Breadth-first serialisation of trees and rational languages

We present here the notion of breadth-first signature and its relationship with numeration system theory. It is the serialisation into an infinite word of an ordered infinite tree of finite degree. We study which class of languages corresponds to which class of words and,more specifically, using a known construction from numeration system theory, we prove that the signature of rational languages are substitutive sequences.Comment: 15 page

Human Dynamics: The Correspondence Patterns of Darwin and Einstein

While living in different historical era, Charles Darwin (1809-1882) and Albert Einstein (1879-1955) were both prolific correspondents: Darwin sent (received) at least 7,591 (6,530) letters during his lifetime while Einstein sent (received) over 14,500 (16,200). Before email scientists were part of an extensive university of letters, the main venue for exchanging new ideas and results. But were the communication patterns of the pre-email times any different from the current era of instant access? Here we show that while the means have changed, the communication dynamics has not: Darwin's and Einstein's pattern of correspondence and today's electronic exchanges follow the same scaling laws. Their communication belongs, however, to a different universality class from email communication, providing evidence for a new class of phenomena capturing human dynamics.Comment: Supplementary Information available at http://www.nd.edu/~network

Waiting time dynamics of priority-queue networks

We study the dynamics of priority-queue networks, generalizations of the binary interacting priority queue model introduced by Oliveira and Vazquez [Physica A {\bf 388}, 187 (2009)]. We found that the original AND-type protocol for interacting tasks is not scalable for the queue networks with loops because the dynamics becomes frozen due to the priority conflicts. We then consider a scalable interaction protocol, an OR-type one, and examine the effects of the network topology and the number of queues on the waiting time distributions of the priority-queue networks, finding that they exhibit power-law tails in all cases considered, yet with model-dependent power-law exponents. We also show that the synchronicity in task executions, giving rise to priority conflicts in the priority-queue networks, is a relevant factor in the queue dynamics that can change the power-law exponent of the waiting time distribution.Comment: 5 pages, 3 figures, minor changes, final published versio

The Critical Exponent is Computable for Automatic Sequences

The critical exponent of an infinite word is defined to be the supremum of the exponent of each of its factors. For k-automatic sequences, we show that this critical exponent is always either a rational number or infinite, and its value is computable. Our results also apply to variants of the critical exponent, such as the initial critical exponent of Berthe, Holton, and Zamboni and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes or recovers previous results of Krieger and others, and is applicable to other situations; e.g., the computation of the optimal recurrence constant for a linearly recurrent k-automatic sequence.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Ultimate periodicity of b-recognisable sets : a quasilinear procedure

It is decidable if a set of numbers, whose representation in a base b is a regular language, is ultimately periodic. This was established by Honkala in 1986. We give here a structural description of minimal automata that accept an ultimately periodic set of numbers. We then show that it can verified in linear time if a given minimal automaton meets this description. This thus yields a O(n log(n)) procedure for deciding whether a general deterministic automaton accepts an ultimately periodic set of numbers.Comment: presented at DLT 201

Optimization in task--completion networks

We discuss the collective behavior of a network of individuals that receive, process and forward to each other tasks. Given costs they store those tasks in buffers, choosing optimally the frequency at which to check and process the buffer. The individual optimizing strategy of each node determines the aggregate behavior of the network. We find that, under general assumptions, the whole system exhibits coexistence of equilibria and hysteresis.Comment: 18 pages, 3 figures, submitted to JSTA

‘Because it’s our culture!’ (Re)negotiating the meaning of lobola in Southern African secondary schools

Payment of bridewealth or lobola is a significant element of marriage among the Basotho of Lesotho and the Shona of Zimbabwe. However, the functions and meanings attached to the practice are constantly changing. In order to gauge the interpretations attached to lobola by young people today, this paper analyses a series of focus group discussions conducted among senior students at two rural secondary schools. It compares the interpretations attached by the students to the practice of lobola with academic interpretations (both historical and contemporary). Among young people the meanings and functions of lobola are hotly contested, but differ markedly from those set out in the academic literature. While many students see lobola as a valued part of ‘African culture’, most also view it as a financial transaction which necessarily disadvantages women. The paper then seeks to explain the young people’s interpretations by reference to discourses of ‘equal rights’ and ‘culture’ prevalent in secondary schools. Young people make use of these discourses in (re)negotiating the meaning of lobola, but the limitations of the discourses restrict the interpretations of lobola available to them

On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases

This article studies the expressive power of finite automata recognizing sets of real numbers encoded in positional notation. We consider Muller automata as well as the restricted class of weak deterministic automata, used as symbolic set representations in actual applications. In previous work, it has been established that the sets of numbers that are recognizable by weak deterministic automata in two bases that do not share the same set of prime factors are exactly those that are definable in the first order additive theory of real and integer numbers. This result extends Cobham's theorem, which characterizes the sets of integer numbers that are recognizable by finite automata in multiple bases. In this article, we first generalize this result to multiplicatively independent bases, which brings it closer to the original statement of Cobham's theorem. Then, we study the sets of reals recognizable by Muller automata in two bases. We show with a counterexample that, in this setting, Cobham's theorem does not generalize to multiplicatively independent bases. Finally, we prove that the sets of reals that are recognizable by Muller automata in two bases that do not share the same set of prime factors are exactly those definable in the first order additive theory of real and integer numbers. These sets are thus also recognizable by weak deterministic automata. This result leads to a precise characterization of the sets of real numbers that are recognizable in multiple bases, and provides a theoretical justification to the use of weak automata as symbolic representations of sets.Comment: 17 page

On the Number of Unbordered Factors

We illustrate a general technique for enumerating factors of k-automatic sequences by proving a conjecture on the number f(n) of unbordered factors of the Thue-Morse sequence. We show that f(n) = 4 and that f(n) = n infinitely often. We also give examples of automatic sequences having exactly 2 unbordered factors of every length

Enumeration and Decidable Properties of Automatic Sequences

We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems