$|x-z| \leq |x-y|+|y-z|$  We know that both LHS and RHS are non negatives. So, I thought of proving this by comparing the squares of both sides but can't advance beyond that step. Any help would be appreciated.
Let $a,b,c$ real numbers. Prove the inequality $|a-b| \leq |a-c| + |c-b|$. Prove that equality holds if and only if $a \leq c \leq b$ or $b \leq c \leq a$.   I've tried starting with just $a \leq c$ and using field properties to reconstruct the inequality, however I haven't been able to make it work. I also tried making the negatives positive and stripping the inequalities and making something happen but again I don't know if that's a proper rule and it didn't seem to get me anywhere.
Timeout on a function call Why so many files in /usr/share/app-install/desktop? For 14.04, I'm trying out Lubuntu LTS.  If I run ls -l /usr/share/app-install/desktop, I get a list of 3085 .desktop files. All except one are dated Apr 18. The exception is:  -rw-r--r-- 1 root root  6328 Mar 28  2013 applications.menu   From the dates, it's clear that all these are installed by default and not by me.   Doesn't each GUI-based program bring in its own .desktop file at the time it's installed (mostly in /usr/share/applications)? Then why are these thousands of other .desktop files needed? From their names, most of them haven't anything to do with Lubuntu. Their names are different from regular .desktop files (such as those in /usr/share/applications): many, not all, are like name:name.desktop whereas the regular ones are just name.desktop. The ones I looked at all have a line with X-AppInstall-Popcon=nnn where nnn is a string of digits. I know that Popcon "gathers statistics determining which packages are the most popular with Ubuntu users". But why is that part of these .desktop files? Infinite Volume but Finite Surface Area Is there any pathological shape that has a finite surface area but an infinite volume? Sort of like the opposite of a Gabriel's horn. Viewing computer screen through 3D glasses, why the asymmetry? Downloading multiple files with youtube-dl I use youtube-dl to download files from YouTube. I have tried downloading multiple files, but it is not working out for me. I have to open multiple terminals and start afresh every time I want to download a video.   Can you help me to download multiple files with a single terminal window by just mentioning all the URLs at once? I use Ubuntu 12.04 64-bit. Prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$ How can I prove $\operatorname{rank}A^TA=\operatorname{rank}A$ for any $A\in M_{m \times n}$?   This is an exercise in my textbook associated with orthogonal projections and Gram-Schmidt process, but I am unsure how they are relevant. Why do all SSH-RSA Keys begin with "AAAAB3NzaC1yc"? How do I automate a key press in AppleScript? Is it possible to automate the key press of a key (such as "a" or "&lt;") in Mozilla Firefox, Google Chrome or Safari (in this order)? I'd like to automate the key press of "a" for example in Mozilla Firefox for 100 times every x seconds. With x being a value between 1 second and 10 seconds chosen randomly at every key press. What Applescript could make the trick?  Also, where can I find a list of all codes for each key of the keyboard? Do you use degrees or radians for trig functions? I was just wondering if you use degrees or radians in trig functions. For example if I have a degree of 0.5 would I do: Sin(0.5) or would I have to convert that to radians?  Or does it not matter either way? Discoveryd Alternative to Replace mDNSResponder Argument? This is a direct follow-on to . , I'd add the "-AlwaysAppendSearchDomains" argument to the mDNSResponder plist file so that all local resolver lookups added the search domains, regardless of whether there was a period in the label to be looked up. For example, whether I try to ssh to hostname foo.bar or foo, the search domain of baz.com should be tried, not only for the latter case, as was (and is) the default behavior.   Now, it looks like the mDNSResponder is gone, replaced with discoveryd. I cannot find a similar option to get it to always appending the search domain. Suggestions? I'd rather not go the route of using a non-default local resolver, but I'd consider it as an option. For now, I'll just type in the long FQDNs, but I'd really rather not.  Thanks, Pete I recently found a bug, which I mentioned on , as well as submitting a case, and after a few days fumbling around with the case, they closed it by saying that they only provide this type of support for Premier Support customers.  Is there a better channel for reporting bugs than by submitting a case, or is there something "special" we have to do? I realize this question is similar to , except that I've already gone the route of , as suggested by the answer there (i.e. submitting a case), to no avail.  