Is spacetime continuous according to string theory? Or is spacetime discrete or has some type of foam?
I would like to understand how Physicists think of space-time in the context of String Theory. I understand that there are $3$ large space dimensions, a time dimension, and $6$ or $7$ (or $22$) extra dimensions, and all these dimensions need to fit together in a way such that the extra dimensions are compactified (with a Calabi-Yau or $G_2$ structure).   My question, however is not about the possible $10$, $11$ or $26$ dimensional manifolds that may be possible, but about whether string theorists consider space-time as somehow quantized (or discrete), or rather as a continuous manifold, or are both options possible? In other words, can strings move continuously through space, or are there a discrete set of locations where strings can be, and does string theory rule out one of the options?  How about the same question in loop quantum gravity (LQG)? Should I think of the spin networks in LQG as describing a discrete space-time?  Thanks for your insight, or any references you may be able to provide!
$m$ doesn't come in the sequence $a_n=[n+\sqrt{n}+\frac{1}{2}]$ iff m is a perfect square Let $(a_n)$ be a sequence of positive integers defined by  $a_n=[n+\sqrt{n}+\frac{1}{2}]$  Show that a natural number $m$ doesn't occur in the sequence iff it is a square. I found this one on UserVoice, and I thought it was fitting to bring it up here for further discussion.  "Stack&nbsp;Overflow already works with Mobile Safari, but the experience is far from optimum. I believe an iPhone-optimized version (or maybe even a native client) would be perfect. I can help with either."  I personally think a some kind of mobile version of Stack&nbsp;Overflow would be great for using as a reference, but using it to ask questions would probably get cumbersome if you had to type up a long question with full description. However, using it to look up an answer when you're away from your desk would be awesome.  But, why not just use the regular Stack&nbsp;Overflow? Well, we all know that the small, touch screens get cumbersome to use when navigating websites built for desktops. How to write a unit-test / test class for trigger? How do I write a unit-test / test class for a trigger?  What are my options?   Inline with the trigger?  New Class? Google page title not matching html  tag Anime with a boy hiring a creature from a stone, meets a man named Dante and starts a journey to collect crystals How do I change the extension of multiple files? I would like to change a file extension from *.txt to *.text. I tried using the basename command, but I'm having trouble on changing more than one file.  Here's my code:  files=`ls -1 *.txt`  for x in $files do     mv $x "`basename $files .txt`.text" done   I'm getting this error:  basename: too many arguments Try basename --help' for more information How could I define function with function such as Coefficient? How exactly do I tell if "A xxx of" is plural or singular? I looked up a few grammar-teaching websites (in Chinese language). The website says many people do not use the phrase "A xxx of" correctly because they do not identify the subject of the phrase correctly.   The website says phrases like "A range of; A high percentage of" is singular, while "a number of; a group of; a couple of; a total of" is plural.  Therefore according to the website, the correct sentences should be as follows.     A wide range of courses is available. ("A wide range" is the subject)      A high percentage of peptides that are made of amino acids is present   in the sample. ("A high percentage" is the subject)      A number/ group/ couple of people were killed in the earthquake.("people" is the subject )       A total of 230 employees were laid off. ("employees" is the subject)   I don't get the pattern here. All of this looks (or should I use "look"?) arbitrary to me. In fact I thought this phrase is supposed to be arbitrary and whether the phrase is plural or singular should depend on the the person's own judgment.   How do I identify the correct subject in the "A xxx of" phrase? How to find historical airfares? Is there a way to find out how much it would have cost to buy a particular airplane ticket at some point in the past?    