Notation for eventually less than - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:00:51Z http://mathoverflow.net/feeds/question/10914 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/10914/notation-for-eventually-less-than Notation for eventually less than Thomas Bloom 2010-01-06T13:13:19Z 2012-12-17T15:29:53Z <p>Is there some existing notation for</p> <blockquote> <p><code>$f(n)\leq g(n)$</code> for sufficiently large n</p> </blockquote> <p>Apart from just writing that itself? I'm thinking of something compact like the Landau notation $f\ll g$.</p> <p>(Apologies if this is too specific for MathOverflow - just close it if so. I was also unsure what tags to add, so just edit it accordingly).</p> http://mathoverflow.net/questions/10914/notation-for-eventually-less-than/10918#10918 Answer by Joel David Hamkins for Notation for eventually less than Joel David Hamkins 2010-01-06T13:49:27Z 2010-01-06T13:55:42Z <p>In logic, this relation is called <em>almost</em> less than or equal, and is denoted with an asterisks on the relation symbol, like this: $f \leq^* g$. </p> <p>For example, the <a href="http://mathoverflow.net/questions/8972#9027" rel="nofollow">bounding number</a> is the size of the smallest family of functions from N to N that is not bounded with respect to this relation. Under CH, the bounding number is the continuum, but it is consistent with the failure of CH that the bounding number is another intermediate value.</p> http://mathoverflow.net/questions/10914/notation-for-eventually-less-than/10927#10927 Answer by Matt Noonan for Notation for eventually less than Matt Noonan 2010-01-06T15:53:48Z 2010-01-06T15:53:48Z <p>Why not just overload $\leq$ when applied to sequences? I don't think there is any opportunity for confusion, and it fits with the notation you would use when extending $\leq$ to an ultraproduct.</p> <p>This is what Jim Henle does in his "non-nonstandard analysis", which uses "eventually" as a replacement for an ultrafilter.</p> http://mathoverflow.net/questions/10914/notation-for-eventually-less-than/116570#116570 Answer by Peter Michor for Notation for eventually less than Peter Michor 2012-12-17T07:31:34Z 2012-12-17T07:31:34Z <p>Good notation should be self-explaining and not require the reader to remember to much. I would write either: $$f\le g\quad\text{eventually}$$ $$f\le g\quad\text{ near }\infty$$ If you use it more than 100 times in a paper you could use something like $$f \preccurlyeq g.$$</p> http://mathoverflow.net/questions/10914/notation-for-eventually-less-than/116616#116616 Answer by Andreas Blass for Notation for eventually less than Andreas Blass 2012-12-17T15:29:53Z 2012-12-17T15:29:53Z <p>I agree with Joel Hamkins's answer, but I don't entirely agree with his comment on that answer. I generally use asterisks to mean "with finitely many exceptions" or "modulo finite sets", so I'd use $f\leq^*g$ and $A\subseteq^*B$ as Joel says. But when working modulo some ideal $I$ other than the ideal of finite sets, I'd ordinarily avoid asterisks and instead write $f\leq_Ig$ and $A\subseteq_IB$.</p> <p>I'd like to protest vigorously against the use of $\ll$ in this situation. To me, $f\ll g$ means that $f$ is a <em>lot</em> smaller than $g$ (at least eventually), whereas here you might have $f(n)=g(n)-1$ for all $n$.</p>