeX-2013), or is this the expected behavior? How to effectively use the man pages? I am looking forward to mastering linux. I have been repeatedly told to refer to man pages when you are in doubt.   Though I am very comfortable with the commandline, I find it very difficult to quickly and efficiently use man to find help with commands. Hence I end up googling up how to get things done. It is even more painful when the system I am using does not have internet access and I have to walk back to my office to get help from the internet.  How does one master man pages? What are the most common tricks that can be used to quickly get whatever you want from the man page? What are the most commonly used keyboard shortcuts? For example,      I like chocolate, vanilla, and lemon and orange ice cream.    Indicating "lemon and orange" is a combined flavor, as an item in the list needing an initial and. Is there a nice solution to get a "presenter mode" for Latex presentations? Catenatives followed by infinitives and gerunds Flag Question Link After 3k Rep I noticed from a question about flagging vs closing (forgive me, I cannot find the link, please post it if you know it), that flags are CONVERTED into close votes for questions automatically. Can somebody confirm this?  If confirmed, I was wondering why the link (again, for questions only) is simply not removed to avoid confusion? mysql_fetch_array does not retrieve all rows From a collection of short stories I read at least 30 years ago. :/  This is going to be incredibly vague, so apologies in advance.  A human and alien (blob/octopus-like...I think) from warring races are captured by an unidentified higher intelligence and put in an enclosure (more like a zoo habitat rather than a prison cell). They are separated by an invisible barrier and can see each other.  I think they were both trying to escape or trying to figure out how they could kill the other first. There was a bulldozer/loader type vehicle they could use at one point to build a shelter (or something).  I believe they were crafting makeshift weapons as well. There was no communication between the two as far as I remember.  The main tension in the story seemed to be was the race to get to the other first.  This has been stuck in my head forever. I know its a very common theme.  The short stories may have included Asimov. C++ handling very large integers I am using the RSA Algorithm for encryption/decryption, and in order to decrypt the files you have to deal with some pretty big values. More specifically, things like   P = C^d % n   = 62^65 % 133   Now that is really the only calculations that ill be doing. I have tried using Matt McCutchen's BigInteger Library, but I am getting a lot of compiler errors during linking, such as:  encryption.o(.text+0x187):encryption.cpp: undefined reference to `BigInteger::BigInteger(int)'  encryption.o(.text+0x302):encryption.cpp: undefined reference to `operator&lt;&lt;(std::ostream&amp;, BigInteger const&amp;)'  encryption.o(.text$_ZNK10BigIntegermlERKS_[BigInteger::operator*(BigInteger const&amp;) const]+0x63):encryption.cpp: undefined reference to `BigInteger::multiply(BigInteger const&amp;, BigInteger const&amp;)'   So I was wondering what would be the best way to go about handling the really big integers that come out of the RSA Algorithm.  I heard that a possibility would be to declare your variables as a double long, so...  long long decryptedCharacter;   but I'm not sure exactly how big of an integer that can store.    Well for example, I try to compile and run the following program using dev C++:  #include iostream  #include "bigint\BigIntegerLibrary.hh"  using namespace std;  int main() {     BigInteger a = 65536;     cout &lt;&lt; (a * a * a * a * a * a * a * a);     return 0; }   then I get those errors.  Derek, I thought that by including the BigIntegerLibrary.hh file, that the compiler would go through and compile all the necessary files that it will use.  How should I try and compile the program above in order to resolve the linking errors? Is there any substantial disadvantage to booking two one-way tickets, versus a round trip ticket?  Booking vacation travel with family of four for winter, from Northeast US to Mexico.  Round trip is pricy and no preferred travel times.  Two one ways is slightly cheaper with much preferred timing.      Different airlines there and back, both reputable national US carriers. Why 'Mrs.' isn't read as 'mistress'? Wasn't Mrs. short for mistress? Why do we read Mrs. as missus(or sometimes missis) instead of mistress? Convert a list to a data frame How to solve ~(P → Q) : P & ~Q by natural deduction? I'm trying to solve the following by natural deduction:  ~(P → Q) : P &amp; ~Q  It's a trivial problem if identities are used, as can be seen by the following:   {1}      1.   ~(P → Q)                    Prem. {1}      2.   ~(~P ∨ Q)                   1 Mat. Impl. {1}      3.   ~~P & ~Q                    2 De Morgan's {1}      4.   P & ~Q                      3 DNE   Starting from step 2 of the previous proof, it's also fairly straightforward to reach a conclusion using natural deduction; i.e. starting from ~(~P ∨ Q), which is an identity of ~(P → Q):  ~(~P ∨ Q) :  P &amp; ~Q   {1}      1.   ~(~P ∨ Q)                   Prem. {2}      2.   ~P                          Assum. {2}      3.   ~P ∨ Q                      2 ∨I {1,2}    4.   ~(~P ∨ Q) & (~P ∨ Q)        1,3 &I {1}      5.   P                           2,4 RAA {6}      6.   Q                           Assum. {6}      7.   ~P ∨ Q                      6 ∨I {1,6}    8.   ~(~P ∨ Q) & (~P ∨ Q)        1,7 &I {1}      9.   ~Q                          6,8 RAA {1}      10.  P & ~Q                      5,9 &I   I also found that it's fairly easy to conclude ~Q:   {1}      1.   ~(P → Q)                    Prem. {2}      2.   Q                           Assum. {3}      3.   P                           Assum. {2,3}    4.   P & Q                       2,3 &I {2,3}    5.   Q                           4 &E {2}      6.   P → Q                       3,5 CP {1,2}    7.   ~(P → Q) & (P → Q)          1,6 &I {1}      8.   ~Q                          2,7 RAA ... {1,2}    12.  P & ~Q   In spite of how easy it looks, I can't figure out how to do it. I believe it can be done in 12 steps. Thanks! G is the product of its Sylow subgroups I want to prove the following:     If every Sylow subgroup of $G$ is a normal subgroup, then $G$ is isomorphic to the product of its Sylow subgroups.   