In terms of proving lower bounds, our knowledge is very limited.

Separate values for the upper and lower bounds may also be used.

These are lower bounds, the typical frequency is expected to be much higher.

A set S is bounded if it has both upper and lower bounds.

There has been some progress on lower bounds, as reported by .

Note that the masses of the planets are (for the most part) lower bounds only.

This hypothesis is used to show very many lower bounds on algorithms in computer science.

The Levenshtein distance has several simple upper and lower bounds.

Moreover, better lower bounds are known for several classes of graph properties.

To find lower bounds for all cases involved solving around 100,000 linear programming problems.

When upper and lower bounds are imposed on the same variable it is important to note that means.

Many of these results also give upper and/or lower bounds on their spanning ratios.

By far, the most common technique for computing these lower bounds has been using communication complexity.

A recursive formula was given by and have provided lower bounds for the growth of this class.

Natural systems show us only lower bounds to the possible, in cell repair as in everything else.

For others investigations have determined upper and lower bounds, but not exact solutions.

And matching lower bounds are known for some specialized models of computation.

It is therefore important to be able to perform discrete sensitivity analysis and to handle lower bounds.

They give strong evidence that the techniques used in these proofs cannot be extended to show stronger lower bounds.

However, one can construct an interval provided by upper and lower bounds on the result of an elementary operation.

Thus as the upper and lower bounds are the same, we have a tight bound for when .

Lower bounds again are defined by inverting the order.

This is relevant because the trace distance provides upper and lower bounds on the fidelity.

The converse is true for lower bounds, which are specified using :

From the mass assignments, the upper and lower bounds of a probability interval can be defined.