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Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ taller than any student or taller than any other student Which of the following sentences is correct?     a)   1. John is taller than any student in his class.   2. John is taller than any other student in his class.        b)   1. No student is taller than John in his class.   2. No other student is taller than John in his class. What is the best way to animate the camera path? I'm modeling a car driving along a road(speeds up, slows down, and stops at traffic lights to wait till the lights turn green, etc.) What is the best way to animate the camera path for the following scenario?   The camera starts moving from point A, and it takes 500 frames to get to point B. During this 500 frames, the camera first gradually speeds up, then moves at a constant speed, and finally slows down to a complete stop at point B. The next 50 frames, the camera will stay at point B. From frame 551, it takes another 500 frames to get to point C from point B. The camera experiences the same motion as described in #1 (speeds up, constant speed, slows down, then stop.) Again, the camera will stay at point C for another 50 frames.  Point C to D is a curve path ( added using Shift + A -> add path). I want the camera to move along the path from frame 1101 to frame 1200. The camera should have the same motion as before(speeds up, constant speed, slows down, then stop.)   What is the best way to animate the camera path in this case? Is there any Python scripts available and can be applied to case like this? Thank you all in advance for your replies. Combinatorics Question Asking How Many Integer Solutions Given Different Restrictions How many integer solutions of $x_1 + x_2 + x_3 + x_4 = 28$ are there with:  (a) $0 ≤ x_i ≤ 12$?  (b) $−10 ≤ x_i ≤ 20$?  (c) $0 ≤ x_i, x_1 ≤ 6, x_2 ≤ 10, x_3 ≤ 15, x_4 ≤ 21$?  \ \  My attempts (I was really only able to solve part (a) but I would appreciate if you could check my work for this one as well):  (a) First I took the total number of integer solutions with no restrictions: There would be ${28+3\choose 3}$ = ${31\choose 3}$ total solutions with no restrictions.   Then I used the inclusion exclusion principle and knew I would have to subtract [P(A) + P(B) + P(C) + P(D)] where A --> $x_1$ is greater than or equal to 13, ..., D--> $x_4$ is greater than or equal to 13. The probabilities of all these cases would equal ${15+3\choose 3}$ = ${18\choose 3}$ because once we assign 13 to one of the $x_i$ there would be $28 - 13 = 15$ remaining to distribute among the $x_i$ values. We would also need to choose which of the $x_i$ to give the 13 to so we would multiply the ${18\choose 3}$ by ${4\choose 1}$ = ${4\choose 1}{18\choose 3}$  We would then have to add back, since we "oversubtracted", the case in which there is an intersection between two of the $x_i$ or in other words two of the $x_i$ receive 13 or more. P$(A\bigcap B)$ = $28-2(13) = 2$ remaining to distribute among the $x_i$ values. So we would also have to choose which of the $x_i$ to give the 13 to so we would have ${4\choose 2}$ ways to choose them. So there would be ${4\choose 2}{2+3\choose 3}$ = ${4\choose 2}{5\choose 3}$ solutions in this case.   Thus the number of solutions would be: ${31\choose 3} - {4\choose 1}{18\choose 3} + {4\choose 2}{5\choose 3}$.  I'm not sure if this is correct and I'm really stuck on b and c so I would appreciate any help or feedback. Thank you! what are all the open subgroups of $(\mathbb{R},+)$ It is easy to find the minimum of $\|Ax-b\|_2$, when $A$ has full column rank. But how is the case when we add the constraint $\|x\|_2=1$? Or, to be explicit,  $$\min_{\|x\|_2=1}\|Ax-b\|_2=?$$  My idea is to construct the corresponding Lagrange function  $$L(x,\lambda) = (Ax-b)^T(Ax-b) + \lambda(x^T x - 1)$$  Differentiate the function with respect to $x,\lambda$, we have  $$A^T A x - A^T b + \lambda x = 0$$  $$x^T x=1$$  However, I find it hard to move forward to get the answer. Could anyone help me? Thanks a lot! What are the numbers / symbols to the right of my reputation counter? Can't execute terminal commands: "the command 'sudo' is available in '/usr/bin/sudo'" I have been using Ubuntu 12.04 for quite a while and have never had a problem come at me out of the blue like this. I'm trying to install python 3.2 on my Ubuntu system and every time I run the code to compile the python source, I get this error:  'Command 'sudo' is available in '/usr/bin/sudo' The command could not be located because '/usr/bin' is not included in the PATH environment variable. sudo: command not found'   I've also tried running gksudo command to view what my environment shows but I pretty much get the same error. Return row only if value doesn't exist Determining the name of the VMware host of a VM guest - from the guest Is it possible to determine the name of the VMware host (ESX or ESXi) that my guest resides in, from within the guest itself?  