What is the process that leads to the volume expansion of the outer layers of a star when it becomes a red giant?
What causes the dimensions of a star increase when its hydrogen fuel is exhausted?
When do we use "at" and "in" with "arrive" talking about place, not time? I'm trying to solve    So far I've done the first part, evaluating the summation ;  where a is just n. I'm not sure where to go from here or what it even means deduce the second summation. I understand that the summation of simply $$\sum\limits_{k=0}^n\cos(kx)$$ is derived from looking at the real part of  $$\sum_{k=0}^n{\rm e}^{ikx} $$I'm guessing they want me to see how the second summation in the question is defined by 'playing around' with the original? \documentclass{article} \usepackage{expl3} \ExplSyntaxOn \begin{document} \tl_to_str:n { testing _ things }  \par  \tl_set:Nn \l_tmpa_tl { testing _ things } \tl_to_str:N \l_tmpa_tl \end{document}   gives a strange result (testing ̇things).  How can I get testing_things?  I basically want verbatim content (with line breaking). Misplaced navigation bar and question title It appears that on Server Fault and Super User, the navigation bar and question title have suddenly become misplaced. The same does not occur on SO and mSO, nor on any of the other SE sites I spot-checked.  This is being seen in both Chrome and Firefox and just began occurring within the last few minutes.      What it should look like:      While this was fixed for a few hours, it has returned as of 22:05. Splitting pieces of a mesh into a new object I've just made a set of armor by using a male body mesh model I downloaded and selecting certain faces from the model and duplicating them to create viable armor pieces.   Since the pieces are just duplicated from the same mesh male model, then it's still part of the same object.   How do I separate the armor piece from the model to make the pieces separate? UK PLAB Visa Refusal due to unexplained bank deposits I had applied for a PLAB (Doctors) Visa for the UK to appear in PLAB test. I got refused on the basis of two recent large deposits in the bank statement I provided (my mothers). In reality, these are my own savings which I had deposited in her account to make it more strong. The only objection I got was that I must provide evidence for the source of these deposits. Other than this, I have previously travelled to the UK and USA as well as other parts of the world and my travel history is fine. Can someone please advise on how to proceed and re-apply? I know this issue has been addressed before with different software, but how do I do it with Shapely?  I have a polygon with a very small gap on the interior of the shape:    Here is a close up:    The polygon is valid so I can't use the buffer trick, and simplifying the polygon also doesn't work. Static Classes In Java With the enthusiastic question of "Who wants ice-cream?", what is the more correct response?         (Not) I.   (Not) me.      Neither response is a sentence. The first response of "(not) I" sounds stuffy, like it should be followed with an indignant sniff. The second sounds like American idiom and acceptable for casual speech.    What do you say? Why is travelling around the speed of light a problem? I don't fully understand what would happen if we could travel at the speed of light. But I saw somewhere here that it would mean events happen out of order. But why is this a problem. It is said that cause has to happen before the effect, but why does this have to be linear?   And why is it that the speed of light is the maximum? I take it that light (photons?) are massless, so why can't they travel faster? prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$ Why we won't extract heat from air temperature? Last two digits of $2^{1000}$ via Chinese Remainder Theorem? I bumped across the aforementioned question in my notes while studying today and I have completely forgot how to do this. I remember using CRT to solve a problem like this on one of my tests, too bad they didn't give back my solutions :(.  Since $\gcd(100,2) = 2$, we can't use the usual Euler's theorem to solve via $\pmod {100}$. So $100 = 5^2 2^2$, and applying Euler's on the 25 gives me $2^{20} \equiv 1 \pmod {25}$. However since $\gcd(2,4) = 2$, we can't do the same for $\bmod 4$. Accordingly how do I set up the other modulo congruence so I can apply CRT? How to create SHA512 password hashes on command line In Linux I can create a SHA1 password hash using sha1pass mypassword. Is there a similar command line tool which lets me create sha512 hashes? Same question for Bcrypt and PBKDF2. How can I rename many files at once? I need to rename a bunch of files in order to fix their sort order, therefore I need to be able to do a sort of "find and replace" so I can replace chunks of the filenames at a time.  