Search results
Results From The WOW.Com Content Network
The upper bound theorem states that if is a simplicial sphere of dimension with vertices, then ((,)) =,, …, The difference between for the dimension of the simplicial sphere, and for the dimension of the cyclic polytope, comes from the fact that the surface of a -dimensional polytope (such as the cyclic polytope) is a ()-dimensional subdivision of a sphere.
Most bounds are greater or equal to one, and are thus not sharp for a polynomial which have only roots of absolute values lower than one. However, such polynomials are very rare, as shown below. Any upper bound on the absolute values of roots provides a corresponding lower bound.
13934 and other numbers x such that x ≥ 13934 would be an upper bound for S. 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 ...
The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix.. This definition extends to the case of infinite graphs with bounded degrees of vertices (i.e. there exists some real number C such that the degree of every vertex of the graph is smaller than C).
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
More generally, one may define upper bound and least upper bound for any subset of a partially ordered set X, with “real number” replaced by “element of X ”. In this case, we say that X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound in X.
An upper bound for R(r, s) can be extracted from the proof of the theorem, and other arguments give lower bounds. (The first exponential lower bound was obtained by Paul Erdős using the probabilistic method.) However, there is a vast gap between the tightest lower bounds and the tightest upper bounds.
The upper bound, proved in Heawood's original short paper, is based on a greedy coloring algorithm. By manipulating the Euler characteristic, one can show that every graph embedded in the given surface must have at least one vertex of degree less than the given bound. If one removes this vertex, and colors the rest of the graph, the small ...