Prove that $\sqrt 2 + \sqrt 3$ is irrational
Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?
Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ Could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$?  Thank you. How to "flatten" a multi-dimensional array to simple one in PHP? It's probably beginner question but I'm going through documentation for longer time already and I can't find any solution. I thought I could use implode for each dimension and then put those strings back together with str_split to make new simple array. However I never know if the join pattern isn't also in values and so after doing str_split my original values could break.  Is there something like combine($array1, $array2) for arrays inside of multi-dimensional array? Copy game data from one PS4 to another? Option clash for package xcolor MacBook Pro (15-inch, 2016) with 2 USB-C to HDMI adapters. The setup works great if I start up in that configuration.   If I connect the adapters while the machine is running or simply sleeping, one of the monitors will never get a signal and stay in low power mode. The OS sees the monitor, it shows in the display arrangement settings and I can move the mouse into it, but until I restart, I can't see anything on it.  Once I have restarted I can let the machine sleep or whatever, and the display still works unless I disconnect the adapter. As soon as I do that a restart is required to use it again.   Using MacOS Sierra (10.12.4)  Using this adapter: What did Auggie's father give to his wife in the movie Wonder? There is a scene roughly in the last third of Wonder where Auggie's parents Nate and Isabel (Owen Wilson and Julia Roberts) spend an evening together while the kids are out. Nate has a surprise gift for his wife. She opens the box just enough to see it, laughs, and closes it again. So I asked myself the question:  What's in the box? Does a bad grade in one course (with otherwise OK record) affect graduate admissions? I'm currently an M. Phil student in Math and I failed a course last semester because I was not able to hand in a home work assignment (which was worth 40% of my grade) on time (I was 30 minutes late).   Apart from that I'd say I'm a fairly decent student, I got an upper second class honours bachelors degree, I have a B+ and A's for all my other courses in my current programme and I'm almost finished with my first paper to be published.   I've however been deeply worried about this failure as it would reflect poorly on my transcript if I were to apply to a Ph.D or another Master's programme.   Will my failing grade affect my chances of getting into a competitive programme or greatly lower my chances of getting a scholarship? Why are all known distributions unimodal? How to read value of variable, where the name of variable is the value of another variable I think an example is in order. Consider I have these two environment variables setup (by a Continuous Delivery tool): PACKAGE_VER=1.2.3 and USE_VERSION_FROM=PACKAGE_VER.  What I want now is a script that can't predict the name of the variable pointing out 1.2.3, only USE_VERSION_FROM. And so, I'm looking for a way to get my script to know the version to use (1.2.3) without knowing anything other than that USE_VERSION_FROM points out the name of the variable that holds the version number.  My naïve non-working first attempt:  #!/bin/bash VERSION_TO_USE="${${USE_VERSION_FROM}}" DES — Can I recover the key when I have both ciphertext and the plaintext? Given a message and DES encrypted form of said message, is it possible to efficiently compute the key used to encrypt the data? I know Blender is free software and bugs happen. But my question is, is it good practice to update Blender every time there is an update? Even if a minor one? (i.e I'd consider 2.56 to 2.56b minor or even 2.63, no?)  I haven't been updating very often for fear that updating frequently causes issues like broken settings, non-compatibility or lost user configs?  Or is updating often recommended to keep up to date and adapted to the new changes, i.e like upgrades in the GUI and functionality? Which databases use real multidimensional indices? Is oracle ever using several indices to get data from tables, or will it always take the one that seems to have the highest selectivity? How about other dbms? Grub rescue - error: unknown filesystem I downloaded Heroku from Heroku toolbelt using  wget -qO- https://toolbelt.heroku.com/install-ubuntu.sh | sh   in the terminal, which they recommended on their site.   My Ubuntu (14.04.1) didn't recognize the -q0-, so I took it out.  It then seemed to download fine - I see the  in the directory I was in when i downloaded it.   Now, however, it still doesn't recognize heroku as a command.  It also won't show me where it is with the 'whereis' command.  But when I look in the current directory with ls -a, it shows up.  It seems to be there.   How do I install it? Proving $\mathbb{Q}$ isn't homeomorphic to $\mathbb{Z}$ Bottom to top explanation of the Mahalanobis distance? I'm studying pattern recognition and statistics and almost every book I open on the subject I bump into the concept of Mahalanobis distance. The books give sort of intuitive explanations, but still not good enough ones for me to actually really understand what is going on. If someone would ask me "What is the Mahalanobis distance?" I could only answer: "It's this nice thing, which measures distance of some kind" :)   The definitions usually also contain eigenvectors and eigenvalues, which I have a little trouble connecting to the Mahalanobis distance. I understand the definition of eigenvectors and eigenvalues, but how are they related to the Mahalanobis distance? Does it have something to do with changing the base in Linear Algebra etc.?  I have also read these former questions on the subject:       I have also read .  The answers are good and pictures nice, but still I don't really get it...I have an idea but it's still in the dark. Can someone give a "How would you explain it to your grandma"-explanation so that I could finally wrap this up and never again wonder what the heck is a Mahalanobis distance? :) Where does it come from, what, why?   UPDATE:   Here is something which helps understanding the Mahalanobis formula: How to interleave the lines of two or more files? I have tried to get the 2 files output in a single file.  I have tried below command:  cat file1 file2   but here file2 data is appending to file1.  What I need is file1 1st line followed by file2 first line, and file1 second line followed by file2 second line etc.  Can anybody tell me how to do this in a simple way? When no model comparison, should I use REML vs ML? I'm running LMM, and I will make no comparison of models. Could I ask which one should I use between REML and ML? How do i prove invertibility of this linear transformation Let $T$ be a linear transformation from a vector space $V$  over reals into $V$ such that $T-T^2=I$. Show that $T$ is invertible How can I get Gnome3 to remember my session?