What tools should I use? I prefer GUI but a command line tool recommendation would also be fine. What does "if not" mean in the given sentence How is it that questions posted to the Stack Overflow show up as #1 in Google search results just minutes after being submitted? What SEO practices are being used to generate this sort of up-to-the-minute accuracy? What is this socket/port/connector on my laptop? I've have this socket on my laptop (Lenovo Z570).  What is this port and what is it used for? And what does that symbol mean (I expected a LOCK symbol)? What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by contradiction, and the other proves the contrapositive, the proofs look almost exactly the same.  For example, say I want to prove: $P \implies Q$ When I want to prove by contradiction, I would say assume this is not true. Assume $Q$ is not true, and $P$ is true. Blabla, but this implies $P$ is not true, which is a contradiction.  When I want to prove the contrapositive, I say. Assume $Q$ is not true. Blabla, this implies $P$ is not true.  The only difference in the proof is that I assume $P$ is true in the beginning, when I want to prove by contradiction. But this feels almost redundant, as in the end I always get that this is not true. The only other way that I could get a contradiction is by proving that $Q$ is true. But this would be the exact same things as a direct proof.   Can somebody enlighten me a little bit here ? For example: Are there proofs that can be proven by contradiction but not proven by proving the contrapositve? Requires installation of untrusted packages? I tried installing VLC Player yesterday, and I got the Requires installation of untrusted packages error.  I searched for hours for a solution to this problem, and finally came upon a suggestion that said to change servers for downloading updates from.  I did this, and it gave me a 404 error when I clicked on "Check" in the Update Manager.  I tried switching my server back to the main one (and after that, several other ones), but I am now still getting a 404 error:   And the other error I get when downloading or updating:   Edit: Output of cat /etc/apt/sources.list:  deb-src http://us.archive.ubuntu.com/ubuntu/ precise main restricted #Added by software-properties  # See http://help.ubuntu.com/community/UpgradeNotes for how to upgrade to # newer versions of the distribution. deb http://us.archive.ubuntu.com/ubuntu/ precise main restricted multiverse deb-src http://us.archive.ubuntu.com/ubuntu/ precise multiverse universe #Added by software-properties  ## Major bug fix updates produced after the final release of the ## distribution. deb http://us.archive.ubuntu.com/ubuntu/ precise-updates main restricted multiverse deb-src http://us.archive.ubuntu.com/ubuntu/ precise-updates restricted main multiverse universe #Added by software-properties  ## N.B. software from this repository is ENTIRELY UNSUPPORTED by the Ubuntu ## team. Also, please note that software in universe WILL NOT receive any ## review or updates from the Ubuntu security team. deb http://us.archive.ubuntu.com/ubuntu/ precise universe deb http://us.archive.ubuntu.com/ubuntu/ precise-updates universe  ## N.B. software from this repository is ENTIRELY UNSUPPORTED by the Ubuntu  ## team, and may not be under a free licence. Please satisfy yourself as to  ## your rights to use the software. Also, please note that software in  ## multiverse WILL NOT receive any review or updates from the Ubuntu ## security team.  ## N.B. software from this repository may not have been tested as ## extensively as that contained in the main release, although it includes ## newer versions of some applications which may provide useful features. ## Also, please note that software in backports WILL NOT receive any review ## or updates from the Ubuntu security team. deb http://us.archive.ubuntu.com/ubuntu/ precise-backports main restricted universe multiverse deb-src http://us.archive.ubuntu.com/ubuntu/ precise-backports main restricted universe multiverse #Added by software-properties  deb http://us.archive.ubuntu.com/ubuntu/ precise-security main restricted multiverse deb-src http://us.archive.ubuntu.com/ubuntu/ precise-security restricted main multiverse universe #Added by software-properties deb http://us.archive.ubuntu.com/ubuntu/ precise-security universe  ## Uncomment the following two lines to add software from Canonical's ## 'partner' repository. ## This software is not part of Ubuntu, but is offered by Canonical and the ## respective vendors as a service to Ubuntu users. deb http://archive.canonical.com/ubuntu precise partner deb-src http://archive.canonical.com/ubuntu precise partner  ## This software is not part of Ubuntu, but is offered by third-party ## developers who want to ship their latest software.