In many cases, only lower and upper bounds are known with a large gap between them.

In this way, both the lower and upper bounds are programmable.

The first upper bounds were based on the 'human' algorithms.

This is an immediate consequence of the upper bounds described below.

This can now be chosen in some arbitrary manner as a point between the lower and upper bounds.

Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative real numbers.

Still, the size of the hypergraph yields first upper bounds.

There is now a clear strategy for finding lower and upper bounds for a 'good' exposure.

This can be seen as a justification for using upper bounds to describe harmonic complexity.

Finding a supremum means to single out one distinguished least element from the set of upper bounds.

It also has least upper bounds of chains.

Like upper bounds, greatest elements may fail to exist.

The problem remains open, but over a sequence of papers researchers have tightened the gap between the known lower and upper bounds.

It may help to point out that the increasing sequences produced by sequential programs all have least upper bounds.

The lower and upper bounds for the listings.

This leads to the definition of upper bounds.

For such a comparison, one may want upper bounds or lower bounds of the accuracy.

We will now solve problem P1 of Section 8.2 again, but with the addition of upper bounds.

"New upper bounds for kissing numbers from semidefinite programming".

The expectation for a function of with respect to the credal set can be characterised only by its lower and upper bounds.

Finally, a set may have many minimal upper bounds without having a least upper bound.

The twentieth century witnessed a great expansion of the upper bounds of the human life span.

The upper bounds on proton decay rate can also be satisfied because the squarks are very heavy as well.

The functions g and h are said to be lower and upper bounds (respectively) of f.

Now define B as a positive constant that upper bounds the first term on the right-hand-side of the above inequality.