What is the argument behind: 1/(1+x) can be approximated by 1-x if x&lt;&lt;1 ?  I couldn't find where this approximation comes from/derived.
A good approximation of $(1+x)^n$ is $1+xn$ when $|x|n &lt;&lt; 1$.  Does this approximation have a name?  Any leads on estimating the error of the approximation?
I use the following table:  \documentclass[a4paper,              12pt,                                        bibliography=totoc,              index=totoc,             abstracton,              headsepline,             ]{scrreprt}  \usepackage{booktabs}    \begin{document}   \begin{table}[H]\hspace*{-1cm} \centering \begin{tabular}{lcclll} \toprule  Company&amp; \multicolumn{2}{c}{exceedances (empirical)}&amp;exeedances (expected)&amp;no. of obs.\\ \cmidrule(r){2-3}\\ &amp;standard normal&amp;standardized Student's-t&amp;&amp;\\ \midrule Company1&amp;1&amp;5&amp;9&amp;13\\ Company2&amp;2&amp;6&amp;10&amp;14\\ Company3&amp;3&amp;11&amp;15\\ Company4&amp;4&amp;8&amp;12&amp;16\\ \bottomrule \multicolumn{4}{l}{\begin{footnotesize}an additional hint about a detail\end{footnotesize}} \end{tabular} \caption{Here is the caption.} \label{labelhere} \end{table}   \end{document}     Which gives    I have two problems:   The space after the \cmidrule(r){2-3} is too big, how can I change this? The table looks not that good, how can I improve it in general? Recursively delete empty directories in Windows Using X11 in OSX 10.8.3 : "ld: library not found for -lX11" show that every continuous real-valued function defined on $S_{\mathbb{\Omega}}$ is eventually constant Google map signed api key errors in Android Is there a formal spelling for the English letter names? Mac turns off randomly and has lines on screen I have the following code that shows either a bug or a misunderstanding on my part.  I sent the same list, but modified over an ObjectOutputStream.  Once as [0] and other as [1].  But when I read it, I get [0] twice.  I think this is caused by the fact that I am sending over the same object and ObjectOutputStream must be caching them somehow.  Is this work as it should, or should I file a bug?   import java.io.*; import java.net.*; import java.util.*;  public class OOS {      public static void main(String[] args) throws Exception {         Thread t1 = new Thread(new Runnable() {             public void run() {                 try {                     ServerSocket ss = new ServerSocket(12344);                     Socket s= ss.accept();                      ObjectOutputStream oos = new ObjectOutputStream(s.getOutputStream());                     List same = new ArrayList();                     same.add(0);                     oos.writeObject(same);                     same.clear();                     same.add(1);                     oos.writeObject(same);                  } catch(Exception e) {                     e.printStackTrace();                 }             }         });         t1.start();          Socket s = new Socket("localhost", 12344);         ObjectInputStream ois = new ObjectInputStream(s.getInputStream());          // outputs [0] as expected         System.out.println(ois.readObject());          // outputs [0], but expected [1]         System.out.println(ois.readObject());         System.exit(0);     } } I'm using Bash on both client and server. When running a command over SSH:   ssh &lt;host&gt; 'declare' gives a list of shell variables. ssh &lt;host&gt; 'mount' gives a list of mountpoints.   However, declare is a Bash builtin, while mount is an external command. How does SSH know which to run if there is a shell builtin and an external command with the same name on the server? Get number of unread sms Omitting 'that' in this sentence Need a MP3 ID3 tagger, and cover fetcher I need to tag my MP3 library, and have tried kid3 (which was manual tagging), when I used Kubuntu 9.10 (I now use Ubuntu Meerkat) Here are the features I am hoping for:   A good and clean UI. Tagging should be automatic, like Winamp's autotag feature, which rocks, btw! It should also embed the cover-art in the mp3, not copy a jpeg file in the folder, because now-a-days all players support displaying cover art. But acceptable if not possible. Rename the files as per some regular expression like %TrackNo - %Artist - %Title.  Should be accurate, and more importantly smart. I want to start tagging at night, and hopefully my collection should be done by the morning, w/o it being stuck at a user prompt at 1%.   If one app cant do all, I am willing to use 3, wouldn't mind exposure to a few more apps ;)  I have used picard or someting, and I didnt like it quite a lot. But wouldn't mind using it, if there is no other alternative.   