I am always confused about the end of the movie Interstellar.  What really happened in that black hole? How could he survive in that black hole and then get out?
In the movie Interstellar,  Cooper (played by ) enters a black hole and sends a message to Murph (his daughter). Later, he is found by rangers, but I don't understand how he gets out of the black hole?  As far as I know, nothing can get out of a black hole, as it has extremely high gravity.
Its very easy to prove $p\mid n^p-n$ for p=3,5,7, it fails for p=9 because $$     (n+1)^9-(n+1)= n^9+9n^8+36n^7+84n^6+126n^5+126n^4+84      n^3+36n^2+8n $$ and $84= 2²\times 3\times7$. Is it true for any $p$ prime? I have checked many primes, but have no idea for the general case. When casting a spell as a bonus action, you can't cast another spell on the same turn except a 1-action cantrip - can the order be "reversed"? Can you cast a cantrip first as a bonus action, then any other spell? If not, would it be broken to allow it?  I'm thinking a Sorcerer casting a Quickened true strike, then casting a leveled spell requiring a touch attack. Seems like a great use of the cantrip + Metamagic feature, but it seems like it is not allowed. You can't see the question owner's special color When the owner of a question comments on their question or an answer to their question or self-answers their question, they get a special color for the box around their username.  The problem is, you can't see the box:        Can you please make this not white on white? I am trying to prove:     $a\mid b \land b\mid c \Rightarrow a\mid c $   $a\mid b$ means that a divides b if there is an integer k, that $b=k\cdot a$  Please give me a hint on how to start, because I have no idea. Network simplex method, leaving and entering variables Visit the UK with an expired French identity card Will using 'var' affect performance? I'm completely new and lost in Ubuntu 12.04, I'm having trouble navigating to my Sublime folder which I know is in my /opt folder. Yet I've tried to no avail to navigate into it via Terminal window. The directory is colored in blue, no idea what this means...but I am able to access the folder using the GUI explorer...?  k@k-Ubuntu:~$ cd /opt k@k-Ubuntu:/opt$ ls Sublime Text 2 k@k-Ubuntu:/opt$ cd Sublime Text 2 bash: cd: Sublime: No such file or directory k@k-Ubuntu:/opt$ cd /Sublime Text 2 bash: cd: /Sublime: No such file or directory Capistrano for Django Let $a\ge1$ and define the sequence $(x_n)$ recursively by: $$x_1 = \sqrt{a}$$ $$x_{n+1}= \sqrt{a+x_n}$$  Here's what I did:  Plugging in some values makes it seem as if the sequence is increasing. I first solve the equation $t=\sqrt{a+t}$, equivalently $t^2 -t -a =0$, to get an idea what's going on. The roots are $$t_{1,2}= \frac{1 \pm \sqrt{1+4a} }{2}.$$ Therefore, between the roots, that is for $$t\in \left( \frac{1 - \sqrt{1+4a} }{2}, \frac{1 + \sqrt{1+4a} }{2}\right)$$  we have $t^2-t-a \le 0 \iff t \le \sqrt{a+t}$. Here's my attempt at showing inductively that $$\forall n\in \mathbb{N}, \ x_n \le \frac{1 + \sqrt{1+4a} }{2}$$  For $n=1$, the statement is true (simple arithmetic). Suppose it's true for $n$. Consider  $$x_{n+1}= \sqrt{a+x_n}\le \sqrt{a+\frac{1 + \sqrt{1+4a} }{2}}=\frac{1 + \sqrt{1+4a} }{2},$$  because $\frac{1 + \sqrt{1+4a} }{2}$ satisfies $x=\sqrt{x+a}$.  So this establishes the sequence $(x_n)$ is bounded. Now, by induction again, I try to show it's monotonically increasing. The base case $x_1\le x_2$ is obvious. Assume (strong induction) $x_{n-1}\le x_n$ for all $n&gt;1$. But since we have $$x_1\le x_n \le \frac{1 + \sqrt{1+4a} }{2}, $$   then $x_n$ lies in the appropriate interval, so $x_n \le \sqrt{a+x_n}=x_{n+1}.$  Is this proof correct? Can the condition on $a$ be relaxed to $a&gt;0$? Maybe even small negative $a$? And could someone provide me with another proof of convergence of $x_n$? I absolutely hate this one! How do I decrease the size of the page file? I just upgraded my laptop RAM to 12GB (W510, 4 DIMM slots), and since I am using a SSD, I find hard drive space to be quite precious. However, since I have more than enough RAM for most stuff, I think I can get by with probably around 4GB of page file. however, on C drive, there is a pagefile.sys which I assume to be reserving space for page files in the OS, how would I decrease the size of this file? “Are” vs. “is” after “parents and the family” Find the equation of the plane passing through a point and a vector orthogonal I have come across this question that I need a tip for.     Find the equation (general form) of the plane passing through the point $P(3,1,6)$ that is orthogonal to the vector $v=(1,7,-2)$.   I would be able to do this if it said "parallel to the vector"  I would set the equation up as $(x,y,x) = (3,1,6) + t(1,7,-2)$ and go from there.  I don't get where I can get an orthogonal vector. Normally when I am finding an orthogonal vector I have two other vectors and do the cross product to find it.  I am thinking somehow I have to get three points on the plane, but I'm not sure how to go about doing that.  Any pointers?  thanks in advance. What was the origin of Star Trek's Borg? How to validate an email address in JavaScript How do regular expressions differ from wildcards used to filter files How to look up a symbol or identify a letter from a math alphabet or other character? I know what my symbol or character looks like, but I don't know what the command is or which math alphabet it came from. How do I go about finding this out? How would you know if your (high dimensional) data exhibits enough clustering so that results from kmeans or other clustering algorithm is actually meaningful?  For k-means algorithm in particular, how much of a reduction in within-cluster variance should there be for the actual clustering results to be meaningful (and not spurious)?  Should clustering be apparent when a dimensionally-reduced form of the data is plotted, and are the results from kmeans (or other methods) meaningless if the clustering cannot be visualized? I started learning Bash a couple of days ago.  I'm trying to obtain an exit status of grep expression into a variable like this:  check=grep -ci 'text' file.sh   and the output that I got is   No command '-ic' found   Should I do it with a pipe command? Why a subspace of a vector space is useful