arXiv subjects
Publications and source records attributed to Maurice Margenstern.
In this paper, we prove that there is a weakly universal cellular automaton on the pentagrid with three states which is rotation invariant and which uses à la Moore neighbourhood. Moreover, at each step of the computation, the set of non quiescent states has always infinitely many cycles.
In this paper, we prove two results. First, there is a family of sequences of embedded quarters of the hyperbolic plane such that any sequence converges to a limit which is an end of the hyperbolic plane. Second, there is no algorithm which would allow us to check whether two given ends are equal or not.
In this paper, we study the contribution of the theory of grossone to the study of infinigons in the hyperbolic plane. We can see that the theory of grossone can help us to obtain much more classification for these objects than in the traditional setting.
The twin primes conjecture is a very old problem. Tacitly it is supposed that the primes it deals with are finite. In the present paper we consider three problems that are not related to finite primes but deal with infinite integers. The main tool of our investigation is a numeral system proposed recently that allows one to express various infinities and infinitesimals easily and by a finite number of symbols. The problems under consideration are the following and for all of them we give affermative answers: (i) do infinite primes exist? (ii) do infinite twin primes exist? (ii) is the set of infinite twin primes infinite? Examples of these three kinds of objects are given.
In this paper, we construct a weakly universal cellular automaton with two states only on the tiling {11,3}. The cellular automaton is rotation invariant and it is a true planar one.
In this paper, we construct a cellular automaton on the heptagrid which is planar, weakly universal and which have three states only. This result improves the best result which was with four states.
In this paper, we construct a cellular automaton on the pentagrid which is planar, weakly universal and which have five states only. This result much improves the best result which was with nine states
In this paper we study the word problem of groups corresponding to tessellations of the hyperbolic plane. In particular using the Fibonacci technology developed by the second author we show that groups corresponding to the pentagrid or the heptagrid are not automatic.
In this paper, we construct a strongly universal cellular automaton on the line with 11 states and the standard neighbourhood. We embed this construction into several tilings of the hyperbolic plane and of the hyperbolic 3D space giving rise to strongly universal cellular automata with 10 states.
In this paper, we try to give the appropriate class of languages to which belong various objects associated with tessellations in the hyperbolic plane.
In this paper, we investigate the possibility to use two tilings of the hyperbolic plane as basic frame for devising a way to input texts in Chinese characters into messages of cellphones, smartphones, ipads and tablets.
In this paper, we construct a family of weakly universal rotation invariant cellular automaton for all grids $\{p,3\}$ of the hyperbolic plane for $p\geq 13$. The scheme is general for $p\geq 17$ and for $13\leq p<17$, we give such a cellular automaton for $p=13$, which is enough. Also, an important property of this family is that the set of cells of the cellular automaton which are subject to changes is actually a planar set. The problem for $p<13$ for a truly planar construction is still open. The best result, for $p=7$, is four states and was obtained by the same author.
We study a Life-like cellular automaton rule $B2/S2345$ where a cell in state `0' takes state `1' if it has exactly two neighbors in state `1' and the cell remains in the state `1' if it has between two and five neighbors in state `1.' This automaton is a discrete analog spatially extended chemical media, combining both properties of sub-excitable and precipitating chemical media. When started from random initial configuration B2/S2345 automaton exhibits chaotic behavior. Configurations with low density of state `1' show emergence of localized propagating patterns and stationary localizations. We construct basic logical gates and elementary arithmetical circuits by simulating logical signals with mobile localizations reaction propagating geometrically restricted by stationary non-destructible localizations. Values of Boolean variables are encoded into two types of patterns --- symmetric (False) and asymmetric (True) patterns --- which compete for the `empty' space when propagate in the channels. Implementations of logical gates and binary adders are illustrated explicitly.
In this paper, we look at how to count the number of elements of a set within the frame of Sergeyev's numeral system. We also look at the connection between the number of elements of a set and the notion of bijection in this new setting. We also show the difference between this new numeral system and the results of the traditional naive set theory.
In this paper, we look at the improvement of our knowledge on a family of tilings of the hyperbolic plane which is brought in by the use of Sergeyev's numeral system based on grossone. It appears that the information we can get by using this new numeral system depends on the way we look at the tilings. The ways are significantly different but they confirm some results which were obtained in the traditional but constructive frame and allow us to obtain an additional precision with respect to this information.
In this paper, we first describe the isothermal constrained loop extension DNA amplification (ICLEDA), which is a new variant of amplification combining the advantages of rolling circle amplification (RCA) and of strand displacement amplification (SDA). Then, we formalize this process in terms of the theory of formal languages and show, on the basis of this formulation, how to manage OR and AND gates. We then explain how to introduce negation, which allows us to prove that, in principle, it is possible to implement the computation of any boolean function on DNA strands using ICLEDA.
This paper introduces a communication system for the tiles of the heptagrid, a tiling of the hyperbolic plane. The method can be extended to other tilings of this plane. The paper focuses on an actual implementation at the programming stage with a short account of two experiments.
In this paper, we look at the possibility to implement the algorithm to construct a discrete line devised by the first author in cellular automata. It turns out that such an implementation is feasible.