Similarly, the reflexive transitive symmetric closure of \ xrightarrow [ R ] { }, denoted \ overset { * } { \ underset R \ leftrightarrow } ( see abstract rewriting system # Basic notions ), is a congruence, meaning it is an equivalence relation ( by definition ) and it is also compatible with string concatenation.
12.
Note also that functions do not in general attain a maximum value, and hence the arg max is sometimes the empty set; for example, \ underset { x \ in \ mathbb { R } } { \ operatorname { arg \, max } } \, x ^ 3 = \ emptyset, since x ^ 3 is unbounded on the real line.
13.
The notation for uniform convergence of f _ n to f is not quite standardized and different authors have used a variety of symbols including ( in rough order of popularity ) f _ n \ rightrightarrows f, \ underset { n \ to \ infty } { \ mathrm { unif \ lim } } f _ n = f, f _ n \ overset { \ mathrm { unif . } } { \ longrightarrow } f.
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