So far, I have gotten to realize that for every prime factor $p_i$ of $|G|$, there is exactly one Sylow subgroup and that these form a partition of $G$. However, I am having trouble finding the isomorphism between $G$ and the product which I will call $G_1 \times \ldots \times G_r$. My idea was just to use the mapping $(g_1, \ldots, g_r) \mapsto g_1\ldots g_r$, which is a bijection, but I don't see whether it is a homomorphism. In general, not all elements of $G_i$ commute with elements of $G_j$, so one cannot simply change the order of the elements in the product. I am looking for an animated series, shown in the 90s. I remember an episode where the boy went into VR and fought a medusa-like monster.  It may be that this was the premise for the whole series: That the children heroes go into VR every time to fight the evil guys.   I am not sure if the whole series were computer-animated but surely there were such elements together with classic animation.  The style of the series was quite dark and cool for teenage boys. I had a Windows 7 installation on my 1TB SSD laptop with 4 primary partitions. I wanted to install Lubuntu Linux but the process failed with an error which kind of said:     cannot write MBR   It gave me an option to put an MBR somewhere else or to boot without MBR. None of these options worked and it became a useless install. After this failed dual boot attempt, Windows would give a BSOD wit Code 007 ending - translated to inaccessible boot device. That is a huge NTFS partition and I don't think I have any other media to back it up before formatting it.  My only option is somehow rebuilding the old MBR. Hence I started creating a whole bunch of live USB boot environments  including Super GRUB Disk and MiniTool Partition Editor Live CD, but none of this could ever help me recover my lost MBR.  Win PE 4.0 reported the partition as bad disc and the latest version did not even have an option for MBR. I managed to create a live USB with Testdisk Running on Ubuntu. Whatever partition was seen as HPFS on quick  search was written in the MBR and after this the disc is totally unreadable. Earlier I could see my data through Linux and now I can see nothing.  Here are my questions:   When some data could be read (but Windows would not boot) the MBR was shown as dynamic MBR. As far as I know dynamic MBR is a Windows concept and my Linux attempt would not have converted it. While installing Linux I tried to delete 2 primary partitions which I found to be not really useful (I backed up their data first). So I am not sure what created the dynamic MBR.  Did my dual install convert those basic MBR to dynamic MBR? How did that happen? Here is a screenshot of my deep search attempt with TestDisk which is still in progress. I can see 4 NTFS partitions and one of these two system partitions should be the original MBR. When I tried to write both these system partitions (after stopping the deep search momentarily) I got an error like "partition not found". The file system utility showed me that there is no backup sector and that may be the reason why it cannot be read.  What other Resorts do I have, other than doing a plain format and then using some recovery tool to recover all my lost data? I am not even sure I will recover the folder names.  What other options should I be trying to restore my old MBR so Windows can boot?      Update  as of last I managed to create a live USB w/ W7  I used AOMEI Partition assistant 6.0 and on deeper  search I can see the 909 gb partition - it wont let me select it for recovery. I am sorry if my earlier Q was not clear. I referred to restoring an MBR from a partition . What happened earlier  - superficial search revealed all old partitions but NOT this one   and the later partitions showed up on Deep search &amp; I can select Other partitions for recovery but for this one. It wont let me select the partition I want  Here are some more screen shots    So in summary all that happened is   4 primary windows 7 partition -->  wanted to install Linux so I deleted 2 non bootable partitions after backing them up  ended up with BSOD ( inaccessible boot device ) and noticed that some windows partitions showed up as Dynamic MBR  (using Live USB Linux,gparted )  Use standard testdisk procedure to write MBR from the superficial NTFS-HDFS partition.  this is where I screwed up . Hindsight I should have really done a deep search and caught the 901gb partition to write MBR. I am not sure if I can undo this action  after this - life gets worse. I am  on a ventilator ! Now C is unreadable by windows and linux both and testdisk cannot list file ( see screenshots ) and I have been using a series of live USB's with some recovery s/w but no luck. ALL I want is to recreate that 901gb partition and the boot partition and put the MBR on the latter so I can get my windows 7 back again Storing C++ template function definitions in a .CPP file I have some template code that I would prefer to have stored in a CPP file instead of inline in the header.  I know this can be done as long as you know which template types will be used.  For example:  .h file  class foo { public:     template &lt;typename T&gt;     void do(const T&amp; t); };   .cpp file  template &lt;typename T&gt; void foo::do(const T&amp; t) {     // Do something with t }  template void foo::do&lt;int&gt;(const int&amp;); template void foo::do&lt;std::string&gt;(const std::string&amp;);   Note the last two lines - the foo::do template function is only used with ints and std::strings, so those definitions mean the app will link.  My question is - is this a nasty hack or will this work with other compilers/linkers? I am only using this code with VS2008 at the moment but will be wanting to port to other environments. Background: I'm making a retro style game for both fun, and an unofficial contest.   I have bought a song, song X. However, I then found notation for song X, and all the instruments. So basically, it's a program recreating song X in a sense.  I took the notation, and used another program to make it sound like a chiptune does (just for the fun of it, and it sounds good :P). However, I would like to use this in my game. It's non-commercial, but is this allowed? I would've said yes, because I'm not actually using the original, copyrighted song.  I think the main point is can I:   Use the music with no reference to the author Use the music with a reference, since it's non-profit. Not use the music at all since the melody and rhythm is copyright.