In 1958 E. Lehmer found an explicit description of those primes p for which a given prime q is a cubic residue. Her result essentially states that if one writes 4p = L + 27M, then q is a cubic residue if and only if M L ≡ t 2 − 1 t3 − 9t mod q for some integer t. In this paper we demonstrate that a similar result may be obtained for cubic nonresidues, yielding a cubic character for fixed p that...

A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set o...

In 1958 E. Lehmer found an explicit description of those primes p for which a given prime q is a cubic residue. Her result states that if one writes 4p = L + 27M, then q is a cubic residue if and only if M/L ≡ (t − 1)/(t − 9t) mod q for some integer t. Recently, Z. Sun has stated a similar result for cubic nonresidues which follows from several corollaries appearing in an earlier paper of his. ...

The phase diagram for a system of model anisotropic particles with six attractive patches in an octahedral arrangement has been computed. This model for a relatively narrow value of the patch width where the lowest-energy configuration of the system is a simple cubic crystal. At this value of the patch width, there is no stable vapor-liquid phase separation, and there are three other crystallin...