It's hard to look up the meaning of a symbol if you don't know what it is called in the context that it is written.
That apostrophe-type symbol in the sequence notation - what is that?
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That's a prime.
In mathematics, statistics, and science, the prime is generally used to generate more variable names for things which are similar, without resorting to subscripts – $x'$ generally means something related to or derived from $x$.
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$\begingroup$ Okay, I think I get it. So in the above example, the writer puts prime there referring to the fact that the a1 above has been modified and the new a1 came from the original set. Thanks! $\endgroup$ – dthree May 24 '14 at 3:22
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Draw the character on Detexify or Shapecatcher. Detexify tells you what (La)TeX commands generates a character that looks like what you drew. Shapecatcher tells you what Unicode characters looks like what you drew.
Mathematical notations tend to be easier to find via Detexify. In my first attempt, I got the correct answer \prime as the second choice, with the other choices obviously not right. Although Unicode does have a U+2032 PRIME character, it didn't come up among the many very similar characters that look like a diagonal line.
Once you have the name, you can look it up. Prime has a standard meaning in some fields (in particular calculus, where it means a derivative), but in general it's a way to generate more symbols: $a$ is the name for something and $a'$ is the name for something else; usually there is a relationship between $a'$ and $a$ but it's up to the surrounding text to explain what that relationship is.
Here, $(a'_1, \ldots, a'_n)$ is the sequence produced as output by the algorithm. It is related to $(a_1, \ldots, a_n)$ in that the two contain the same elements but in a different order (i.e. it's a permutation).
answered Jul 24 '14 at 8:48
Gilles 'SO- stop being evil'Gilles 'SO- stop being evil'
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The idea here is that the sequence $a'_1,\ldots,a'_n$ is a rearrangement of $a_1,\ldots,a_n$ in non-decreasing order. For example, if the original sequence were $3,1,2,5,4,3$, then the new primed sequence would be $1,2,3,3,4,5$.
The prime ($^\prime$) doesn't have a fixed meaning, and in this particular case, the author could just as well have used a completely new letter for the new sequence, say $b_1,\ldots,b_n$. One common use for the prime is in analysis, where it designates the derivative with respect to some "understood" variable. Another common usage is in linear algebra, where it is sometimes used to denote the transpose (one of several common notations for the transpose, including $^t$ and $^T$).
answered May 24 '14 at 4:16
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As well as "prime", well covered by the other answers, the symbol is often pronounced "dash" or "dashed", as in "$x$-dash(ed)" and "$x$-double-dash(ed)" for $x'$ and $x''$, respectively. I've no idea why this is, since the symbol is completely unrelated to the punctuation mark known as a dash ("–").
answered Jul 24 '14 at 20:51
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As Mowji already mentioned, it means a one thing is derived from another.
Think of it as what it come from, the prime used in mathematics, for the a kind of function, named derivative of a function, as in: $f(x) = 2 \times x, f'(x)$ = 2 or another example $f(x) = sin(x), f'(x) = cos(x)$.
The keyword to have in mind sawing this sign is, “derivative” and/or “derived”.
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$\begingroup$ I don't see what this adds to the other answers. Giving examples of the derivative isn't really relevant, since the primes in the question clearly aren't derivatives. Everything else is already covered by the other answers. $\endgroup$ – David Richerby Jul 24 '14 at 7:03
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$\begingroup$ It says where this symbol come from. Every one know what is the derivative of a function. That say, people here really knows how to welcome answers… :-/ . $\endgroup$ – Hibou57 Jul 24 '14 at 12:54
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$\begingroup$ Everyone knows what the derivative of a function is and this question isn't about differentiation. That's two good reasons not to give examples of derivatives. And, as I said, everything else in your answer is covered in the other ones. This site works because people vote answers up or down according to how good they think the answers are. If you don't want the possibility of your answers being voted down, don't post them here. The fact that your other answer so far is at +3 clearly indicates that it's nothing personal. $\endgroup$ – David Richerby Jul 24 '14 at 18:22
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$\begingroup$ @David Richerby: where did I suggest I though it's “personal”? I just said the point was missed. Well… forget about it… $\endgroup$ – Hibou57 Jul 24 '14 at 20:46
