Generally, the definition of prime numbers is all those natural numbers greater than 1, having only two divisiors, the number itself and 1. But, can the negative integers also be thought of in the same way?   For example: if we take the case of -1. Then it is divisible by 1 and itself. Can we call it prime? Why? Or why not?
Do we have negative prime numbers?   $..., -7, -5, -3, -2, ...$
Proving the Cauchy-Schwarz inequality by induction It seems to me that tuples are just as elementary as sets. Can tuples be reduced to sets? How can I prove this inequality in a triangle: $\frac{1}{(b+c-a)} +\frac{1}{(c+a-b)} +\frac{1}{(a+b-c)} \gt \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$? If $a,b,c$ are the sides of a triangle then show that- $\frac{1}{(b+c-a)} +\frac{1}{(c+a-b)} +\frac{1}{(a+b-c)} \gt \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$  I got this problem from an old book on algebra. I've been trying for a long time but can't reach the answer.  My try on the question-  We know that in a triangle, the sum of two sides is always greater than the third side  $b+c\gt a$ $\implies$ $b+c-a \gt 0$  $c+a\gt b$ $\implies$ $c+a-b\gt 0$  $a+b \gt c$ $\implies$ $a+b-c\gt 0$  Also,  $\frac {1}{(b+c)} \lt \frac{1}{a}$  $\frac{1}{(c+a)} \lt \frac{1}{b}$  $\frac{1}{(a+b)} \lt \frac{1}{c}$  Adding,  $\frac{1}{(b+c)} + \frac{1}{(c+a)} +\frac{1}{(a+b)} \lt \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$  How do I proceed further? All genuine answers are welcome :) How can I export object's field schema information to excel? How can I export the fields of an object from Salesforce into an Excel sheet? I'm only looking for the field schema information, not the objects records. I am using NCR18650 batteries with the tp4056 charging module.  When I connect the charger at 2.8&nbsp;V (fully discharged), the tp4056 IC is getting hot at a 1&nbsp;A charge current. Is it OK to reduce this charge current by changing the charge current selecting resistor? So the tp4056 IC heat would be reduced, wouldn't it?  For my application I don't need a quick charge. With a low current, charging time will be increased, but it does not matter for my application.  Will reducing the charge current affect the battery's lifetime? Or will it damage the battery?  I am trying to reduce the heat of the tp4056 IC by reducing the charging current, maybe from 1&nbsp;A to 500&nbsp;mA or less. The 18650 battery never heats for a 1&nbsp;A charge current. Showing a compact metric space has a countable dense subset Sufficient Identification for Intra Schengen flights If one takes a direct flight from one Schengen member state to another, what's the minimum sufficient ID that needs to be provided to board the flight? I'm trying to install a Matlab R2011a launcher for Unity in Ubuntu 12.04. I've tried  (although I know it's for 11.10 and mentions that even 11.10 is an unsupported OS for Matlab R2011a) but without any satisfactory solution.  This is my launcher file, /usr/share/applications/matlab.desktop:  #!/usr/bin/env xdg-open [Desktop Entry] Type=Application Icon=/usr/share/icons/matlab.png Name=MATLAB R2011a Comment=Start MATLAB - The Language of Technical Computing Exec=matlab -desktop Categories=Development;   I open the dash panel and search for "matlab". This launcher is found among applications. I click it, and Matlab's splash screen shows up, but when it disappears the program doesn't start. (I've verified with htop that no matlab-processes are running in the background either.)  If I add  Terminal=true   to the launcher file, the program starts OK, and opens a terminal as well as Matlab. However, both the terminal and Matlab itself show up in the Launcher area, with the Matlab icon, so it looks like I have two Matlab instances running when really it's only one. (Actually, they show up as two different programs, and not just two instances of the same - the icons are independent, not grouped together.)  This is definitely not optimal. I had hoped to create a launcher I can lock to the launcher area, and then that same icon will be the icon for the active Matlab instance when the program is running.  How do I create a launcher for Matlab that works as expected?  Update: I was apparently a bit unclear on my symptoms, I'll try to clarify a little. I've also tried some suggestions from the answers, and further investigated what happens. My current setup (a launcher file with Terminal=true and Exec=matlab -desktop -nosplash &amp;) renders the following behavior:   I open Dash by pressing the Windows key on my laptop, and search for "matlab". It finds the launcher named "MATLAB R2011a". I click it. A terminal window opens, using the icon I referred to in the launcher file. Almost immediately, MATLAB's splash screen also opens, using the same icon (and thus being grouped with the terminal window in the launcher). The splash screen disappears and, so does one of the icons in the launcher. The MATLAB desktop environment opens, using a different version of the icon which is displayed next to the icon for the terminal window (not grouped with it).   