Thanks for your time! I've seen many variants of this problem online, but not quite the same as this, so I don't believe this is a duplicate.  The famous differential equation problem models a skydiver jumping out of a plane as  $$m \frac{dv}{dt} = -mg-cv$$ where m is the mass of the skydiver, g is acceleration due to gravity (9.81 m/s) and c is the coefficient of air resistance.  Here is my question:     Suppose the skydiver exits the stationary helicopter at an altitude of 2000m and opens the parachute at 500m. For this exercise, the parachute opens instantaneously. Suppose m= 70kgs, the coefficient of air resistance is c = 14 for a free falling body and c = 105 after the parachute opens. When does the skydiver open the parachute? When does the Skydiver hit the ground?   So my attempt at a solution kinda starts like this, but this is where I get stuck:  Let's assume that s(t) is the distance the body falls in time t from its initial point of release, then $v = \frac{ds}{dt}$ and $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$. Thus we have a second order differential equation: $m \frac{d^2s}{dt^2} = -mg-c\frac{ds}{dt}$. By simply manipulation, we get $m \frac{d^2s}{dt^2} + c\frac{ds}{dt} = mg$. I originally started thinking of finding the integrating factor, however, this seems more complicated and won't help produce the equation that I need.  Would anyone be able to help? Much thanks in advanced! Prove 24 divides $u^3-u$ for all odd natural numbers $u$ At our college, a professor told us to prove by a semi-formal demonstration (without complete induction):   For every odd natural: $24\mid(u^3-u)$   He said that that example was taken from a high school book - maybe I din't get something, but I really have no idea to prove that without using complete induction.  Any idea of smart demonstration?  Thanks for your help. (Excuse my bad English.) Why are my SVG imports so small? I'm definitely a newbie with blender still, as I've dipped my toe in a bit being a video editor.  My issue is importing .svg saved from illustrator.  Just simple vectors with not too many paths.  Just a thick text company logo.  I want to bring in and extrude, light, etc.  I've done it before in older versions of blender with no problem.  Am I not doing something right? Can someone tell me how to get an eps saved as an svg from illustrator to size correctly? Can a single vertex be rotated? In the duplication options for the dupli verts there is a option for Rotation. Witch brings me to my question, is there anyway to rotate a vertex? Vertex's have normals, is there any way to control which way the normal of a single unconnected vertex points? Obtaining remote host name in bash script Sorry my knowledge of Linux bash commands is pretty basic, I've been searching for a while but I'm not 100% sure what I need to search for.  I was wondering if there's a way to grab the current logged in users remote host name in a Linux bash script? I have a script in which I need to log each time a user runs it. I'm obtaining the date like so:  cdat=`/bin/date +%a' '%d' '%h' '%Y', '%H':'%M`;   I now need to add the users remote host (not the username they logged in with). I'm not 100% sure I'm using the right terminology here either, just to clarify; by 'remote host name' I mean the same output that prints on the screen on most servers I've logged into over ssh, for example:  Last login: Mon Jul 22 16:35:09 2013 from win7-i7-stuart.my.domain.com   I'm looking for the win7-i7-stuart.my.domain.com bit. Why did the writers select the date November 5, 1955 in Back to the Future? Is there any significance to the date November 5, 1955 that was chosen in Back to the Future? Is it one of the directors birthday or something like that or were there specific real life events that day in history?   I am not looking for in-universe answers. SOQL results returning RecordTypeId which is not included in query I am running this script from developer console.  List&lt;Account&gt; lstAccount = [SELECT Id, Name, BillingStreet From Account LIMIT 1]; System.debug(lstAccount);   Debug Log:     23:47:04:012 USER_DEBUG [5]|DEBUG|(Account:{Id=0019000001nNLGEAA4, Name=Acct_nm001, BillingStreet=xyz Road, RecordTypeId=01290000000iaNiAAI})   I never queried for RecordtypeId in SOQL, but it is included in the list.  Although, if I run the query from the query editor, it is not showing RecordtypeId column.   Is this by design? Then why not Salesforce is not providing us audit fields in the first scenario. How can I go back to the original interface layout? I don't know what the hell I did I just want to get back to modelling