What does this order symbol mean?

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The order symbol discussed is represented as \mathcal{O}, indicating "order of magnitude." It signifies that a quantity is on the order of 1, meaning it is greater than 0.1 and less than 10. In mathematical contexts, such as x \sim \mathcal{O}(\lambda^n), it describes the size relationship between variables. The term "O(1)" specifically implies a range of values, while "O(1032) years" denotes a proton's lifetime being between 10^31 and 10^33 years. The discussion also touches on the relationship of similar symbols in TeX bitmap naming.
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It means "Order" as in "order of magnitude". \mathcal{O}(1), when referring to some quantity, means that the order of magnitude of the quantity is on the order of the number 1 - it's larger than 0.1 and less than 10. Typically we take order of things as powers of some number.

e.g. x \sim \mathcal{O}(\lambda^n) means the variable x is of the same order of size as the parameter \lambda to the nth power - it's between \lambda^{n-1} and \lambda^{n+1} in size.

See http://en.wikipedia.org/wiki/Big_O_notation
 
That is "on the order of". Its most general mathematics definition can be complicated but in this application saying "O(1)" really means "a few": less than 10 but more than 0.1. Saying that the lifetime of a proton is "O(1032) years" means that the lifetime of a proton is certainly greater than 1031 years but less than 1033 years.
 
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Thank you oh so very very much. I've noticed that there are letters that take a similar tex bitmap name; are they related?
 
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