I can lock the terminal window's icon to the launcher and successfully start MATLAB by clicking it, but it doesn't feel optimal that I start the program with one icon, and switch to it with another. I've also tried the following:   Exec without the ampersand &amp; in the launcher command, but it didn't make a difference. Executing matlab -nosplash manually from a terminal still shows the splash screen. (What, then, does the nosplash option really do?) I am looking for a way to clean up the mess when my top-level script exits.  Especially if I want to use set -e, I wish the background process would die when the script exits. Check differentiability of multivariable functions I am trying to determine if the following functions are (totally)   differentiable at $(x,y)=(0,0)$.      $$f(x,y)=\sqrt{\vert xy \vert}$$      $$g(x,y)=e^{{{\vert x \vert}^3}y}$$   However I am not particularly sure where to start. The concept of the total derivative being the Jacobian matrix makes sense but I don't really understand how to work with functions that may not have a Jacobian matrix. Any help would be greatly appreciated! How do I install Ubuntu? I would like to see a full how-to guide on how to install Ubuntu. Within a set of directories, how do I find a file whose name includes a given string, such as "abcde"? One of my associates had their Hotmail account broadcasting spam, and presumably hacked. They managed to change the password, but after that, the account got locked; Hotmail asks you for a phone number, and they're supposed to call you with a code.  This person requested the code several times, with different numbers (even two of mine!), but never received any call with the confirmation code.  I referred them to the full-length process, which includes answering the security question; apparently, they forgot their question (having entered it over a decade ago and never reviewed it since).  What options, if any, exist to restore access to this person's account? They only want to grab their contacts; they plan to operate full-force on GMail going forward. How to prove that a set is invariant? Not able to add 3 JPanels to a main panel How do I resize partitions using command line without using a GUI on a server? I only have access to the server via a terminal and I can't use graphical tools such as GParted! I want to create a new partition from a part of the root (about 768mb) for swap.  # df -h Filesystem      Size  Used Avail Use% Mounted on /dev/vda         20G  1.3G   18G   7% / udev             10M     0   10M   0% /dev tmpfs           199M  4.9M  194M   3% /run tmpfs           100M   12K  100M   1% /run/user tmpfs           5.0M  4.0K  5.0M   1% /run/lock I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm:  given $x\in[1,2)$, do the following:  L1: [Initialize] Set $y\leftarrow0$, $z\leftarrow x/2$, $k\leftarrow1$.  L2: [Test for end] If $x=1$, stop.  L3: [Compare] If $x-z&lt;1$, go to L5.  L4: [Reduce values] Set $x\leftarrow x-z$, $z\leftarrow x/2^k$, $y\leftarrow y+\log_b(2^k/(2^k-1))$, and go to L2.  L5: [Shift] Set $z\leftarrow z/2$, $k\leftarrow k+1$, and go to L2.  (The algorithm needs an auxiliary table which stores $\log_b2$, $\log_b(4/3)$, and so on, to as many values as the precision of the computer.  Then Knuth concludes: this exercise is to explain why the above algorithm will terminate and why it computes an approximation of $y=\log_b x$.  Well, I see that it works, but I am not able to explain why... Renaming user name I made a mistake in the username at the time of installation. So I changed the name of the user after logging in by creating a guest account with SUDO rights. But when I checked the list of users using ls /home command I am not seeing the new name for the user instead Old name is appearing. But I use the new name to login.   Is it like I need to change the name inside Home directory separately? Also , if there is any other changes needs to be done .. ? is there a way to find suitable ratio for matches to controls (using "ratio" in Matchit package in R or other programs) I am using matchit package in R to find a control set for my treated units and I am testing the model with different ratio to get suitable control set. Is there a way to find suitable ratio and I will use that to find my control set. Combining the entries of two lists