Darko Veljan

The distance matrix in chemistry

The graph-theoretical (topological) distance matrix and the geometric (topographic) distance matrix and their invariants (polynomials, spectra, determinants and Wiener numbers) are presented. Methods of computing these quantities are discussed. The uses of the distance matrix in both forms and the related invariants in chemistry are surveyed. Special attention is paid to the 2D and 3D Wiener numbers, defined respectively as one half of the sum of entries in the topological distance matrix and the topographic distance matrix. These numbers appear to be very valuable molecular descriptors in the structure property correlations.

Enumerative aspects of secondary structures

Abstract A 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.

Logarithmic behavior of some combinatorial sequences

Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove logarithmic behavior of some combinatorially relevant sequences, such as Motzkin and Schroder numbers, sequences of values of some classic orthogonal polynomials and many others. The calculus method extends also to numbers indexed by two or more parameters.

The sine theorem and inequalities for volumes of simplices and determinants

Abstract We present two proofs of the sine theorem for a simplex and derive a well-known majorization of its volume in terms of its faces. We apply this to obtain some other volume and determinant majorizations.

Thébault's theorem

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Incenter Distances in a Triangulation: 11116

Solution by Philipp Lampe, Bonn, Germany. Let T be a triangular cell in A. The center M of the circle O is the circumcenter of T. According to Eulers formula, the distance d from M to the incenter of T is given by d2 = R2 ? 2Rr, where R denotes the circumradius of T (which is equal to the radius of O) and r denotes the inradius of T. Let x, y, and z be the signed distances from M to the sides of T, where the sign of the distance is positive if and only if M lies on the same half-plane as the triangle T. Carnots theorem states that R + r=x + y + z. It follows that

Inequalities for Volumes of Simplices in Terms of Their Faces

By generalizing some well-known results, we first obtain an inequality involving the volume and product of s-contents of s-faces of an n-simplex. Using this we generalize two inequalities maximizing the volume of one or two simplices in terms of their edge lengths.