Apsotrophe for plural possessive noun. E.g. Company Name
What is the correct possessive for nouns ending in "‑s"?
I'm reading through a proof in my lecture notes:     Every convergent sequence of a metric space has a unique limit.   Suppose $\lim_{n \to \infty} x_n = l$ and $\lim_{n \to \infty} x_n = m.$ Then for every $\epsilon &gt; 0,$ there exists $N$ such that for every $n \geqslant N,$  $$d(l,m) \leqslant d(l,x_n) + d(x_n,m) &lt; \epsilon + \epsilon = 2 \epsilon. $$  So $(\star) \hspace{1mm} 0 \leqslant \frac{d(l,m)}{2} &lt; \epsilon \Longrightarrow \hspace{1mm} (\star^\prime) \hspace{1mm} \frac{d(l,m)}{2} =0 \Longrightarrow l = m.$  I can follow the proof up until the jump from $(\star)$ to $(\star^\prime)$. Why does $\frac{d(l,m)}{2}=0$? Why is there such a big difference between "Size" and "Size on disk"? I read recently that if you hold a bullet in one hand and a pistol in the other, both hands at the same height, and subsequently fired the pistol at the same time as dropping the bullet, both bullets would hit the ground at the same time (assuming the fired bullet encounters no obstacles).  Is this true, and if so can you explain the reasoning behind it with as few equations as possible? I assume it is essentially that gravity is unaffected by momentum, but it just sounds counter-intuitive. $3^{n+1}$ divides $2^{3^n}+1$ Describe all positive integers,n such that $3^{n+1}$divides $2^{3^n}+1$. I am little confused about what the question asks-if it asks me to find all such positive integers, or if it asks me to prove that for every positive integer n,$3^{n+1}$ divides $2^{3^n}+1$. Kindly clarify this doubt and if it's the former part, please verify my solution-n=1. I have the below list of files  aro_tty-mIF-45875564pmo_opt aro_tty-mIF-45875664pmo_opt aro_tty-mIF-45875964pmo_opt aro_tty-mIF-45875514pmo_opt aro_tty-mIF-45875524pmo_opt   that I need to rename to  aro_tty-mImpFRA-45875564pmo_opt aro_tty-mImpFRA-45875664pmo_opt aro_tty-mImpFRA-45875964pmo_opt aro_tty-mImpFRA-45875514pmo_opt aro_tty-mImpFRA-45875524pmo_opt Multiple bibliographies (reference lists) using embedded bibliography database I would like to have multiple bibliographies (reference lists) as described in   ().   However, I'd like to use with a single .tex file, which would contain the bibliography database embedded, as in  ...   Now, I want to use letter documentclass, for which the bibliography needs to be defined ()   So what I end up is with this example (code below, compiled with just pdflatex, no bibtex(which I'd like to avoid in this case)):     ... obviously, something works (multibib recognized primary and secondary cites, assigning them number "1" as start) -- however I don't get two, separate primary and secondary list of references; would that be possible in this context? (doesn't have to be with multibib, but I'd like to avoid a separate .bib file and a bibtex call)  EDIT: I'd also like separate cite keys (i.e. rendered as [a1] and [b1]; not yet sure if multibib can do that; edit: no it cant,  says "You can also edit the .bbl files created by BibTEX and add 'a', 'b',. . . to the labels, but your changes will be overwritten by subsequent BibTEX runs." and recommends \cite[a]{key}) - how possible would that be?  &nbsp;  The code is this:   \documentclass{letter}  % Environment thebibliography undefined when using letter: http://www.latex-community.org/forum/viewtopic.php?f=4&amp;t=3359 \makeatletter \setlength \labelsep {.5em} \newcommand\newblock{\hskip   .11em\@plus.33em\@minus.07em} \let\@openbib@code\@empty \newcommand\refname{References} \newcommand\section{\@startsection   {section}{1}{\z@}%   {-3.5ex \@plus -1ex \@minus -.2ex}%   {2.3ex \@plus.2ex}%   {\normalfont\Large\bfseries}} \newenvironment{thebibliography}[1] {\section*{\refname}%   \@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}%   \list{\@biblabel{\@arabic\c@enumiv}}%   {\settowidth\labelwidth{\@biblabel{#1}}%     \leftmargin\labelwidth \advance\leftmargin\labelsep \@openbib@code     \usecounter{enumiv}%     \let\p@enumiv\@empty \renewcommand\theenumiv{\@arabic\c@enumiv}}%   \sloppy \clubpenalty4000 \@clubpenalty \clubpenalty   \widowpenalty4000%   \sfcode`\.\@m} {\def\@noitemerr {\@latex@warning{Empty       `thebibliography' environment}}%       \endlist} \makeatother  \usepackage{multibib} \newcites{pri}{Primary sources} \newcites{sec}{Secondary sources}  \begin{document}   \begin{letter}{Addressee} Blah, blah - content...  Primary cite: \citepri{lamport94} ...  Secondary cite: \citesec{lamport94} ...   \begin{thebibliography}{9}  \bibitem{lamport94}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1994.  \bibitem{lamport95}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1995.  \end{thebibliography}  \end{letter}   \end{document} How to search based on part of file name in Google Drive I'd like to be able to find all files in my Google drive that contain the string "_INSTRUCTION_" in the file name. Is there a way to do this?  EDIT 1: When I search for "RMA2013", I should get a dropdown with several options. As you can see, there are several Word and .pdf files with the phrase "RMA2013". Still I get no dropdown help:   And even if I click the search button, I get nothing: Sharing Open Source Python/ArcPy Code and Models for ArcGIS Desktop? I will be writing scripts for ArcGIS Desktop in Python with ArcPy.   Is there a community or open source project where code and models can be shared? Given the length ot the sides $a , b$ and $c$  of $ \triangle ABC$.  What is the length of the radius of the circumcribed circle?  After some formula substitution I came to the monster formula:  $$ \frac {a b c}{\sqrt{2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4}} $$  Can this formula be simplified? This code in JS gives me a popup saying "i think null is a number", which I find slightly disturbing. What am I missing?    