Bounded

Bounded non-determinism An NTM has the property of bounded non-determinism, i.e., if an NTM always halts on a given input tape T then it halts in a bounded number of steps, and therefore can only have a bounded number of possible configurations.

If a sequence is both bounded from above and bounded from below, then the sequence is said to be bounded.

An infinite subset of the rational numbers may or may not be bounded from below and may or may not be bounded from above.

An interval is said to be left-bounded or right-bounded if there is some real number that is, respectively, smaller than or larger than all its elements.

A set S of real numbers is called bounded from above if there is a real number k such that k ≥ s for all s in S. The number k is called an upper bound of S. The terms bounded from below and lower bound are similarly defined.

Bounded and totally bounded spaces Diameter of a set.

However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps.

In engineering applications, a function corresponding to a linear time-invariant (LTI) system is stable if every bounded input produces a bounded output.

In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is locally bounded as well.

Intervals that are bounded at only one end are said to be half-bounded.

Moreover, a function ƒ defined on a bounded interval is Riemann-integrable if and only if it is bounded and the set of points where f is discontinuous has Lebesgue measure zero.

Note that this concept of boundedness has nothing to do with finite size, and that a subset S of a bounded poset P with as order the restriction of the order on P is not necessarily a bounded poset.

Properties and uses seeAlso Equivalence to linear bounded automaton A formal language can be described by a context-sensitive grammar if and only if it is accepted by some linear bounded automaton (LBA).

The converse does not hold, since any infinite set can be given the discrete metric (one of the examples above) under which it is bounded and yet not totally bounded.

The image of a totally bounded subset under a uniformly continuous function is totally bounded.

The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds.

Variants Bounded Pareto distribution seeAlso infobox The bounded (or truncated) Pareto distribution has three parameters α, L and H. As in the standard Pareto distribution α determines the shape.

After a brief pause the band bounded on stage for the encore, Keith exclaiming “We gave you long enough to understand what KOKO looks like without We Are Scientists- it’s dark!”, epitomising their madcap mantra perfectly.

As the buzzer sounded, Caruso bounded off the court, clenching his fists as he tipped his head back and let out a scream.

It is a bounded concept.