scholarly journals Central sets generated by uniformly recurrent words

2013 ◽  
Vol 35 (3) ◽  
pp. 714-736 ◽  
Author(s):  
MICHELANGELO BUCCI ◽  
SVETLANA PUZYNINA ◽  
LUCA Q. ZAMBONI

AbstractA subset $A$ of $ \mathbb{N} $ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $\mathop{({x}_{n} )}\nolimits_{n\in \mathbb{N} } $ of natural numbers. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of IP-sets possessing rich combinatorial properties: each central set contains arbitrarily long arithmetic progressions and solutions to all partition regular systems of homogeneous linear equations. In this paper we investigate central sets in the framework of combinatorics on words. Using various families of uniformly recurrent words, including Sturmian words, the Thue–Morse word and fixed points of weak mixing substitutions, we generate an assortment of central sets which reflect the rich combinatorial structure of the underlying words. The results in this paper rely on interactions between different areas of mathematics, some of which have not previously been directly linked. They include the general theory of combinatorics on words, abstract numeration systems, and the beautiful theory, developed by Hindman, Strauss and others, linking IP-sets and central sets to the algebraic/topological properties of the Stone-Čech compactification of $ \mathbb{N} $.

2011 ◽  
Vol 346 ◽  
pp. 324-331
Author(s):  
Wei Jiang ◽  
Xin Luo ◽  
Wen Chuan Jia ◽  
Yuan Tai Hu ◽  
Hong Ping Hu

A new algorithm is presented to calculate the degrees of freedom (DOFs) of spatial complex mechanisms by using the coefficient matrix of the linear constraint equations. A joint constraint matrix is firstly put forward for each kind of joint to formulate linear constraint equations in terms of spatial fine displacements of joint acting point with respect to joint frame. Two kinds of transformation are then proposed to rewrite all the constraint equations in terms of a set of fine displacements of all bodies and it leads to a set of homogeneous linear equations. The rank of the resulting coefficient matrix stands for the number of effective constraints and therefore the DOFs of the mechanism can be easily figured out. The proposed method can be widely used to solve the problem of DOFs for many spatial complex mechanisms, which may not be correctly solved with traditional approaches. Besides, the proposed method is very easy for implementation.


1913 ◽  
Vol 12 ◽  
pp. 137-138
Author(s):  
John Dougall

A system of n non-homogeneous linear equations in n variables has one and only one solution if the homogeneous system obtained from the given system by putting all the constant terms equal to zero has no solution except the null solution.This may be proved independently by similar reasoning to that given for Theorem I., or it may be deduced from that theorem. We follow the latter method.


2002 ◽  
Vol 12 (01n02) ◽  
pp. 371-385 ◽  
Author(s):  
JEAN BERSTEL

Sturmian words are infinite words over a two-letter alphabet that admit a great number of equivalent definitions. Most of them have been given in the past ten years. Among several extensions of Sturmian words to larger alphabets, the Arnoux–Rauzy words appear to share many of the properties of Sturmian words. In this survey, combinatorial properties of these two families are considered and compared.


2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
Véronique Bruyère ◽  
Michel Rigo

Held at the Institute of Mathematics of the University of Liège, Liège, September 8―11, 2004 International audience This special issue of Discrete Mathematics & Theoretical Computer Science is dedicated to the tenth "Journées montoises d'informatique théorique" conference (Mons theoretical computer science days) which was held, for the first time, at the Institute of Mathematics of the University of Liège, Belgium, From 8th to 11th September 2004. Previous editions of this conference took place in Mons 1990, 1992, 1994, 1998, in Rouen 1991, in Bordeaux 1993, Marseille 1995, Marne-La-Vallée 2000 and Montpellier 2002.<p> This tenth edition can be considered as a widely international one. We were lucky to have almost 85 participants from fourteen different countries: Austria, Belgium, Burkina Faso, Canada, Czech republic, Finland, France, Germany, Israel, Italy, Japan, Norway, Poland and Portugal. The main proportion of researchers participating to this event was coming from France and Italy where a long tradition of combinatorics on words is well established. During four days, 42 contributed talks and 7 invited talks were given, the main topics being combinatorics on words, numeration systems, automata and formal languages theory, coding theory, verification, bio-informatics, number theory, grammars, text algorithms, symbolic dynamics and tilings. The invited speakers were: J. Cassaigne (CNRS, Luminy-Marseille), D. Caucal (IRISIA-CNRS, Rennes), C. Frougny (LIAFA, Université Paris 8), T. Helleseth (University of Bergen), S. Langerman (FNRS, Université Libre de Bruxelles), F. Neven (Limburgs Universitair Centrum, Diepenbeek), M.-F. Sagot (Inria Rhône-Alpes, Université Lyon I).<p> We would like to thanks all the participants, the invited speakers and the anonymous referees who made possible this event and special issue. Each paper has been refereed using high scientific standard by two independent referees. Readers of this special issue may wonder why it took so long to obtain it. We have encountered some problems with the formerly chosen journal and for the benefit of the contributors to this issue, we have chosen Discrete Mathematics & Theoretical Computer Science to publish their work.


2014 ◽  
Vol 23 (6) ◽  
pp. 1114-1147 ◽  
Author(s):  
PAWEŁ HITCZENKO ◽  
SVANTE JANSON

This paper concerns a relatively new combinatorial structure called staircase tableaux. They were introduced in the context of the asymmetric exclusion process and Askey–Wilson polynomials; however, their purely combinatorial properties have gained considerable interest in the past few years.In this paper we further study combinatorial properties of staircase tableaux. We consider a general model of random staircase tableaux in which symbols (Greek letters) that appear in staircase tableaux may have arbitrary positive weights. (We consider only the case with the parametersu=q= 1.) Under this general model we derive a number of results. Some of our results concern the limiting laws for the number of appearances of symbols in a random staircase tableaux. They generalize and subsume earlier results that were obtained for specific values of the weights.One advantage of our generality is that we may let the weights approach extreme values of zero or infinity, which covers further special cases appearing earlier in the literature. Furthermore, our generality allows us to analyse the structure of random staircase tableaux, and we obtain several results in this direction.One of the tools we use is the generating functions of the parameters of interest. This leads us to a two-parameter family of polynomials, generalizing the classical Eulerian polynomials.We also briefly discuss the relation of staircase tableaux to the asymmetric exclusion process, to other recently introduced types of tableaux, and to an urn model studied by a number of researchers, including Philippe Flajolet.


Sign in / Sign up

Export Citation Format

Share Document