if (isNaN(null)) {   alert("null is not a number"); } else {   alert("i think null is a number"); }     I'm using Firefox 3. Is that a browser bug?  Other tests:    console.log(null == NaN);   // false console.log(isNaN("text")); // true console.log(NaN == "text"); // false     So, the problem seems not to be an exact comparison with NaN?  Edit: Now the question has been answered, I have cleaned up my post to have a better version for the archive. However, this renders some comments and even some answers a little incomprehensible. Don't blame their authors. Among the things I changed was:   Removed a note saying that I had screwed up the headline in the first place by reverting its meaning Earlier answers showed that I didn't state clearly enough why I thought the behaviour was weird, so I added the examples that check a string and do a manual comparison. How can I remove a key from a Python dictionary? What are the ways to position things absolutely on the page? I know that there is tikz, but also eso-pic and textpos.  I'd like to see a list of possible ways (not necessarily packages) for absolute positioning of things (which I believe amounts to "boxes") on the page, together with their advantages and disadvantages (for instance: tikz's current page node is very easy to use, but usually requires two LaTeX runs, and overhead from loading tikz if you don't need it for any other things is considerable).  Both LaTeX and ConTeXt solutions are welcome Why don't metals bond when touched together? It is my understanding that metals are a crystal lattice of ions, held together by delocalized electrons, which move freely through the lattice (and conduct electricity, heat, etc.).   If two pieces of the same metal are touched together, why don't they bond?   It seems to me the delocalized electrons would move from one metal to the other, and extend the bond, holding the two pieces together. If the electrons don't move freely from one piece to the other, why would this not happen when a current is applied (through the two pieces)? There's a problem statement:     For each $m \in \mathbb{N}$, we construct a sequence $m_0$, $m_1$, $m_2,\dots$ denoted $S_m$, recursively via $m_0=m$ and      $$m_{i+1} = m_i + \left\lfloor \sqrt{m_i} \right\rfloor$$      for all $i \ge 0$.   Here, $\lfloor x \rfloor$ is the floor of $x$, the greatest integer less than or equal to $x$.   Hence, we have $\left\lfloor \sqrt{10} \right\rfloor=3$ and $\left\lfloor \sqrt{29} \right\rfloor=5$.      Show that for each positive integer $m$, the sequence $S_m$ contains the square of some integer.   I'm pretty certain that this can be proved with induction. I am just not quite sure what to induct on. Examining examples shows that $S_m$ always results in a perfect square eventually, though I'm not sure how to prove it. .fbx export why there are no materials or textures? As I understand these two statements:   An electromagnetic field gives particles charge A photon is a quantum of electromagnetic field   It must mean that a photon carries charge. But I guess it isn't true. Why not? Question: Let $E$ be a normed space. Let $G$ be a closed subspace of $E$ and let $F$ be a finite dimensional subspace of $E$. Show that $F+G$ is a subspace of $E$ and is closed.  I'm having trouble in showing $F+G$ to be closed. I know that $F$ is itself closed and complete, as it is a finite dimensional subspace of a normed space, and that if $F$ were compact that $G+F$ would be closed. I also know that the closed unit ball of any finite dimensional normed space is compact.  I tried two methods. One was to take a convergent sequence $(x_n)_{n \in \mathbb N}$ in $G+F$. Then we can write $x_n = f_n + g_n$ where $f_n$ and $g_n$ are sequences in $F$ and $G$ respectively, and I attempted to find a way to force the individual components $f_n$ and $g_n$ inside the unit ball which would enable me to say that they had convergent subsequences. I couldn't see how to do this, however.   The other thought was to try and show that $F$ is compact, but I don't see a way to do this as I can't imagine it to be simply true without some other conditions on $F$.  Are one of these methods the right way to go? Or should I go another direction? I would appreciate any help I can get, although I would prefer not to be presented with a full proof so that I can do some work for myself. Thanks! I'd like to find out why  \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = 6 \end{align}  I tried to rewrite it into a geometric series  \begin{align} \sum_{n=0}^{\infty} \frac{n^2}{2^n} = \sum_{n=0}^{\infty} \Big(\frac{1}{2}\Big)^nn^2 \end{align}  But I don't know what to do with the $n^2$. How do rich people guarantee the safety of their money, when savings exceed the FDIC limit? I am curious how a millionaire would guarantee the safety of his money, given that the FDIC only insures up to $250K of an individual's deposits at a bank.  If someone had $3 million that they wanted to put into the bank, would they have to open up 12 different bank accounts and deposit $250K into each one, so that all of his money is insured by the FDIC?  Or is there an easier way to guarantee the safety of all his money? What metadata does a LibreOffice file contain and how to remove it? I am using LibreOffice on Linux. One way I know of viewing and removing metadata from a LibreOffice file is going to File -&gt; Properties -&gt; General, unchecking  “apply user data” and click reset. Then go to “Description” and “Custom” Properties and remove information from there, if exists.   Does it truly remove all the metadata or something is left and what is it and what other methods can I use to remove it?   Also, if I unzip a LibreOffice file, is contents.xml the only place where metadata may be found?  If I forgot to remove any metadata, is it possible that the username is transmitted?