Quasiperiodicity and non-computability in tilings

We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the fixed point construction; we improve this general technique and make it enforce the property of local regularity of tilings needed for quasiperiodicity. We prove also a stronger result: any effectively closed set can be recursively transformed into a tile set so that the Turing degrees of the resulted tilings consists exactly of the upper cone based on the Turing degrees of the later.Comment: v3: the version accepted to MFCS 201

Tracing Communications and Computational Workload in LJS (Lennard-Jones with Spatial Decomposition)

LJS (Lennard-Jones with Spatial decomposition) is a molecular dynamics application developed by Steve Plimpton at Sandia National Laboratories [1]. It performs thermodynamic simulations of a system containing fixed large number (millions) of atoms or molecules confined within a regular, three-dimensional domain. Since the simulations model interactions on atomic scale, the computations carried out in a single timestep (iteration) correspond to femtoseconds of the real time. Hence, a meaningful simulation of the evolution of the system's state typically requires a large number (thousands and more) of timesteps. The particles in LJS are represented as material points subjected to forces resulting from interactions with other particles. While the general case involves N-body solvers, LJS implements only pair-wise material point interactions using derivative of Lennard-Jones potential energy for each particle pair to evaluate the acting forces. The velocities and positions of particles are updated by integrating Newton's equations (classical molecular dynamics). The interaction range depends on the modeled problem type; LJS focuses on short-range forces, implementing a cutoff distance rc outside which the interactions are ignored. The computational complexity of O(N2), characteristic for systems with long-range interactions, is therefore substantially alleviated. LJS deploys spatial decomposition of the domain volume to distribute the computations across the available processors on a parallel computer. The decomposition process uniformly divides parallelepiped containing all particles into volumes equal in size and as close in shape to a cube as possible, assigning each of such formed cells to a CPU. The correctness of computations requires the positions of some particles (depending on the value of rc) residing in the neighboring cells to be known to the local process. This information is exchanged in every timestep via explicit communication with the neighbor nodes in all three dimensions (for details see [2]). LJS also takes the advantage of the third Newton's law to calculate the force only once per particle pair; if the involved particles belong to cells located on different processors, the results are forwarded to the other node in a "reverse communication" phase. Besides communications occurring in every iteration, additional messages are sent once every preset number of timesteps. Their purpose is to adjust cell assignments of particles due to their movement. To minimize the overhead of the construction of particle neighbor lists, LJS replaces rc with extended cutoff radius rs (rs > rc), which accounts for possible particle movement before any list updates need to be carried out. Due to a relatively small impact of that phase on the overall behavior of the application, we ignored it in our analysis