Just now, when reviewing close votes I voted to close a question and saw that there were four other votes cast for off-topic     but clicked off-topic I could only see one other vote, the other 3 was missing.    I have been seeing this issue many times today.
I've found strange behavior when closing this  - I definitely see that there are 3 closing votes and 2 of them are at "Off-topic" section:   -but there are no explanations on section itself:   -while, normally, it's expected that it will appear. In custom comments section there are also no closing-votes displayed.   Perhaps it's just my browser (which is FF 26 x64 under Win) - so marking this as a "possible bug". Thank you for your attention.
Without appealing to choice, can we prove that if $X$ is well-orderable, then so too is $2^X$? Without appealing to the axiom of choice, it can be shown that (Proposition:) if $X$ is well-orderable, then $2^X$ is totally-orderable.  Question: can we show the stronger result that if $X$ is well-orderable, then so too is $2^X$?  Proof of Proposition. Pick any well-ordering of $X$. Then the lexicographic order totally orders $2^X$.  More explicitly: for any two $f,g \in 2^X$, define $f &lt; g$ iff   there exists $x \in X$ such that $f(x) \neq g(x)$, and if $x \in X$ is minimal such that $f(x) \neq g(x)$, then $f(x)=0$ and $g(x)=1$.   It can be shown that $&lt;$ totally-orders $2^X$. How can I find $\gcd(a^m+1,a^n+1)$ with $a,m,n$ positive integers?    I have this idea:  Let $d=\gcd(m,n)$. Then there exist positive integers $x,y$ such that $mx-ny=d$ (WLOG). We shall find $G=\gcd(a^m+1,a^n+1)$.  If $m,n$ are odd, then $d$ is odd, therefore one and only one of $x,y$ is even. We have:  $$a^{ny}(a^d+1)=a^{mx}+a^{ny}=(a^{mx}-1)+(a^{ny}+1)=(a^{mx}+1)+(a^{ny}-1).$$  If $x$ is even and $y$ is odd, then $a^{m}+1\mid a^{mx}-1$ and $a^{n}+1\mid a^{ny}+1$, therefore $G\mid a^{ny}(a^d+1)$. If $x$ is odd and $y$ is even, then $a^{m}+1\mid a^{mx}+1$ and $a^{n}+1\mid a^{ny}-1$, thus $G\mid a^{ny}(a^d+1)$. However, since $\gcd(a^m+1,a^{ny})=\gcd(a^n+1,a^{ny})=1$, so $\gcd(G,a^{ny})=1$, hence $G\mid a^d+1$. We also have $a^d+1\mid a^{m}+1$ and $a^d+1\mid a^{n}+1$, so $a^d+1\mid G$. Thus $G=a^d+1$.  If $v_2(m)=v_2(n)=v_2(d)=k&gt;1$, then there exist some odd numbers $m_1,n_1,d_1$ such that $m=2^km_1,n=2^kn_1,d=2^kd_1$. We shall have $m_1x-n_1y=d_1$, so one and only one of $x,y$ is even, and we can use the same argument when $m,n$ are odd, so $G=a^d+1$.  However, if $v_2(m) \neq v_2(n)$, I cannot find any solutions for this. I think that $G \in \{1,2\}$, but I cannot prove or disprove it. How can I find $G=\gcd(a^m+1,a^n+1)$ if $v_2(m) \neq v_2(n)$ ? Moreover, is there anything from my arguments that can be improved ? Running programs in the background from terminal How do I run a program in the background of a shell, with the ability to close the shell while leaving the program running? Lets say my UI is having problems or for some reason, I need to boot up a program from the terminal window, say, nm-applet:  nm-applet   When it's started, it occupies the foreground of the terminal window.  Is there any simple way to run the program in the background without needing to leave the terminal open or have it occupy the whole terminal?  On that note, I did find a way to run programs from the terminal and have it allow for other inputs, by appending an ampersand (&amp;) to the command as such:  nm-applet &amp;   But this isn't much use as any processes started in the terminal are killed once the terminal is closed. How to execute multiple statements in a MATLAB anonymous function? I'd like to do something like this:  &gt;&gt; foo = @() functionCall1() functionCall2()   So that when I said:  &gt;&gt; foo()   It would execute functionCall1() and then execute functionCall2().  (I feel that I need something like the )  EDIT:  functionCall1 and functionCall2 are not necessarily functions that return values. Ubuntu 14.04, Intel Chromebook, no sound Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e. This is a corollary to a proof in Bass, but I don't understand why it follows from the proof he gives. I follow everything up until the last statement. Why is it that proving that the integral is $0$ for all Borel measurable sets guarantees it is true for all Lebesgue measurable sets? Am I missing something vital about the relationship between Lebesgue measurable sets and Borel measurable sets? Statement: Let $m$ be Lebesgue measure and $a \in\mathbb R$. Suppose that $f : \mathbb R \to\mathbb R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e. Proof: We prove the conclusion first for intervals, then for unions of intervals, then for open sets, then for Borel measurable sets. For any interval $(c,d)$, we have $$\int_c^df = \int_a^d f - \int^c_a f = 0.