Entropy rate of higher-dimensional cellular automata

We introduce the entropy rate of multidimensional cellular automata. This number is invariant under shift-commuting isomorphisms; as opposed to the entropy of such CA, it is always finite. The invariance property and the finiteness of the entropy rate result from basic results about the entropy of partitions of multidimensional cellular automata. We prove several results that show that entropy rate of 2-dimensional automata preserve similar properties of the entropy of one dimensional cellular automata. In particular we establish an inequality which involves the entropy rate, the radius of the cellular automaton and the entropy of the d-dimensional shift. We also compute the entropy rate of permutative bi-dimensional cellular automata and show that the finite value of the entropy rate (like the standard entropy of for one-dimensional CA) depends on the number of permutative sites. Finally we define the topological entropy rate and prove that it is an invariant for topological shift-commuting conjugacy and establish some relations between topological and measure-theoretic entropy rates

Occam's hammer: a link between randomized learning and multiple testing FDR control

We establish a generic theoretical tool to construct probabilistic bounds for algorithms where the output is a subset of objects from an initial pool of candidates (or more generally, a probability distribution on said pool). This general device, dubbed "Occam's hammer'', acts as a meta layer when a probabilistic bound is already known on the objects of the pool taken individually, and aims at controlling the proportion of the objects in the set output not satisfying their individual bound. In this regard, it can be seen as a non-trivial generalization of the "union bound with a prior'' ("Occam's razor''), a familiar tool in learning theory. We give applications of this principle to randomized classifiers (providing an interesting alternative approach to PAC-Bayes bounds) and multiple testing (where it allows to retrieve exactly and extend the so-called Benjamini-Yekutieli testing procedure).Comment: 13 pages -- conference communication type forma

Constant-length substitutions and countable scrambled sets

In this paper we provide examples of topological dynamical systems having either finite or countable scrambled sets. In particular we study conditions for the existence of Li-Yorke, asymptotic and distal pairs in constant--length substitution dynamical systems. Starting from a circle rotation we also construct a dynamical system having Li--Yorke pairs, none of which is recurrent

Les mots et les images en mathématiques et ailleurs

Among the obstacles that hinder communication between mathematics and experimental sciences, some are linguistic by nature. Indeed, because of its very peculiar character and of the fact that it relies solely on deduction, the language of mathematics cannot grasp the notion of experiment; it also induces a significant number of mathematicians to deny metaphore and metonymy any place in elaborate scientific language. While this denial is rather legitimate in their own field, it is certainly not so in physics, chemistry, biology and elsewhere. Hence frequent misunderstandings between mathematicians and other researchers. However, misunderstanding is far from being a general rule, as shown by the existing interactions between sciences. Here we introduce the notion of {\it intermediate complex}, in order to account for situations in which misunderstandings and understandings both occur. An intermediate complex is a word or a phrase from common language, together with its various meanings, that has been chosen in one or several fields to express a scientific question that has not yet been formulated properly. An intermediate complex can be interpreted in different ways, some scientific and others not. It is a communication medium which allows different sciences to talk and work together.Parmi les obstacles qui empêchent ou ralentissent la communication entre les mathématiciens et les expérimentalistes, nous relevons que certains sont de nature sémantique. En effet, de par son extrême spécificité et son caractère déductif, le langage mathématique est incapable d'appréhender la notion d'expé\-rience ; il pousse un certain nombre de mathématiciens à refuser à la métaphore et à la métonymie toute place dans un discours scientifique achevé, ce qui est (relativement) légitime dans le leur, mais certainement pas dans ceux de la physique, de la chimie, de la biologie et des autres domaines scientifiques. D'où des malentendus fréquents entre mathématiciens et spécialistes d'autres disciplines. Ces malentendus ne sont toutefois pas une norme, puisque la communication se fait malgré tout. La notion de {\it complexe intermédiaire}, introduite ici, vise à rendre compte à la fois des malentendus et du dialogue existants. Il s'agit d'une expression tirée du langage courant, porteuse de sens variés et reprise par une ou plusieurs disciplines pour symboliser une question scientifique encore mal formulée. Un complexe intermédiaire se prête à des interprétations divergentes, certaines scientifiques et d'autres non; c'est aussi un moyen de communication approximative qui permet à diverses sciences de dialoguer dans une perspective de travail commun

Language complexity of rotations and Sturmian sequences

AbstractGiven a rotation of the circle, we study the complexity of formal languages that are generated by the itineraries of interval covers. These languages are regular iff the rotation is rational. In the case of irrational rotations, our study reduces to that of the language complexity of the corresponding Sturmian sequences. We show that for a large class of irrationals, including e, all quadratic numbers and more generally all Hurwitz numbers, the corresponding languages can be recognized by a nondeterministic Turing machine in linear time (in other words, belongs to NLIN)

Pratiques langagières et processus dialogiques d’identification sur les réseaux socionumériques : le cas de la langue bretonne

Internet and social digital networks (SDN) are, for the Breton language, a recent setting for social practices inwhich forms of recontextualization of a minority language in a post‐diglossic situation occur. The purpose ofthis thesis is to describe the transformations using a dialogical model of identity development. Such model focuses on a three‐dimensional analysis that encompassing the institutionalization forms of a language in a society, social representations of a language and social practices resulting in expressions of belonging. This dialogical process model, whose design is grounded in scholar works in the social history field, is first instantiated from a sociolinguistic perspective to describe SDN glottopolitical intervention characteristics in the post‐diglossia context. This conceptual model is then applied to in analysis of extended forms of sociability enabled, facilitated and structured by SDN both in a media and public communication context. Finally, the model allows the juxtaposition of the sociopolitical analysis of the Breton claim and the sociopolitical theory of public space on the three dimensions on which SDN have an effect: construction of social problems such as institutionalization, symbolic construction of territorial identity and citizenship as a social practice and way ofbelonging. Liberal glottopolitical interventions developed around SDN create forms institution of language based on both the market force and the autonomy capacity of social actors to build a regional territoryInternet et les réseaux socionumériques (RSN) constituent, pour la langue bretonne, un contexte récent dans les pratiques sociales à partir duquel peuvent s’observer des formes de recontextualisation d’une langue minorée en situation de post‐diglossie. Cette thèse propose d’en décrire les évolutions à l’aide d’un modèle dialogique d’élaboration d’identité qui offre trois pôles d’analyse : les formes de l’institutionnalisation de la langue dans la société, les représentations sociales de la langue et les pratiques sociales constituant des expressions d’appartenance. Ce modèle dialogique de processus, dont la conception est étayée par des travaux d’histoire sociale, est d’abord instancié au plan sociolinguistique, afin de montrer les conditions de l’intervention glottopolitique des RSN dans le contexte post‐diglossique. Le modèle conceptuel est ensuite exploité dans l’analyse étendue des formes de sociabilité que les RSN organisent, facilitent et structurent ycompris dans le champ des médias et de la communication publique. Enfin, le modèle permet de juxtaposer l’analyse sociopolitique de la revendication bretonne à la théorie sociopolitique de l’espace public sur les trois pôles d’analyse de la place des RSN : la construction de problèmes publics comme institutionnalisation, la construction symbolique de l’identité territoriale et la citoyenneté comme pratique sociale et forme d’appartenance. Les interventions glottopolitiques libérales développées autour des RSN concourent à des formes d’institution de la langue fondées à la fois sur la capacité d’autonomie des acteurs sociaux à construire l’espace régional mais aussi sur les conditions du march