Distributing Labels on Infinite Trees

Sturmian words are infinite binary words with many equivalent definitions: They have a minimal factor complexity among all aperiodic sequences; they are balanced sequences (the labels 0 and 1 are as evenly distributed as possible) and they can be constructed using a mechanical definition. All this properties make them good candidates for being extremal points in scheduling problems over two processors. In this paper, we consider the problem of generalizing Sturmian words to trees. The problem is to evenly distribute labels 0 and 1 over infinite trees. We show that (strongly) balanced trees exist and can also be constructed using a mechanical process as long as the tree is irrational. Such trees also have a minimal factor complexity. Therefore they bring the hope that extremal scheduling properties of Sturmian words can be extended to such trees, as least partially. Such possible extensions are illustrated by one such example.Comment: 30 pages, use pgf/tik

The vertex-transitive TLF-planar graphs

We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs, a class containing in particular all the one-ended planar Cayley graphs and the normal transitive tilings. We characterize these graphs with a finite local representation and a special kind of finite state automaton named labeling scheme. As a result, we are able to enumerate and describe all TLF-planar vertex-transitive graphs of any given degree. Also, we are able decide to whether any TLF-planar transitive graph is Cayley or not.Comment: Article : 23 pages, 15 figures Appendix : 13 pages, 72 figures Submitted to Discrete Mathematics The appendix is accessible at http://www.labri.fr/~renault/research/research.htm

Optimal path planning for surveillance with temporal-logic constraints

In this paper we present a method for automatically generating optimal robot paths satisfying high-level mission specifications. The motion of the robot in the environment is modeled as a weighted transition system. The mission is specified by an arbitrary linear temporal-logic (LTL) formula over propositions satisfied at the regions of a partitioned environment. The mission specification contains an optimizing proposition, which must be repeatedly satisfied. The cost function that we seek to minimize is the maximum time between satisfying instances of the optimizing proposition. For every environment model, and for every formula, our method computes a robot path that minimizes the cost function. The problem is motivated by applications in robotic monitoring and data-gathering. In this setting, the optimizing proposition is satisfied at all locations where data can be uploaded, and the LTL formula specifies a complex data-collection mission. Our method utilizes Büchi automata to produce an automaton (which can be thought of as a graph) whose runs satisfy the temporal-logic specification. We then present a graph algorithm that computes a run corresponding to the optimal robot path. We present an implementation for a robot performing data collection in a road-network platform.This material is based upon work supported in part by ONR-MURI (award N00014-09-1-1051), ARO (award W911NF-09-1-0088), and Masaryk University (grant numbers LH11065 and GD102/09/H042), and other funding sources (AFOSR YIP FA9550-09-1-0209, NSF CNS-1035588, NSF CNS-0834260). (N00014-09-1-1051 - ONR-MURI; W911NF-09-1-0088 - ARO; LH11065 - Masaryk University; GD102/09/H042 - Masaryk University; FA9550-09-1-0209 - AFOSR YIP; CNS-1035588 - NSF; CNS-0834260 - NSF

FRACTAL GEOMETRY IN AGRICULTURAL CASH PRICE DYNAMICS

Agricultural prices are determined by natural and socio-economic factors that are known to be self-similar at different time scales and to follow non-periodic cyclical patterns. These properties are most easily understood using Mandelbrot's fractal geometry, in which a jagged time series is treated as a jagged coastline or any other natural phenomenon. The fractal market hypothesis provides the theory needed to explain why fractal structure exists in agricultural prices. Empirical evidence confirms theoretical predictions.Demand and Price Analysis,

Wavelet analysis on symbolic sequences and two-fold de Bruijn sequences

The concept of symbolic sequences play important role in study of complex systems. In the work we are interested in ultrametric structure of the set of cyclic sequences naturally arising in theory of dynamical systems. Aimed at construction of analytic and numerical methods for investigation of clusters we introduce operator language on the space of symbolic sequences and propose an approach based on wavelet analysis for study of the cluster hierarchy. The analytic power of the approach is demonstrated by derivation of a formula for counting of {\it two-fold de Bruijn sequences}, the extension of the notion of de Bruijn sequences. Possible advantages of the developed description is also discussed in context of applied

Evolutionary games on graphs

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.Comment: Review, final version, 133 pages, 65 figure