How can I make sure that a bug like this can find its way to someone capable of at least confirming that the behavior was expected? This behavior isn't documented anywhere, and so I assume that it's a bug that needs to be addressed. To show that Fermat number $F_{5}$ is divisible by $641$. How can I show that  $F_{5}=2^{2^5}+1$ is divisible by $641$. What is the proper way to handle data between scenes? I am developing my first 2D game in Unity and I have come across what seems an important question.  How do I handle data between scenes?  There seems to be different answers to this:   Someone mention using , while other people told me this should be used to store other things like screen brightness and so on. Someone told me that the best way was to make sure to write everything into a savegame everytime that I changed scenes, and to make sure that when the new scene loads, get the info from the savegame again. This seemed to me wasteful in performance. Was I wrong? The other solution, which is the one I have implemented so far is to have a global game object that isn't destroyed between scenes, handling all the data between scenes. So when the game starts, I load a Start Scene where this object is loaded. After this ends, it loads the first real game scene, usually a main menu.   This is my implementation:  using UnityEngine; using UnityEngine.UI; using System.Collections;  public class GameController : MonoBehaviour {      // Make global     public static GameController Instance {         get;         set;     }      void Awake () {         DontDestroyOnLoad (transform.gameObject);         Instance = this;     }      void Start() {         //Load first game scene (probably main menu)         Application.LoadLevel(2);     }      // Data persisted between scenes     public int exp = 0;     public int armor = 0;     public int weapon = 0;     //... }   This object can be handled on my other classes like this:  private GameController gameController = GameController.Instance;   While this has worked so far, it presents me with one big problem: If I want to load directly a scene, let's say for instance the final level of the game, I can't load it directly, since that scene does not contain this global game object.  Am I handling this problem the wrong way? Are there better practices for this kind of challenge? I would love to hear your opinions, thoughts and suggestions on this issue.  Thanks How do you interpret a significant but weak correlation? Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic. Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic.   What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve this and showing that $|G|$ order element in $G$ exists is enough. Why do we tag questions? My question—my meta-question as the case may be—concerns the purpose of these words referred to as "tags". Let me be clear: I use tags. The site supports them, so why shouldn't I? My understanding of their reason for existing is tragically incomplete.   Are they used for searching? Are they useful in any other way, or do they merely act as decoration?     For more information, see "" and "" in the . I am a Nigerian citizen.  In 2007 I was twice denied a visa to the UK.  Recently I was filling out an application for another UK visitor visa and by mistake I answered no to the question whether I had ever been denied a UK visa, instead of yes.  When reviewing my application, the visa officer found out about the previous visa rejections and as a result I got a 10 year ban on travel to the UK.  What can I do to lift my 10 year ban?   Will this ban affect my chances of obtaining a visitor visa to the USA? Cursor hitting "border" between differently sized monitors I'm not sure if I'll be to explain this right. I have a dual monitor display on Windows 7 and my left monitor (laptop) has a lower vertical resolution than the right monitor (regular LCD monitor). So obviously, if I stick these monitors together, there's a part in the upper-left corner of my LCD monitor where cursor cannot go to the other monitor. I remember there was a trick to make this gap disappear, and make it seem like the monitors are of equal height.  Any suggestions?  By the way, I am also using UltraMon. Where is the denoise option in 2.90? I'm using python\pyodbc and would like to access the second result set of a stored procedure. As near as I can tell, pyodbc does not support multiple result sets. Additionally, I can't modify the stored procedure. Are there any options to access the second result set using SQL or some other work-around? Perhaps create a second stored procedure that only returns the second result set of the first?