Let $\mathbf X = \Bbb R$ with distance function defined by $d(x,y) = {|x-y|}^\alpha$ , where $\alpha \in \Bbb R$  $(0&lt;\alpha\le1)$.  Prove that $(\Bbb R , d)$ is a metric space.  The first three properties are easy, so I only need the triangle inequality. I tried to use Bernoulli's inequality, but did not lead me to the proof.  $d(x,y) \le d(x,z) + d(z,y)$  Please help me, Spanish is my language.
I am curious about whether $|x|^p$ with $0&lt;p&lt;1$ satisfy $|x+y|^p\leq|x|^p+|y|^p$ for $x,y\in\mathbb{R}$.   So far my trials show that this seems to be right... So can anybody confirm whether this is right or wrong and give a proof or counterexample?
Replacement character being used question URL Some questions, for example    Instead of using a correct URL, such as  http://meta.stackexchange.com/questions/79543/٩●̮̮̃-̃۶-forbid-unicode   are using  https://superuser.com/questions/516363/what-is-the-��-character   What is causing this behavior? I am developing firefox extension and wanted to count total number of text box of webpage so how to count text box? Flirtatious Primes Here's a possibly interesting prime puzzle. Call a prime $p$ flirtatious if the sum of its digits is also prime. Are there finitely many flirtatious primes, or infinitely many? What are the consequences of "copying" a data set for OLS? I'm trying to setup my multi-finger tap in the following way:   Single finger - Left click Two fingers - Middle click Three fingers - Right click   What I currently have is:   Single finger - Left click Two fingers - Right click Three fingers - none   I've looked up several questions here on Ask Ubuntu, as well as the Ubuntu wiki on the subject (plus some general Google searching), but couldn't find anything to help me out. I know of xinput, but I don't know what to change there and how.  If anyone is able to point me to the right direction, I'd be grateful and award him with 25 shiny rep points, a large green tick mark, and some cookies. How do I split a string, breaking at a particular character? What kind of power management circuit do I need to embedded a rechargeable battery in my electronics project? I am doing a project where my electronics runs on 5V which I wanted to power from a 7.2v lipo regulated down to that voltage. I'd really like to include the opportunity to charge the lipo without removing it from the project. What kind of circuit/components do I need to do it properly?  Say, would putting a battery charging ic at the battery's terminals in parallel to my system be enough to achieve what I need? If there are 5 points on the surface of a sphere, then there is a closed half sphere, containing at least 4 of them.  It's in a pigeonhole list of problems. But, I think I have to use rotations in more than 1 dimension.  Regards Alternative IDE for VB6 and VBA accounting for LED resistance Can only move object origin in object mode, cant move the mesh I can only move the origin point of the mesh that I selected in object mode, but not the whole mesh Did I set anything up by accident? Why is everything in minecraft opaque red and blue? With ksh I'm using read as a convenient way to separate values: $ echo 1 2 3 4 5 | read a b dump $ echo $b $a  2 1 $  But it fails in Bash: $ echo 1 2 3 4 5 | read a b dump $ echo $b $a     $  I didn't find a reason in the man page why it fails, any idea? How to prove that $d \sin(x)/dx = \cos(x)$ without circular logic such as L'Hôpital's rule? I'm wondering where a new path has to be added to the PATH environment variable. I know this can be accomplished by editing .bashrc (for example), but it's not clear how to do this.  This way:  export PATH=~/opt/bin:$PATH   or this?    export PATH=$PATH:~/opt/bin I need to make a converter to go from RS422 full duplex to my Arduino. I have found lots of different chips which I think can do this, but am slightly confused by the terminology. The RS422 device is a laser distance sensor ().  It has RX+, RX-, TX+, TX-. So I have two lines for input and two lines for output.  I could use two RS485 line drivers to achieve the full duplex comms which I need.  There are a lot of chips to do that.  But then on my Arduino (clone: chipkit max32) would I need to use two UART channels? And then alternate which one I am sending each byte on?  The device uses 4 byte instructions. So if I wanted to send an instruction would I send Serial1(firstByte) then Serial2(secondByte), Serial1(thirdByte), Serial2(fourthByte)?   And then receive data the same way? I feel like I am missing something and can do this on one UART channel but am not sure how. I'm trying to wire up 6 RGB LEDs in parallel, all controlled from a single source (well, three sources, one for each colour). The LEDs came supplied with resistors to limit the current of 270 Ohm for a 5v supply.  The problem is, 6 LEDs x 3 colours = 18 resistors, which is a lot, and means I need a much bigger board and a lot more soldering.  So, can I instead wire the LEDs in parallel with each other, with a single resistor protecting all six? (3 resistors in total, one for each colour). How do I calculate the value of that resistor?  More details:  The LEDs are being driven from a  to supply a bit of current, which is in turn controlled by a Netduino providing a PWM signal on the three channels.  . If I've correctly understood the data sheet they want 20mA of current, and forward voltages of 2, 3, 3 volts (for R,G and B respectively?). The supplied resistors were all 270 Ohm, so the channels may not be balanced quite right.  For extra credit: I'm only using 3 of the transistors in my driver chip, which has 8 in total. could I wire the PWM from the netduino to a second trio of transistors, and split the LEDs into two groups of three? Is it worth the effort?  PS I don't have any diagramming tools to hand, but I can provide a diagram (drawn in paint) if it would help clarify my question. (see also ) Word for the satisfaction of curiosity There is a good feeling you get when your curiosity is satisfied, especially if is about a question you found interesting or important, or have spent a lot of time and energy trying to answer. Depending on how much I wanted to know, this feeling can be quite pleasing. Is there a word specifically for either the satisfaction of a curiosity, or the feeling that results? Title vanishes when editing a question Why is the gap between the translation and the last line of the gloss so wide in my interlinear gloss?