International audienceIn this paper we construct fixed finite tile systems that assemble into particular classes of shapes. Moreover, given an arbitrary n, we show how to calculate the tile concentrations in order to ensure that the expected size of the produced shape is n. For rectangles and squares our constructions are optimal (with respect to the size of the systems). We also introduce the notion of parallel time, which is a good approximation of the classical asynchronous time. We prove that our tile systems produce the rectangles and squares in linear parallel time (with respect to the diameter). Those results are optimal. Finally, we introduce the class of diamonds. For these shapes we construct a non trivial tile system having also a linear parallel time complexity
In this paper we ask which properties of a distributed network can be computed from a few amount of local information provided by its nodes. The distributed model we consider is a restriction of the classical CON GEST (distributed) model and it is close to the simultaneous messages (communication complexity) model defined by Babai, Kimmel and Lokam. More precisely, each of these n nodes -which only knows its own ID and the IDs of its neighbors-is allowed to send a message of O(log n) bits to some central entity, called the referee. Is it possible for the referee to decide some basic structural properties of the network topology G? We show that simple questions like, "does G contain a square?", "does G contain a triangle?" or "Is the diameter of G at most 3?" cannot be solved in general. On the other hand, the referee can decode the messages in order to have full knowledge of G when G belongs to many graph classes such as planar graphs, bounded treewidth graphs and, more generally, bounded degeneracy graphs. We leave open questions related to the connectivity of arbitrary graphs.
International audienceWe present a novel way to design self-assembling systems using a notion of signal (or ray) akin to what is used in analyzing the behavior of cellular automata. This allows purely geometrical constructions, with a smaller specification and easier analysis. We show how to design a system of signals for a given set of shapes, and how to transform these signals into a set of tiles which self-assemble into the desired shapes. We show how to use this technique on three examples : squares (with optimal assembly time and a small number of tiles), general polygons, and a quasi periodic pattern : Robinson tiling
Cellular Automata have been used since their introduction as a discrete tool of modelization. In many of the physical processes one may modelize thus (such as bootstrap percolation, forest fire or epidemic propagation models, life without death, etc), each local change is irreversible. The class of freezing Cellular Automata (FCA) captures this feature. In a freezing cellular automaton the states are ordered and the cells can only decrease their state according to this "freezing-order". We investigate the dynamics of such systems through the questions of simulation and universality in this class: is there a Freezing Cellular Automaton (FCA) that is able to simulate any Freezing Cellular Automata, i.e. an intrinsically universal FCA? We show that the answer to that question is sensitive to both the number of changes cells are allowed to make, and geometric features of the space. In dimension 1, there is no universal FCA. In dimension 2, if either the number of changes is at least 2, or the neighborhood is Moore, then there are universal FCA. On the other hand, there is no universal FCA with one change and Von Neumann neighborhood. We also show that monotonicity of the local rule with respect to the freezing-order (a common feature of bootstrap percolation) is also an obstacle to universality.Context-sensitive simulation can get us over this hurdle as we show below; it is akin to the notion of conjugacy in symbolic dynamics [14].
In this paper we study distributed algorithms on massive graphs where links represent a particular relationship between nodes (for instance, nodes may represent phone numbers and links may indicate telephone calls). Since such graphs are massive they need to be processed in a distributed and streaming way. When computing graph-theoretic properties, nodes become natural units for distributed computation. Links do not necessarily represent communication channels between the computing units and therefore do not restrict the communication flow. Our goal is to model and analyze the computational power of such distributed systems where one computing unit is assigned to each node. Communication takes place on a whiteboard where each node is allowed to write at most one message. Every node can read the contents of the whiteboard and, when activated, can write one small message based on its local knowledge. When the protocol terminates its output is computed from the final contents of the whiteboard. We describe four synchronization models for accessing the whiteboard. We show that message size and synchronization power constitute two orthogonal hierarchies for these systems. We exhibit problems that separate these models, i.e., that can be solved in one model but not in a weaker one, even with increased message size. These problems are related to maximal independent set and connectivity. We also exhibit problems that require a given message size independently of the synchronization model.
If multi-phase machines equipped with toothconcentrated winding with half a slot per pole and per phase offer interesting characteristics (simplified manufacturing, no space subharmonic, fault-tolerant ability), their low fundamental winding factors make their designs and controls challenging. The paper addresses the case of a seven-phase Surface-mounted Permanent Magnet (SPM) machine which has a fundamental winding factor lower than the third. This so-called bi-harmonic specificity is considered in order to achieve good torque quality (average value and ripples). Regarding the design, the magnet layer is segmented into two identical radially magnetized tiles that cover about three-quarters the pole arc. Regarding the control, the rated Maximum Torque Per Ampere (MTPA) supply strategy (h1h3 control) aims at generating a third harmonic current component greater than the fundamental. A prototype has been manufactured: the ability of the machine to provide smooth torque is experimentally confirmed through the implementation of a simple MTPA control which copes with high distortion in no-load voltage.
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