Casimir energy and entropy in the sphere-sphere geometry

Sphere geometry and invariants

A finite abstract simplicial complex G defines two finite simple graphs: the Barycentric refinement G1, connecting two simplices if one is a subset of the other and the connection graph G′, connecting two simplices if they intersect. We prove that the Poincaré-Hopf value i(x) = 1−χ(S(x)), where χ(S(x)) is the Euler characteristics of the unit sphere S(x) of a vertex x in G1, agrees with the Gre...

Random Geometry on the Sphere

We introduce and study a universal model of random geometry in two dimensions. To this end, we start from a discrete graph drawn on the sphere, which is chosen uniformly at random in a certain class of graphs with a given size n, for instance the class of all triangulations of the sphere with n faces. We equip the vertex set of the graph with the usual graph distance rescaled by the factor n−1/...

Public sphere and intellectual discourses in Iran

Distance Geometry on the Sphere

The Distance Geometry Problem asks whether a given weighted graph has a realization in a target Euclidean space R which ensures that the Euclidean distance between two realized vertices incident to a same edge is equal to the given edge weight. In this paper we look at the setting where the target space is the surface of the sphere SK−1. We show that the DGP is almost the same in this setting, ...

Casimir effect for a D-dimensional sphere

The Casimir force on a D-dimensional sphere due to the confinement of a massless scalar field is computed as a function of D, where D is a continuous variable that ranges from −∞ to ∞. The dependence of the force on the dimension is obtained using a simple and straightforward Green’s function technique. We find that the Casimir force vanishes as D → +∞ (D non-even integer) and also vanishes whe...