Raziskovalni matematični seminar - Arhiv

Na seminarju bodo predstavljeni nekateri novejši rezultati v zvezi z ohranjevalci na tenzorskem produktu kompleksnih matrik.

The concepts of adjacency and distance are two of the most basic terms in graph theory. A natural generalization leads to the notion of distance degree which tells us the number of vertices at some distance from the chosen vertex. When studying distance degrees we can limit ourselves to the so called distance degree regular graphs, where all the distance degrees for all the vertices are the same, and self-median graphs, where the sum of distances from a chosen vertex to all the other vertices is independent of the selected vertex. It was shown that the former correspond to strongly distance-balanced graphs while the latter correspond to distance-balanced graphs.

The talk will focus on the properties and known families of the mentioned types of graphs, the complexity of obtaining them and their use in practical applications. It will present an overview of the known results and pose some still unanswered questions.

Roughly speaking an end of a graph G(V,E) is an equivalence class of one-way infinite paths such that any two different equivalence classes can be separated from each other by removing a finite subset S of V(G). In this talk we consider infinite subsets of V(G) with certain growth rates which can be separated from each other by a connected subgraph of G with smaller growth rate. Following the ideas presented in a paper of Stallings we not only generalize the concept of ends, but also some of the well-known results about ends of graphs.

In this talk, we will consider the Hermite interpolation with PH spline curve. In particular, at each spline segment the C^1 data at two end points are interpolated by PH cubic biarc and the two arcs are joined together at some unknown common point sharing the same unknown tangent vector. The obtained biarc is expressed in a closed form with three free parameters. The problem of specifying two free angular parameters has been introduced in the literature for characterizing Hermite interpolation based on PH quintics and the strategy to determine these two parameters easily and suitably is identified by the acronym CC. We will export this strategy to our field. More precisely, we will present a method which requires only the evaluation of two analytic explicit expressions and which still guarantees that, when the PH cubic interpolant to Hermite data exists, it is reproduced by our algorithm. The asymptotic behavior of solution will be studied and the shape-preserving properties, particularly the torsion, will be presented. The resulting algorithm is implemented in Mathematica and somenumerical experiments that confirm the theoretical results will be shown.

These results were obtained during the visit of the speaker at the University of Florence last April, and are joint work with Alessandra Sestini (Università degli Studi di Firenze), Carla Manni (Università di Roma “Tor Vergata”) and Maria Lucia Sampoli (Università degli Studi di Siena).