$$ By linearity, if $G$ is the finite union of disjoint open intervals, then $\int_G f = 0$. Now, if $G$ is open, then $G$ is the countable union of disjoint, open intervals, $G = \bigcup^n_{i=1}I_i$. We have $$\int_G f = \int \lim_{n\to\infty} f \cdot \chi_{\bigcup^n_{i=1} I_i}$$ and since $f_n$ is integrable for all $n$, and $|f_n| \le f$ for all $n$, by the dominated convergence theorem, $$\int_G f = \lim_{n\to\infty} \int_{\bigcup^n_{i=1}I_i} f =0.$$ Now, if $G_n$ is a decreasing sequence of open sets converging to $H$, since $|f \cdot \chi_{G_n}| \le f \cdot \chi_{G_1}$ for all $n$, we again have by the DCT that $\int_H f = 0 = \lim_{n\to\infty} \int_{G_n}f = 0$. Finally, if $G$ is a Borel measurable set, then we know that there exists a sequence $G_n$ of decreasing, open sets, that converge to some set $H$, where $H \setminus G$ is a null set. Thus, $$\int_G f = \int_H f = 0.$$ We have found that for every Borel measurable set $G$, $\int_G f =0$. Since $f$ is real valued and measurable, $f = 0$ a.e. I've been having trouble defeating this monster. Even with lots of buff potions, health potions, high tier armor (Molten), and ranged weapons with plenty of ammo (Minishark), I can't seem to kill this boss.  Worse, when I get it down to about 2000 hp, even if I'm fairly far away, it'll suck me in and continually rise towards the skybox, which doesn't allow me to do anything for some reason.  How do I defeat the Wall of Flesh? What gear should I be using and what kind of potions do I need? How are echo and print different in PHP? Possible Duplicate:          Is there any major and fundamental difference between these two functions in PHP? What is the probability that all three tosses result in the same outcome? A coin is tossed three times: The probability of zero heads is 1/8 and the probability of zero tails is 1/8.  And my question is: What is the probability that all three tosses result in the same outcome?   So, if P(zero heads)= 1/8 , then that should be the same of p(all tails)?  If so, we could use the Addition Rule which is $P(A \cup B) = P(A) + P(B)$  where A and B are disjoint events, i.e. $A \cap B = \emptyset$, A is the event of tossing all heads and B is the event of tossing all tails.  I'm not sure how to continue after that... would the complement be used? Reference request: Standard textbook for first-order predicate logic in english I am writing my bachelor thesis in english, but I have studied logic in my different native language. Since a rely on elementary results from first-order predicate logic (especially satisfaction in structures), I should reference the reader to a textbook for details.   Is there a standard textbook that starts with elementary propositional logic and goes to Incompleteness theorems? Meaning of Simplify's tolerance parameter The standard parameters of Douglas-Peucker's simplify algorithm are geometry and tolerance (e.g  in PostGIS). What's the meaning of the tolerance parameter? I know that the bigger the value, the coarser the geometry will be. But does the number has any unit or it is just arbitrary? How to disable javascript modal popup ads Nginx Subdirectory returning 404 I currently have a bunch of working static files at the domain name khairulslt.me (from NameCheap). Recently, I've tried setting up a subdirectory (khairulslt.me/RGBGame) as seen in the code below; However, I keep getting 404 errors. What am i missing out?  server {    server_name khairulslt.me www.khairulslt.me;    autoindex off;   location / {     root /var/www/khairulslt.me;     index circles.html;   }   listen 443 ssl; # managed by Certbot ssl_certificate /etc/letsencrypt/live/khairulslt.me/fullchain.pem; # managed by Certbot ssl_certificate_key /etc/letsencrypt/live/khairulslt.me/privkey.pem; # managed by Certbot include /etc/letsencrypt/options-ssl-nginx.conf; # managed by Certbot ssl_dhparam /etc/letsencrypt/ssl-dhparams.pem; # managed by Certbot  location /RGBGame {     root /var/www/khairulslt.me/RGBGame;     index colorGame.html;     try_files $uri $uri/ /var/www/RGBGame/colorGame.html?q=$uri&amp;$args;     autoindex off;   }  location /robots.txt { return 200 "User-agent: *\nDisallow: /\n";    }  }   server { if ($host = www.khairulslt.me) {     return 301 https://$host$request_uri;   } # managed by Certbot   if ($host = khairulslt.me) {     return 301 https://$host$request_uri;   } # managed by Certbot   listen 80; server_name khairulslt.me www.khairulslt.me; return 404; # managed by Certbot }   I've tried all the different settings like (alias vs root), try_files, autoindex(on, off). My main website www.khairulslt.me works fine and is serving the html/css/js files in /var/www/khairulslt.me perfectly fine. I'm just not sure why I can't replicate the settings for my other app (html/css/js files in folder /RGBGame). Could it be that I've missed out something that isn't related to the nginx config? From official :     Due to a head injury, Cyclops is unable to shut off his optic blasts   at will and must therefore wear ruby quartz lenses to block the beams.   