Search results
Results From The WOW.Com Content Network
For instance, the Szemerédi–Trotter theorem, an upper bound on the number of incidences that are possible between given numbers of points and lines in the plane, follows by constructing a graph whose vertices are the points and whose edges are the segments of lines between incident points. If there were more incidences than the Szemerédi ...
This is a hard computational problem; if we are not able to solve it exactly, we can instead try to find lower and upper bounds on the size of the minimum feedback vertex set. One approach to find lower bounds is to find a collection of vertex-disjoint cycles in a graph. For example, consider the graph in Figure 1.
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above.
The Edwards-Erdős gives a lower bound on b(G) for every connected signed graph G. Bound (a) was improved for special classes of graphs: triangle-free graphs, graphs of given maximum degree, H-free graphs, etc., see e.g. [5] [6] [7] Poljak and Turzik [8] extended the Edwards-Erdős bound to weighted Max-Cut:
Thus, the infimum or meet of a collection of subsets is the greatest lower bound while the supremum or join is the least upper bound. In this context, the inner limit, lim inf X n, is the largest meeting of tails of the sequence, and the outer limit, lim sup X n, is the smallest joining of tails of the sequence. The following makes this precise.
Get AOL Mail for FREE! Manage your email like never before with travel, photo & document views. Personalize your inbox with themes & tabs. You've Got Mail!
If the midpoint is smaller, one can set it as the lower bound of the next interval +, and if the midpoint is larger, one can set it as the upper bound of the next interval. This guarantees that x ∈ I n + 1 {\displaystyle {\sqrt {x}}\in I_{n+1}} .