Factors of disconnected graphs and polynomials with nonnegative integer coefficients

We investigate the uniqueness of factorisation of possibly disconnected finite graphs with respect to the Cartesian, the strong and the direct product. It is proved that if a graph has $n$ connected components, where $n$ is prime, or $n=1,4,8,9$, and satisfies some additional conditions, it factors uniquely under the given products. If, on the contrary, $n=6$ or 10, all cases of nonunique factorisation are described precisely.Comment: 14 page

mathematische

The well known binary, decimal,..., number systems in the integers admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when we choose a digit set that does not contain 0. We prove that such digit sets exist and we provide infinitely many examples for every base b with |b | ≥ 4, and for b = −2. For the special case b = −2, we give a full characterisation of all valid digit sets. Key words: Radix systems 1 Introduction an

The Casas-Alvero conjecture for infinitely many degrees

Over a field of characteristic zero, it is clear that a polynomial of the form (X-a)^d has a non-trivial common factor with each of its d-1 first derivatives. The converse has been conjectured by Casas-Alvero. Up to now there have only been some computational verifications for small degrees d. In this paper the conjecture is proved in the case where the degree of the polynomial is a power of a prime number, or twice such a power. Moreover, for each positive characteristic p, we give an example of a polynomial of degree d which is not a dth power but which has a common factor with each of its first d-1 derivatives. This shows that the assumption of characteristic zero is essential for the converse statement to hold.Comment: 7 pages; v2: corrected some typos and references, and added section on computational aspect

The Cartesian product of graphs with loops

We extend the definition of the Cartesian product to graphs with loops and show that the Sabidussi-Vizing unique factorization theorem for connected finite simple graphs still holds in this context for all connected finite graphs with at least one unlooped vertex. We also prove that this factorization can be computed in O(m) time, where m is the number of edges of the given graph.Comment: 8 pages, 1 figur

Unusual Hybrid Closure of Ventricular Septal Defects

A planned combined perventricular and “open heart” surgical closure of multiple ventricular septal defects had to be modified intraoperatively due to a technical fault disabling echocardiographic guidance. Through an atriotomy, device closure of a muscular defect and patch closure of a perimembranous ventricular septal defect were performed. In unusual situations, collaboration of the surgical and interventional team is crucial

Deterministic equation solving over finite fields

It is shown how to solve diagonal forms in many variables over finite fields by means of a deterministic efficient algorithm. Applications to norm equations, quadratic forms, and elliptic curves are given.Thomas Stieltjes Institute for MathematicsUBL - phd migration 201