Relative concentrations of six isomeric Eu@C 72 -one based on the IPR C 72 cage (i.e., obeying the isolated-pentagon rule, IPR), two cages with a pentagon-pentagon junction (symmetries C 2 and C 2 v ), a cage with one heptagon, a cage with two heptagons, and a cage with two pentagon-pentagon fusions-are DFT computed using the Gibbs energy in a broad temperature interval. It is shown that the tw...

We classify coproducts on matrix algebra in terms of solutions to some modification of pentagon equation. The construction of Baaj and Skan-dalis describing finite dimensional unitary solutions of pentagon equation is extended to the non-unitary case. We establish the relation between Hopf-Galois algebras and solutions to modified pentagon equation.

A new method for evaluating hand preshaping during reaching-to-grasp movement is proposed. The method makes use of all five fingers in estimation of prehension. The investigation was performed on six healthy subjects grasping three different objects at various positions and orientations. The objects were presented to the subjects by means of a robot, which also induced perturbations in both obj...

An almost forgotten gem of Gauss tells us how to compute the area of a pentagon by just going around it and measuring areas of each vertex triangles (i.e. triangles whose vertices are three consecutive vertices of the pentagon). We give several proofs and extensions of this beautiful formula to hexagon etc. and consider special cases of affine–regular polygons. The Gauss pentagon formula is, in...

Let Kn denote the complete undirected graph on n vertices. A Steiner pentagon covering design (SPCD) of order n is a pair (Kn;B), where B is a collection of c(n)= n=5 n−1=2 pentagons from Kn such that any two vertices are joined by a path of length 1 in at least one pentagon of B, and also by a path of length 2 in at least one pentagon of B. The existence of SPCDs is investigated. The main appr...

We perform a detailed analysis of the contact force network in a dense confined packing of pentagonal particles simulated by means of the contact dynamics method. The effect of particle shape is evidenced by comparing the data from pentagon packing and from a packing with identical characteristics, except for the circular shape of the particles. A counterintuitive finding of this work is that, ...

An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328` points in the plane contains an empty pentagon or ` collinear points. This is optimal up to a constant factor since the (` − 1) × (` − 1) grid contains no empty pentagon and no ` collinear points. ...

Naive male Hooded Lister rats (Rattus norvegicus) were required to find a submerged platform in a right-angled corner between a long and a short wall of a pool in the shape of an irregular pentagon. Tests in a rectangular pool revealed a preference for the corners that corresponded with the correct corner in the pentagon. These findings indicate that rats identified the correct corner in the pe...

By employing the theoretical method based on tight-binding, we study electronic band structure of single-wall carbon nanotube (CNT) superlattices, which the system is the made of the junction between the zigzag and armchair carbon nanotubes. Exactly at the place of connection, it is appeared the pentagon–heptagon pairs as topological defect in carbon hexagonal network. The calculations are base...

We prove that for any convex pentagon in the plane there are two disks, among the five disks having a side of the pentagon as diameter, that do not intersect. All our arguments are from classical Euclidean geometry.