I would expect this to be possible via VMware Tools, but am not sure where to look. jquery mobile external panel not taking on styling wireless network adapter supported on ubuntu? Why isn't an infinite direct product of copies of $\Bbb Z$ a free module?   Actually I was asked to show that it's not projective, but as $\Bbb{Z}$ is a PID, so it suffices to show it's not free.   I am stuck here. I saw some questions in SE, but there is no satisfactory answer at all. I've found ,  and  which are releated to my question, but no longer seem to apply since Drupal 8.1 and the deprecation of .  Also popular guides such as  appear to no longer apply as they also mention composer manager.   The other solution opposed to composer manager, alter the core composer.json file feels like too much core hacking and would probably break with every Drush update of Drupal core(?).  More specifically, I am attempting to update  from Drupal 7 to 8. I have created my composer.json file as follows:  {   "name": "drupal/views_vcards",   "description": "Allows creation of vCards using the fields provided by the Views module.",   "type": "drupal-module",   "license": "GPL-2.0+",   "homepage": "https://drupal.org/project/views_vcards",   "require": {     "maennchen/zipstream-php": "0.3.*"   } }   But if I put a composer.json file in my module folder, how do I make Drupal aware that that file is there, and how do I make sure the required zipstream-php library is downloaded?  Simply running composer update from the Drupal root does update a lot of Drupal dependencies, but it does not include the composer.json files that are in the modules folders. I also don't assume I should be calling composer install from inside all modules with dependencies.  How do I make Drupal aware of a module's dependency without using composer manager and/or hacking core?  UPDATE:  The  appears to have support a wildcard path:  {     "require": {         "wikimedia/composer-merge-plugin": "dev-master"     },     "extra": {         "merge-plugin": {             "include": [                 "composer.local.json",                 "extensions/*/composer.json" // &lt; ---- THIS LINE             ],             "require": [                 "submodule/composer.json"             ],             "recurse": true,             "replace": false,             "merge-dev": true,             "merge-extra": false         }     } }   Why doesn't core merge modules/*/composer.json, that would solve everything right?  Update 2:  THe reasoning for not supporting this is covered in  (which also has been quiet for some time now). What is the most concise and efficient way to find out if a JavaScript array contains a value? This is the only way I know to do it: function contains(a, obj) {     for (var i = 0; i &lt; a.length; i++) {         if (a[i] === obj) {             return true;         }     }     return false; }  Is there a better and more concise way to accomplish this? z-score and Normal Distribution Are $T\mathbb{S}^2$ and $\mathbb{S}^2 \times \mathbb{R}^2$ different? I have seen the claim that $T\mathbb{S}^2$ and $\mathbb{S}^2 \times \mathbb{R}^2$ are not diffeomorphic, but I have only ever seen the proof that they are not isomorphic as vector bundles (which is a cute application of the hairy ball theorem). How would I prove that they are indeed not diffeomorphic (or better, not homeomorphic)? Better way of saying "those who are curious"? Do the *Result objects in Database have a common ancestor? I'm doing some polymorphic programming, and I decided that I wanted to be able to gracefully handle partial saves using SaveResult without blocking the user experience. The code looks roughly like this:  static void logDmlErrors(SObject[] src, SObject[] dest, SaveResult[] results, String message) {     // Transform results into Chatter error messages, send them to admins }   I then went about writing the utility methods that would use this function:  public static void insertRecords(SObject[] src, SObject[] dest, String message) {     logDmlErrors(src, dest, Database.insert(dest, false), message); }   Everything went well, until I came across the delete and undelete methods, which instead use DeleteResult and UndeleteResult. Both of those two classes contain the exact same methods as SaveResult, and yet don't appear to inherit from a common ancestor. This means that I can't just cast to a super-type to get to this information.  Thankfully, Summer 16 apparently introduced support for serializing (and deserializing) these classes as JSON, so I do have a fairly expensive (but accurate) workaround if I want proper polymorphic behavior, but I thought I'd throw it out there and see if someone has an alternative solution to this problem. mbstring is missing for phpmyadmin in ubuntu 16.04 When I installed phpmyadmin in Ubuntu 16.04 LTS, it gives me an error mentioned below.     The mbstring extension is missing. Please check your PHP configuration   then I tried to install php-mbstring, it says mbstring is up to date, Can anyone help me to solve this?