On stretching the interval simplex-permutohedron

A family of polytopes introduced by E.M. Feichtner, A. Postnikov, and B. Sturmfels, which were named nestohedra, consists in each dimension of an interval of polytopes starting with a simplex and ending with a permutohedron. This paper investigates a problem of changing and extending the boundaries of these intervals. An iterative application of Feichtner–Kozlov procedure of forming complexes of nested sets is a solution of this problem. By using a simple algebraic presentation of members of nested sets it is possible to avoid the problem of increasing the complexity of the structure of nested curly braces in elements of the produced simplicial complexes.     

Hypergraph polytopes

We investigate a family of polytopes introduced by E.M. Feichtner, A. Postnikov and B. Sturmfels, which were named nestohedra. The vertices of these polytopes may intuitively be understood as constructions of hypergraphs. Limit cases in this family of polytopes are, on the one end, simplices, and, on the other end, permutohedra. In between, as notable members one finds associahedra and cyclohedra. The polytopes in this family are investigated here both as abstract polytopes and as realized in Euclidean spaces of all finite dimensions. The later realizations are inspired by J.D. Stasheff ?s and S. Shnider?s realizations of associahedra. In these realizations, passing from simplices to permutohedra, via associahedra, cyclohedra and other interesting polytopes, involves truncating vertices, edges and other faces. The results presented here reformulate, systematize and extend previously obtained results, and in particular those concerning polytopes based on constructions of graphs, which were introduced by M. Carr and S.L. Devadoss.     

A generalization of the original Jordan–von Neumann theorem

The problem of representability of quadratic functionals (acting on modules over unital complex ∗-algebras), by sesquilinear forms, is generalized by weakening the homogeneity equation. The corresponding representation theorem can be considered as a generalization of (the original form of) the classical Jordan–von Neumann characterization of complex inner product spaces.     

Strong isochronism of two-dimensional reversible cubic systems

We obtain the maximum order of strong isochronism of reversible cubic systems on the plane.     

Dynamics of a noncommutative monopole

We consider an electrically charged particle simultaneously interacting with a magnetic monopole and a dual monopole in the momentum space. It is a prototype of a three-dimensional system involving noncommuting and/or noncanonical variables but having geometric and also gauge symmetries in both the position and momentum spaces. We discuss the main features of the motions and conservation laws and the analogies to the case of planar motion.     

Maximal chains of isomorphic subgraphs of the Rado graph

The partial order 〈E(R)∪{?},?〉, where E(R) is the set of isomorphic subgraphs of the Rado graph R, is investigated. The order types of maximal chains in this poset are characterized as the order types of compact sets of reals having the minimum non-isolated.