Refined Euler\u27s inequalities in plane geometries and spaces

Refined famous Euler\u27s inequalities R ≥ nr of an n-dimensional simplex for n = 2, 3 and 4 as well as of non-Euclidean triangles in terms of symmetric functions of edge lengths of a triangle or a simplex in question are shown. Here R is the circumradius and r the inradius of the simplex. We also provide an application to geometric probabilities of our results and an example from astrophysics to the position of a planet within the space of four stars. We briefly discuss a recursive algorithm to get similar inequalities in higher dimensions

The Distance Matrix for a Simplex

In this paper we consider some geometric questions relating to a higher dimensional analogon of a triangle, called simplex. In particular, weshall be concerned with the distance matrix of its vertices. We shall also majorize the volume of a simplex in terms of the distances between vertices. As consequences, we shall derive some inequalities for determinants and, in particular, an improvement of the well-known Hadamard\u27s inequality. We shall also point to some possible applications to the chemical graph theory

On some primary and secondary structures in combinatorics

A possible upgrade of a curriculum in undergraduate course in combinatorics is presented by giving more bijective proofs in the standard (or primary) combinatorics and by adding some topics on more refined (or secondary) combinatorics, including Dyck and Motzkin paths, Catalan, Narayana and Motzkin numbers and secondary structures coming from biology. Some log-convexity properties and asymptotics of these numbers are also presented

Enumerative aspects of secondary structures

AbstractA secondary structure is a planar, labeled graph on the vertex set {1,…,n} having two kind of edges: the segments [i,i+1], for 1⩽i⩽n−1 and arcs in the upper half-plane connecting some vertices i,j, i⩽j, where j−i>l, for some fixed integer l. Any two arcs must be totally disjoint. We enumerate secondary structures with respect to their size n, rank l and order k (number of arcs), obtaining recursions and, in some cases, explicit formulae in terms of Motzkin, Catalan, and Narayana numbers. We give the asymptotics for the enumerating sequences and prove their log-convexity, log-concavity and unimodality. It is shown how these structures are connected with hypergeometric functions and orthogonal polynomials