The Lattice of integer partitions and its infinite extension

In this paper, we use a simple discrete dynamical system to study the integers partitions and their lattice. The set of the reachable configurations equiped with the order induced by the transitions of the system is exactly the lattice of integer partitions equiped with the dominance ordering. We first explain how this lattice can be constructed, by showing its strong self-similarity property. Then, we define a natural extension of the system to infinity. Using a self-similar tree, we obtain an efficient coding of the obtained lattice. This approach gives an interesting recursive formula for the number of partitions of an integer, where no closed formula have ever been found. It also gives informations on special sets of partitions, such as length bounded partitions.Comment: To appear in LNCS special issue, proceedings of ORDAL'99. See http://www.liafa.jussieu.fr/~latap

Influence of change of regulation on the goals achieved in futsal

Este artículo analiza la hipótesis de que los cambios introducidos en la normativa del fútbol sala modifican su lógica interna disminuyendo los goles totales y modificando su forma de ejecución. El objetivo de estudio es analizar los goles y manera de realizarlos en una temporada anterior y posterior al cambio de reglamentación para establecer cuantitativamente cómo el cambio de reglas del 2.006 afecta al juego. Se analizaron 3.126 goles en 442 partidos, 1.771 goles en 232 partidos en la temporada 2.002-2.003 y 1.355 goles en 210 partidos en la temporada 2.013-2.014. El método utilizado fue la metodología observacional, se utilizó el programa Lince vl.2.1. Los resultados muestran una reducción estadísticamente significativa en el número de goles de una temporada a otra. En la temporada 2.002-2.003 se lograron 1.927 goles con un promedio por equipo de 120,38 ± 28,58, y en la temporada 2.013-2.014 1.355 goles con un promedio de 90,40 ± 27,72This article analyses the hypothesis that the changes introduced in the regulation of futsal modify the inner logic of the game, what turns into a reduction of total goals and a significant variation in the form of execution. Therefore, it is set out as aim of study to analyze the goals and the way to achieve them in a previous season and in one subsequent to the change of regulation in order to establish quantitatively how the 2006 rules change has affected the game. Totally, 3126 goals were analyzed, scored in 442 matches, distributed in 1771 goals in 232 matches in season 2002-2003 and 1355 goals in 210 matches in season 2103-2014. The method used in this study was observational methodology. For the observational process, it has been used the observational software Lince vl.2.1. It has been carried out using the IBM SPSS 19.0.0 program. The results show a statistically significant reduction in the number of goals from one season to another. In season 2002-2003, 1927 goals were achieved with an average by team of 120.38±28.58, by 1355 goals in season 2013-2014 of 90.40±27.7

The Lattice structure of Chip Firing Games and Related Models

In this paper, we study a famous discrete dynamical system, the Chip Firing Game, used as a model in physics, economics and computer science. We use order theory and show that the set of reachable states (i.e. the configuration space) of such a system started in any configuration is a lattice, which implies strong structural properties. The lattice structure of the configuration space of a dynamical system is of great interest since it implies convergence (and more) if the configuration space is finite. If it is infinite, this property implies another kind of convergence: all the configurations reachable from two given configurations are reachable from their infimum. In other words, there is a unique first configuration which is reachable from two given configurations. Moreover, the Chip Firing Game is a very general model, and we show how known models can be encoded as Chip Firing Games, and how some results about them can be deduced from this paper. Finally, we define a new model, which is a generalization of the Chip Firing Game, and about which many interesting questions arise.Comment: See http://www.liafa.jussieu.fr/~latap

On Conservative and Monotone One-dimensional Cellular Automata and Their Particle Representation

Number-conserving (or {\em conservative}) cellular automata have been used in several contexts, in particular traffic models, where it is natural to think about them as systems of interacting particles. In this article we consider several issues concerning one-dimensional cellular automata which are conservative, monotone (specially ``non-increasing''), or that allow a weaker kind of conservative dynamics. We introduce a formalism of ``particle automata'', and discuss several properties that they may exhibit, some of which, like anticipation and momentum preservation, happen to be intrinsic to the conservative CA they represent. For monotone CA we give a characterization, and then show that they too are equivalent to the corresponding class of particle automata. Finally, we show how to determine, for a given CA and a given integer $b$, whether its states admit a $b$-neighborhood-dependent relabelling whose sum is conserved by the CA iteration; this can be used to uncover conservative principles and particle-like behavior underlying the dynamics of some CA. Complements at {\tt http://www.dim.uchile.cl/\verb' 'anmoreir/ncca}Comment: 38 pages, 2 figures. To appear in Theo. Comp. Sc. Several changes throughout the text; major change in section 4.

Sticky Seeding in Discrete-Time Reversible-Threshold Networks

When nodes can repeatedly update their behavior (as in agent-based models from computational social science or repeated-game play settings) the problem of optimal network seeding becomes very complex. For a popular spreading-phenomena model of binary-behavior updating based on thresholds of adoption among neighbors, we consider several planning problems in the design of \textit{Sticky Interventions}: when adoption decisions are reversible, the planner aims to find a Seed Set where temporary intervention leads to long-term behavior change. We prove that completely converting a network at minimum cost is $\Omega(\ln (OPT) )$-hard to approximate and that maximizing conversion subject to a budget is $(1-\frac{1}{e})$-hard to approximate. Optimization heuristics which rely on many objective function evaluations may still be practical, particularly in relatively-sparse networks: we prove that the long-term impact of a Seed Set can be evaluated in $O(|E|^2)$ operations. For a more descriptive model variant in which some neighbors may be more influential than others, we show that under integer edge weights from $\{0,1,2,...,k\}$ objective function evaluation requires only $O(k|E|^2)$ operations. These operation bounds are based on improvements we give for bounds on time-steps-to-convergence under discrete-time reversible-threshold updates in networks.Comment: 19 pages, 2 figure

On the effects of firing memory in the dynamics of conjunctive networks

Boolean networks are one of the most studied discrete models in the context of the study of gene expression. In order to define the dynamics associated to a Boolean network, there are several \emph{update schemes} that range from parallel or \emph{synchronous} to \emph{asynchronous.} However, studying each possible dynamics defined by different update schemes might not be efficient. In this context, considering some type of temporal delay in the dynamics of Boolean networks emerges as an alternative approach. In this paper, we focus in studying the effect of a particular type of delay called \emph{firing memory} in the dynamics of Boolean networks. Particularly, we focus in symmetric (non-directed) conjunctive networks and we show that there exist examples that exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in determinate if some vertex will eventually change its state, given an initial condition. We prove that this problem is {\bf PSPACE}-complete

Parallel Chip Firing Game associated with n-cube orientations

We study the cycles generated by the chip firing game associated with n-cube orientations. We show the existence of the cycles generated by parallel evolutions of even lengths from 2 to $2^n$ on $H_n$ (n >= 1), and of odd lengths different from 3 and ranging from 1 to $2^{n-1}-1$ on $H_n$ (n >= 4)