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$A < B$ means A is smaller than B.

$A \ll B$ means A is some orders of magnitude smaller than B (see also this question for a more in-depth discussion). In modelling, it may mean that A can be neglected ($A + B \approx B$).

In the LaTeX amssymb symbol list, section Binary relations, I found the symbols $\lll$ and $\ggg$, spelt as \lll and \ggg, respectively. What does $A \lll B$ mean? An order of magnitude of order of magnitudes smaller? Does it mean $A \cdot B \approx B$ even if $A \gg 1$? Like in this example?

$B=10^{10^{10}}$ and $A=10^{10}$, then $A \cdot B$ = $10^{10^{10}} \cdot 10^{10} \approx 10^{10^{10}+10} \approx 10^{10^{10}} \approx B$

...or does it mean something else?

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Can you provide an example of its use? – Andrew Uzzell Feb 11 '13 at 13:36
If this symbol is actually used anywhere (I've never seen it), then certainly the author would have defined it previously. – David Mitra Feb 11 '13 at 13:48
I've never seen the symbol either (and I use the second symbol differently than you do). I suspect it was defined in whatever text you are reading. (edit: In other words, I agree with David Mitra) – mixedmath Feb 11 '13 at 13:50
I've found it in the LaTeX symbol list, and added a link in the question. – gerrit Feb 11 '13 at 13:52
It means "very much less than"; what that means depends on the situation. – AakashM Feb 11 '13 at 15:07
up vote 12 down vote accepted

Quoting Mariano Suárez-Alvarez:

The symbol denotes whatever the author tells you it will denote in his comments about notation, and there is a special place in hell for users of unexplained notation.

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Good answer, but why is it Community Wiki? – gerrit Feb 12 '13 at 8:34
@gerrit Because the answerer felt he/she didn't deserve rep points – Rustyn Mar 26 '14 at 20:49
+1 for the fantastic quote – user285523 Nov 8 '15 at 16:10

Mariano Suárez-Alvarez's comment gives the correct answer that the usage of $\lll$ or $\ggg$ is nonstandard and will have to be defined in-context, but it might be of interest that several sources use these symbols to denote bitwise shifts. See examples here, here, and here.

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Java has a >>> operator, but does not have <<<. In any case, I doubt that's the intended use in the OP's case, as >>> is not the same as . (Also, your links are behind a pay-wall - can you find any examples that are free to view?) – Darrel Hoffman Apr 3 at 14:55

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