Upper and Lower Bounds


Also found in: Dictionary, Thesaurus, Wikipedia.

Bounds, Upper and Lower

 

(in mathematics), important characteristics of sets on a number line.

The upper bound of a set E of real numbers is the smallest number of all numbers A, which possess the property such that for any x of E the inequality x ≤ A is satisfied. In other words, the upper bound of set E is that number a such that for any x of E the inequality x ≤ A is satisfied and that for any a′ < a a number x0 of E will be found for which x0 < a′. In this definition it is assumed that set E is not empty. For the existence of an upper bound it is necessary and sufficient that set E be bounded from above—that is, that numbers A exist such that x ≤ A for any x of E. This proposition is one of the forms of the principle of continuity of a number line (the so-called Weierstrass principle of continuity). If among the numbers of set E there is one greater than any of the others, then it is the upper bound of E. If, however, there is no such greatest number among the numbers of E, this set may still have an upper bound. For example, the upper bound of all negative numbers is equal to zero. The set of all positive numbers is not bounded from above and therefore has no upper bound; it is sometimes said that its upper bound is equal to + ∞.

The lower bound of set E is defined analogously to the upper bound as the greatest of the numbers B, which possess the property such that for any x of E the inequality x ≥ B is satisfied. The upper bound of set E is designated sup E (from the Latin supremum—“the highest”); the lower bound is designated inf E (from the Latin infimum—“the lowest”). The importance of the concepts of upper and lower bounds was explained by the German mathematician K. Weier-strass; they are basic for the rigorous exposition of the fundamentals of mathematical analysis. Analogous to the concept of upper and lower bounds for sets of numbers has been the introduction of the concepts of upper and lower bounds for any partially ordered sets.

REFERENCE

Fikhtengol’ts, G. M. Kurs differentsial’nogo i integral’ nogo ischisleniia, 6th ed., vol. 1. Moscow, 1966.

S. B. STECHKIN

References in periodicals archive ?
Sharpness of the unconstrained upper and lower bounds requires that the portfolio sum is constant on the upper and lower parts of the distribution, respectively.
To make this index able to evaluate the lower and upper bounds of the interval cluster we compute the following equations to the upper and lower bounds, respectively:
The upper and lower bounds of the corresponding radiated array pattern are deduced according to the rules of the interval arithmetic.
In view of majorization [22]-[26] and Minkowski theory [27][28], the analytical upper and lower bounds on the ergodic capacity of D-MIMO systems are derived.
With the estimated lower and upper bounds of the delamination area created during the mode I fracture test in the DCB specimens, the average AE signal amplitude and the respective upper and lower bound for the average crack area created per average AE signal (with respect to signal amplitude) can be estimated.
A state estimation approach is proposed by which the initial state [x.sub.0] is estimated by solving its tightest upper and lower bounds. LP optimization techniques are used to obtain the bounds and the cases for upper and lower bounds which are dealt with separately.
The upper and lower bounds of the correct classification, considering all M targets, are given by
Since the amount of distributional information for a ratio varied among the three experimental groups, the upper and lower bounds set by individuals in the median group should differ from those set by individuals in the quartile group, which in turn, should vary from those set by individuals in the decile group.
Table 3 shows nine cases with assumed parameter values for Equation 6 and the resulting upper and lower bounds. For Case 1, the parameters are chosen to give upper and lower limits equivalent to the 1993 second-quarter averages for the upper and lower limits for small banks, shown in Figure 8.
The result that, with the exception of the June 1983, June 1984, December 1984, March 1986, June 1986, September 1987, December 1987, March 1988, June 1988, December 1989, March 1991, June 1994 and September 1994, no one contract exhibits both upper and lower bound violations is consistent with this implication.
To identify the strengths and weaknesses of two regression techniques in estimating the slopes of the upper and lower bounds of scatter diagrams, we selected six independent data sets from a larger body of data illustrating the relationship between the body sizes of predators and prey.
We show that under mild restrictions the constant factors implicit in our upper and lower bounds are often equal.
Full browser ?