arXiv subjects

Hassan Douzi

Publications and source records attributed to Hassan Douzi.

3 recordsLinked to original sources

arXiv2022

Convergence of some perturbed sequences of rational powers and application to syracuse problem

Sequences of rational powers \left( ξ\left( \frac{p}{q} \right)^{n} \right)_{n\ge 0}, especially in the case \frac{p}{q}=\frac{3}{2}, have a connection with many important combinatorics and number theory problems as for example Syracuse, Z-number and waring problems. Conjectures from such problems are known to be intractable and only few partial results exist until now. In this paper, we study a family of perturbed sequences of rational powers called 'Branch sequences' of the form \left( S_{n}=\left( ξ+Σ_{n} \right)\left( \frac{p^{n}}{q^{n+e_{n}}} \right) \right)_{n\ge 0}. Under the assumption that such sequences are deterministic and they have controlled positive perturbations, we establish the convergence result: min_{n\ge 0}(S_{n})\le q^{2}. As an application, we show that Syracuse sequences are 'Branch sequences' with all the required conditions for convergence and therefore this confirms the Collatz conjecture. Keywords: Sequences of rational powers, Syracuse conjecture, Collatz problem, 3x+1 problem.

math.GM↗

arXiv2015

A Contribution in the Rotor-router model

In this paper I propose to approach the Rotor-router problem by considering it as one example of a big family of many other similar models. The study of some specific samples of them may contribute, in my opinion, at a more understanding of the J.Propp model. In fact we can easily generalize the Rotor-router to many other models, with different regular geometric shapes, by slightly changing the ants' displacements rules. The two directions Rotor-Router (RR2) is particularly interesting because in it's Abelian version it is generated by an easy mathematical explicit scheme. Moreover we can also generate the J.Propp Rotor-Router using a similar iterative explicit algorithm. The study of RR2 establishes also a relationship between the J.Propp model and a family of symmetrical models which generates the same round forms.

math.DS↗

arXiv2015

Combinatorics of a fractal tiling family

In this paper, we propose to enumerate all different configurations belonging to a specific class of fractals: A binary initial tile is selected and a finite recursive tiling process is engaged to produce auto-similar binary patterns. For each initial tile choice the number of possible configurations is finite. This combinatorial problem recalls the famous Escher tiling problem [2]. By using the Burnside lemma we show that there are exactly 232 really different fractals when the initial tile is a particular 2x2 matrix. Partial results are also presented in the 3x3 case when the initial tile presents some symmetry properties.

math.CO↗