Wikipedia:Reference desk/Archives/Mathematics/2010 December 21

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December 21

Question about Limits and Functions

Do two functions $f$ and $g$ exist such that $\lim _{x\rightarrow \infty }(f(x)-g(x))=0$ but it is not true that $\lim _{x\rightarrow \infty }{\frac {f(x)}{g(x)}}=1$, besides "obvious" cases ($f=0$ and $g=0$ unconditionally, for one example)? I thought of this question after seeing the converse was true in the Prime number theorem article, for $f(x)=\pi (x)$ and $g(x)={\frac {x}{\ln x}}$. If anyone can help with this curiosity of mine, it would be greatly appreciated! JamesMazur22 (talk) 02:57, 21 December 2010 (UTC)

How about $f(x)=1/x$, $g(x)=1/x^{2}$? —Bkell (talk) 04:25, 21 December 2010 (UTC)

Or $f(x)={\frac {1}{x}}$, $g(x)={\frac {1}{2x}}$, for a finite limit. --COVIZAPIBETEFOKY (talk) 15:41, 21 December 2010 (UTC)

Haha maybe I should have actually thought about it myself. JamesMazur22 (talk) 22:00, 23 December 2010 (UTC)

Resolved

statistics symbol

Hi. What is the meaning of a symbol that looks like the "independent" thumbtack symbol on table of mathematical symbols, but with two vertical lines? It looks like an "entails" symbol turned through 90 degrees anticlockwise, or a pi symbol upside-down. From context, it seems to mean something like (statistically) independent, but perhaps a slightly weaker condition. Thanks, 81.129.133.123 (talk) 21:40, 21 December 2010 (UTC)

Do you mean the coproduct, \coprod, $\coprod$? It is most frequently used to mean disjoint union of sets. Eric. 82.139.80.114 (talk) 23:05, 21 December 2010 (UTC)

An example is adjunction space. Concretely, suppose T is a torus with a hole with boundary B cut into it, and we would like to make a double torus. We do this by taking two copies of T and identifying the boundaries of the two holes. Let $T_{i}=T,\ B_{i}=B,\ i=1,2$ be our two copies of the torus, then loosely speaking the double torus is $(\coprod _{i}T_{i})/\sim$, where $\sim \$ is the relation identifying $B_{1}$ with $B_{2}$. The purpose of using the disjoint union is to make the two copies $T_{1},\ T_{2}$ "different" from each other; since in fact $T_{1}=T=T_{2}$, if we just used the ordinary union $\cup$ the two copies would still be same and we would just get $T$ in the end, instead of a double torus. Eric. 82.139.80.114 (talk) 23:17, 21 December 2010 (UTC)

Oh, wait, it's a statistics symbol? Can you give us an example of the context in which you saw it? Eric. 82.139.80.114 (talk) 23:23, 21 December 2010 (UTC)

Is this the one you mean: ⫫? I found that on unicode mathematical operators; I don't know it's meaning. 86.135.251.208 (talk) 23:36, 21 December 2010 (UTC)

In addition to \coprod ($\textstyle \coprod$ or $\coprod$) and \amalg ($\amalg$), LaTeX also includes the symbol \sqcup ($\sqcup$) and a corresponding \bigsqcup ($\textstyle \bigsqcup$ or $\bigsqcup$). Perhaps the symbol you're referring to is one of these? —Bkell (talk) 23:39, 21 December 2010 (UTC)

I see it written in TeX like this:

$X\bot \!\!\!\bot Y\,$

meaning the two random variables X and Y are mutually independent. That's not the same as the coproduct symbol. I've never heard of any particular name for the typographical symbol. Michael Hardy (talk) 05:26, 22 December 2010 (UTC)

(OP here). Michael Hardy above has the correct symbol. In context, the symbol is certainly something like independence. But why doesn't the author use the standard thumbtack symbol with a single vertical line? I am guessing that it might refer either to pairwise independence but not total independence, or exchangeability, or some other variant of independence. The RVs in question are Gaussian Processes, if it helps. Thanks, 131.111.23.223 (talk) 08:19, 22 December 2010 (UTC)

There was some discussion of this earlier this year on the WikiProject Statistics talk page. See Wikipedia talk:WikiProject Statistics/Archive 3#New-ish notation. Qwfp (talk) 10:51, 22 December 2010 (UTC)

Thanks, Qwfp!

Resolved

. The meaning was "conditionally independent", which makes sense of the article. Kia Ora, 131.111.23.223 (talk) 13:41, 22 December 2010 (UTC)