Class Of Simplices Research Articles

Can Various Classes of Atomic Configurations (Delaunay Simplices) be Distinguished in Random Dense Packings of Spherical Particles?

Abstract One-dimensional and two-dimensional distributions of the characteristics introduced previously for the forms of Delaunay simplices - tetrahedricity and octahedricity - have been investigated in computer models of a crystal, a liquid and an amorphous solid. It has been established that in the absence of thermal perturbations (in the proper structure of the liquid and in an amorphous substance) there exists a distinguishable class of simplices with five almost equal edges and the sixth being longer. This class of simplices named isopentacmons, in turn includes the types of good tetrahedra and good quartoctahedra (a quarter of octahedron). In disordered systems the fraction of tetrahedra relative to quartoctahedra exceeds substantially that in the FCC crystal.

Two Notes on Metric Geometry

If $\mu ,\smallint d\mu = 1$, is a signed Borel measure on the unit ball in ${E^3}$, it is shown that ${\sup _\mu }\smallint \smallint | {p - q} |d\mu (p)d\mu (q) = 2$ with no extremal measure existing. Also, a class of simplices which generalizes the notion of acute triangle is studied. The results are applied to prove inequalities for determinants of the Cayley-Menger type.

Two notes on metric geometry

If μ , ∫ d μ = 1 \mu ,\smallint d\mu = 1 , is a signed Borel measure on the unit ball in E 3 {E^3} , it is shown that sup μ ∫ ∫ | p − q | d μ ( p ) d μ ( q ) = 2 {\sup _\mu }\smallint \smallint | {p - q} |d\mu (p)d\mu (q) = 2 with no extremal measure existing. Also, a class of simplices which generalizes the notion of acute triangle is studied. The results are applied to prove inequalities for determinants of the Cayley-Menger type.