For example, how could I find out how much it would have cost to buy a one-way itinerary from airport A to airport B on October 1, 2011 for travel on November 1, 2011? When I am writing I often come across the words "around" and "round." I was wondering what is different between them, and how they would be used in different contexts. How to install Flash player on Ubuntu? Even though I have the latest Adobe Flash player 11.2 for Linux in Firefox -   - a particular online flash game still requests that I update to the latest flash player.  I've tried this online game on my laptop (Windows 8 with Flash Player 11.8) and it has no problems.  Is there a way to trick Ubuntu or this game into thinking Flash Player 11.8 is "installed"? Will the below code cause memory leak in c++ class someclass {};  class base {     int a;     int *pint;     someclass objsomeclass;     someclass* psomeclass; public:     base()     {         objsomeclass = someclass();         psomeclass = new someclass();         pint = new int();          throw "constructor failed";         a = 43;     } }  int main() {     base temp(); }   In the above code, the constructor throws. Which objects will be leaked, and how can the memory leaks be avoided?  int main() {     base *temp = new base(); }   How about in the above code? How can the memory leaks be avoided after the constructor throws? Is there a point of beating Vault of Glass raid on Hard after finishing on Normal? Good GameBoy Advance Emulator? Is there any good GameBoy Advance Emulator for ubuntu which supports loading gameshark or action replay saves? Sending an HTTP request using QNetworkAccessManager I am using QGIS and I created a vector grid on a crop field in order to cut it into plots. I want to save each grid cell into individual polygon shapefiles automatically.  Here is an example of a vector grid I would like to split into 117 shapefiles:    Eventually, I did it manually with the selection tool and "Save as" but it is really time-consuming when there are more than 100 plots to save.    Is there any way to do it faster and more automatically?  Each plot as an attribute named "Plot" that can discriminate against them (basically being id + 1). It's the attribute used here to label them. Upgrade to the latest PHP version in Ubuntu I am using  in my server and NextCloud informs me that:     You are currently running PHP 7.0.32-0ubuntu0.16.04.1. Upgrade your   PHP version to take advantage of performance and security updates   provided by the PHP Group as soon as your distribution supports it.   So, I tried to upgrade me OS (Ubuntu 16.04), by executing:  apt-get update  apt-get dist-update  Then I restarted the server and I re-executed the above commands. After all these, when I connect from terminal I am getting:  Welcome to Ubuntu 16.04.5 LTS (GNU/Linux 4.4.0-83-generic x86_64)   * Documentation:  https://help.ubuntu.com  * Management:     https://landscape.canonical.com  * Support:        https://ubuntu.com/advantage  0 packages can be updated. 0 updates are security updates.   and my PHP version is still PHP 7.0.32-0ubuntu0.16.04.1 (cli) ( NTS ).  This means that PHP 7.0.32 is the latest possible version that I can install in ubuntu 16.04; I used VIRTUALBOX to install Ubuntu 12.04 on my windows pc. I used apt-get to install Python 2.7 that shipped with 12.04. I followed the directives I saw on this site to install Numpy, Scipy, Matplot all failed. I download tar source file of these modules. How do I install them manually? Someone pls help, I'm getting frustrated. I need to find a way of proving that the square roots of a finite set    of different primes are linearly independent over the field of    rationals.    I've tried to solve the problem using elementary algebra  and also using the theory of field extensions, without success. To  prove linear independence of two primes is easy but then my problems  arise. I would be very thankful for an answer to this question. Maximum area of isosceles triangle with perimeter 6 cm Problem. What is the side lengths of an isosceles triangle with $6cm$ perimeter which has the largest area? I solved this problem as below and I would like to learn  that if there is anything wrong: Let us say that the side lengths are $a$ and the bottom length is $6-2a$. By using pythagorean theorem we can conclude that the height $h$ is equal to $\sqrt{6a-9}$. Then, the area of the triangle can be given as $f(a)=(3-a)(\sqrt{6a-9})$. So $f'(a)=-\sqrt{6a-9}+\cfrac{3}{\sqrt{6a-9}}$. Finally, $f'(a)=\cfrac{12-6a}{\sqrt{6a-9}}$ and $f'(a)=0 \iff a =2$.