I am currently reading through JC Lagarias' "The $3x+1$ Problem and its Generalizations" and have come across some notation reading :
$$\sup_{K \ge 0} T^{(K)}(N)$$
Now I assume that this means "suppose that $K$ is greater than or equal to $0$", however I want to understand any equations I am potentially going to use in a project and thus don't want to rely on assumptions. This may seem a very simple question but I appreciate any help.
On
The abbreviation sup usually stands for supremum. You can read more about that here: http://en.wikipedia.org/wiki/Infimum_and_supremum
On
"$\sup$" and "$\inf$" stand for "supremum" and "infimum". When dealing with real-valued expressions, these are just what you may have seen called the "least upper bound" and "greatest lower bound", respectively.
The stuff below the symbol you wrote describes the set over which $K$ is allowed to roam, and as it does so you are looking to find the least upper bound of the stuff following the "$\sup$" symbol.
So, this expression represents the least upper bound of $T^K(N)$ as $K$ ranges over all nonnegative real numbers.
"$\sup$" and "$\inf$" can often be thought of as generalizations of "$\max$" and "$\min$", respectively.
Sup ("supremum") means, basically, the largest. So this: $$\sup_{k\ge0}T^{(k)}(N)$$ refers to the largest value $T^{(k)}(N)$ could get to as $k$ varies.
It's technically a bit different than the maximum—it's the smallest number that is greater-than-or-equal to every number in the set.
So, for example, the interval $[0,1)$ has no maximum value, but $1$ is the supremum of the interval, because it's greater-than-or-equal-to everything in that interval, and because it's the smallest number with that property. (Note that $(0,1),(0,1],[0,1),[0,1]$ all have the same supremums.)