From the same page:     Cyclops possesses the mutant ability to project a beam of heatless   ruby-colored concussive force from his eyes, which act as   inter-dimensional apertures between this universe and another.   When his eyes are always blocked by ruby-colored beam of concussive force, how does he see? Correct way to use include and includeonly when writing a large document like a thesis I am writing a large document with several chapters and I would like to work on each chapter separately. In my main.tex I include the chapters either like   \include{chap1.tex}  \include{chap2.tex}  \include{chap3.tex}   or   \input{chap1.tex}  \input{chap2.tex}  \input{chap3.tex}   I think \include is preferable if I want things to be referenced and cited correctly? but doing it the \include way latex main.tex I don't get any dvi output whereas with \input things compile fine to dvi. What's going wrong here?  I would then like to use \includeonly (or \excludeonly ) to work on individual sections of the document whilst maintaining the referencing etc.  A mwe can be handy:  \documentclass[a4paper,10pt]{article} \includeonly{chap1}  \begin{document}   \include{chap1}   \include{chap2}   \include{chap3} \end{document} Downloading US Census data? How to force a table into page width? Is there a way to see questions with a certain tag from all Stack Exchange sites? For example, I see there are questions with the  tag on Stack Overflow, but there are also such questions on Super User, Unix/Linux, and other sites in the network.  Is there a way to follow a tag across all of the Stack Exchange sites, so that one can see the global activity of a topic without visiting all the sites separately? Can I create a Controller that simply returns an image asset?  I would like to route this logic through a controller, whenever a URL such as the following is requested:  www.mywebsite.com/resource/image/topbanner   The controller will look up topbanner.png and send that image directly back to the client.  I've seen examples of this where you have to create a View - I don't want to use a View. I want to do it all with just the Controller.  Is this possible? How can I launch a GameObject at a target if I am given everything except for its launch angle? I am trying to launch an object at a target, given its position, its target position, the launch speed, and the gravity.  I am following this formula from :  $$ \theta = arctan \bigg( \frac{v^2 \pm \sqrt{v^4-g(gx^2 + 2yv^2)}}{gx} \bigg) $$  I have simplified the code to the best of my ability, but I still cannot consistently hit the target.  I am only considering the taller trajectory, of the two available from the +- choice in the formula.  Does anyone know what I am doing wrong?  using UnityEngine;  public class Launcher : MonoBehaviour {     public float speed = 10.0f;      void Start()     {         Launch(GameObject.Find("Target").transform);     }      public void Launch(Transform target)     {         float angle = GetAngle(transform.position, target.position, speed, -Physics2D.gravity.y);         var forceToAdd = new Vector2(Mathf.Cos(angle), Mathf.Sin(angle)) * speed;         GetComponent&lt;Rigidbody2D&gt;().AddForce(forceToAdd, ForceMode2D.Impulse);     }      private float GetAngle(Vector2 origin, Vector2 destination, float speed, float gravity)     {         float angle = 0.0f;          //Labeling variables to match formula         float x = Mathf.Abs(destination.x - origin.x);         float y = Mathf.Abs(destination.y - origin.y);         float v = speed;         float g = gravity;          //Formula seen above         float valueToBeSquareRooted = Mathf.Pow(v, 4) - g * (g * Mathf.Pow(x, 2) + 2 * y * Mathf.Pow(v, 2));         if (valueToBeSquareRooted &gt;= 0)         {             angle = Mathf.Atan((Mathf.Pow(v, 2) + Mathf.Sqrt(valueToBeSquareRooted)) / g * x);         }         else         {             //Destination is out of range         }          return angle;     } }