We present transition metal-embedded (T@Gan) endohedral Ga clusters as a favorable structural motif for superconductivity and develop empirical, molecule-based, electron counting rules that govern the hierarchical architectures that the clusters assume in binary phases. Among the binary T@Gan endohedral cluster systems, Mo8Ga41, Mo6Ga31, Rh2Ga9, and Ir2 Ga9 are all previously known superconductors. The well-known exotic superconductor...

Let T be a T-set, i.e., a finite set of nonnegative integers satisfying 0 ∈ T, and G be a graph. In the paper we study relations between the T-edge spans espT (G) and espd⊙T (G), where d is a positive integer and d ⊙ T = {0 ≤ t ≤ d (max T + 1): d |t ⇒ t/d ∈ T} . We show that espd⊙T (G) = d espT (G) − r, where r, 0 ≤ r ≤ d − 1, is an integer that depends on T and G. Next we focus on the case T = {0} and show that espd⊙{0} (G) =...

We consider deformations of an elastic body having initially a spherical shape. Assumed deformation energy depends on the first and second gradient of displacements. We apply an equatorial line density of dead loads, that are forces per unit line length directed in radial direction and applied along the equator of the sphere. We restrict ourselves our analysis to the case of linearized second strain gradient isotropic elasticity...