Wikipedia:Reference desk/Archives/Mathematics/2007 May 27

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May 27

Syndrome

I read the definition of syndrome in the context of coding theory at Syndrome (disambiguation). The sentence is pretty difficult to understand, though – is anyone able to explain it more clearly? For example, if I've received 26 bits of which 16 are data and 10 check bits, what is the syndrome? Thanks. ›mysid (☎∆) 15:49, 27 May 2007 (UTC)

That page is only (as the title suggests) a disambiguation page, whose purpose is to direct the reader to the appropriate place, in this case the article on Decoding methods, and in particular the section Syndrome decoding. That article is actually also not easy to understand, but at least you should give it a try. You will also need to understand what a linear code is, and what a parity-check matrix is. Your "for example" question can only be answered by someone who knows the code that was used, for example in the form of a generating matrix. The roles of the bits (data versus check) is not relevant. Please read these other articles first; if, after reading them, there are things you don't understand, we'll be happy to try to explain them.  --Lambiam^Talk 18:30, 27 May 2007 (UTC)

What is this limit?

$\lim _{x\rightarrow 0}{\frac {1-x}{x}}^{\frac {1-x}{x}}\,$

202.168.50.40 23:06, 27 May 2007 (UTC)

Infinite. The expression is close to 1/x - 1. So says my computer and computers are always right *cough*.

After seeing the unparsed expression, I guess you mean $\lim _{x\rightarrow 0}\left(\left({\frac {1-x}{x}}\right)^{\frac {1-x}{x}}\right)$. In that case it is even more infinite than the first expression. ^^

Note $\lim _{x\rightarrow 0}{\frac {1}{x}}$ doesn't really have a limit in normal circumstances (approaching from the left it is $-\infty$, approaching from the right it is $+\infty$, here we have the assumption $-\infty \neq \infty$... if we instead assume $-\infty =+\infty$ for whatever purposes (complex analysis? point at infinity?), then, yeah, that's infinite, and so would $\lim _{x\rightarrow 0}{\left({\frac {1}{x}}-1\right)}^{({\frac {1}{x}}-1)}\,$. Otherwise we have a undefinable limit to an undefinable limit, and I don't think any sort of standard manipulation would make it definable. Root⁴(one) 05:07, 28 May 2007 (UTC)

Well, allowing x to approach 0 from below opens a big can of worms, with negative numbers being raised to arbitrary powers and all. Still, if we are working with the Riemann sphere, for any interpretation of exponentitation, the limit will be infinity. -- Meni Rosenfeld (talk) 05:19, 28 May 2007 (UTC)

Honestly, I do not see this big can of worms. The function to study can be rewritten as $e^{\left({\frac {1}{x}}-1\right)log\left({\frac {1}{x}}-1\right)}$ when x is close enough to 0, no matter if it is positive or negative. With this writing it is clear that the limit is $+\infty$. Cthulhu.mythos 09:39, 30 May 2007 (UTC)

I just wanted to say, after playing with Maxima and especially this Java complex calculator applet, I severely underestimated my understanding of ${(1/z)}^{1/z}$ and related functions when considering the complex plane.

1. Meni, you are exactly right, letting x approach from the negative side of the real number line does open one big can of worms. Maybe a can of worms of infinite volume. I wasn't even thinking about the fact we're mixing in complex numbers and the like. I certainly messed up by thinking (or rather lazily guessing and not proving because I wasn't sure how to prove) it would approach infinity from the left only along the real line (at least in magnitude). Actually, It appears I have a little bit of egg on my face. I found it appears to approach 0. This makes sense if you consider a related limits. $\lim _{x\rightarrow 0^{-}}|1/x|^{1/x}$ or $\lim _{x\rightarrow +\infty }x^{-x}$. (I'm not using using anything related to the Riemann sphere because I wouldn't know how to prove limits from any particular direction on there or if finding limits from a particular direction makes any sense)
2. $z^{-1/z}$ is one complex and fascinating critter, especially on the Riemann sphere! If this doesn't have a name, I swear it ought to. It blows my mind! As mentioned, DIRECTLY from the left, it converges to zero. If you take the path DIRECTLY from the right, it converges to "inf". as expected. If you take circles in the complex plane and try to map them out... weird things happen. circles appear to wrap into amazing patterns. I don't think what I am seeing. I don't quite understand it. I don't think what I'm seeing in terms of path is related to any sampling problems (although the problems certainly do crop up). I've been playing with the calculator for the past two hours or so because I don't even remotely know how to approach understanding it analytically, or I don't even know what I'd want to understand about it. Even if I am only seeing artifacts related to problematic sampling of an infinite range of points, what I see is still... just... awesome! Root⁴(one) 04:07, 29 May 2007 (UTC)

You are right, I was also under the false impression that at least the magnitude of the expression converges to infinity. Indeed, what we have here is an essential singularity, which I know can be a weird phenomenon, but do not understand it completely myself or have done such experimentation with. For this purpose, $(1/z)^{1/z}$ is equivalent to $z^{-1/z}$ or just $z^{1/z}$. I am not sure if it exhibits anything more interesting than the cannonical example of an essential singulariy, $e^{1/z}$. -- Meni Rosenfeld (talk) 09:40, 29 May 2007 (UTC)

To Cthulhu: When x is a very small negative number, 1/x is a very large negative number, and you're taking the log of a negative. Not good. Black Carrot 10:05, 30 May 2007 (UTC)

Yeppp. I always mess those signs up. Cthulhu.mythos 10:09, 30 May 2007 (UTC)

There is nothing inherently wrong with taking the log of a negative number, but it requires choosing a specific branch beforehand, and it will give a purely imaginary number, which, when exponentiated, gives an oscillatory result, rather than a convergent one. -- Meni Rosenfeld (talk) 13:59, 30 May 2007 (UTC)