Selfsimilarity, Simulation and Spacetime Symmetries

We study intrinsic simulations between cellular automata and introduce a new necessary condition for a CA to simulate another one. Although expressed for general CA, this condition is targeted towards surjective CA and especially linear ones. Following the approach introduced by the first author in an earlier paper, we develop proof techniques to tell whether some linear CA can simulate another linear CA. Besides rigorous proofs, the necessary condition for the simulation to occur can be heuristically checked via simple observations of typical space-time diagrams generated from finite configurations. As an illustration, we give an example of linear reversible CA which cannot simulate the identity and which is 'time-asymmetric', i.e. which can neither simulate its own inverse, nor the mirror of its own inverse

On Self-Similar Global Textures in an expanding Universe

We discuss self-similar solutions to $O(4)$ textures in Minkowski space and in flat Friedmann-Robertson-Walker backgrounds. We show that in the Minkowski case there exist no solutions with winding number greater than unity. However, we find besides the known solution with unit winding number also previously unknown solutions corresponding to winding number less than one. The validity of the non-linear sigma model approximation is discussed. We point out that no spherically symmetric exactly self-similar solutions exist for radiation or matter dominated FRW cosmologies, but we find a way to relax the assumptions of self-similarity that give us approximative solutions valid on intermediate scales.Comment: 12 pages (LaTeX) + 1 postscript figure. To appear in Phys. Lett

Intrinsic universality in tile self-assembly requires cooperation

We prove a negative result on the power of a model of algorithmic self-assembly for which it has been notoriously difficult to find general techniques and results. Specifically, we prove that Winfree's abstract Tile Assembly Model, when restricted to use noncooperative tile binding, is not intrinsically universal. This stands in stark contrast to the recent result that, via cooperative binding, the abstract Tile Assembly Model is indeed intrinsically universal. Noncooperative self-assembly, also known as "temperature 1", is where tiles bind to each other if they match on one or more sides, whereas cooperative binding requires binding on multiple sides. Our result shows that the change from single- to multi-sided binding qualitatively improves the kinds of dynamics and behavior that these models of nanoscale self-assembly are capable of. Our lower bound on simulation power holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. On the positive side, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. Our negative result can be interpreted to mean that Turing universal algorithmic behavior in self-assembly does not imply the ability to simulate arbitrary algorithmic self-assembly processes.Comment: Added references. Improved presentation of definitions and proofs. This article uses definitions from arXiv:1212.4756. arXiv admin note: text overlap with arXiv:1006.2897 by other author

Phase transitions in quantum chromodynamics

The current understanding of finite temperature phase transitions in QCD is reviewed. A critical discussion of refined phase transition criteria in numerical lattice simulations and of analytical tools going beyond the mean-field level in effective continuum models for QCD is presented. Theoretical predictions about the order of the transitions are compared with possible experimental manifestations in heavy-ion collisions. Various places in phenomenological descriptions are pointed out, where more reliable data for QCD's equation of state would help in selecting the most realistic scenario among those proposed. Unanswered questions are raised about the relevance of calculations which assume thermodynamic equilibrium. Promising new approaches to implement nonequilibrium aspects in the thermodynamics of heavy-ion collisions are described.Comment: 156 pages, RevTex. Tables II,VIII,IX and Fig.s 1-38 are not included as postscript files. I would like to ask the requestors to copy the missing tables and figures from the corresponding journal-referenc

The matter power spectrum in redshift space using effective field theory

The use of Eulerian 'standard perturbation theory' to describe mass assembly in the early universe has traditionally been limited to modes with k <= 0.1 h/Mpc at z=0. At larger k the SPT power spectrum deviates from measurements made using N-body simulations. Recently, there has been progress in extending the reach of perturbation theory to larger k using ideas borrowed from effective field theory. We revisit the computation of the redshift-space matter power spectrum within this framework, including for the first time for the full one-loop time dependence. We use a resummation scheme proposed by Vlah et al. to account for damping of the baryonic acoustic oscillations due to large-scale random motions and show that this has a significant effect on the multipole power spectra. We renormalize by comparison to a suite of custom N-body simulations matching the MultiDark MDR1 cosmology. At z=0 and for scales k <~ 0.4 h/Mpc we find that the EFT furnishes a description of the real-space power spectrum up to ~ 2%, for the ell=0 mode up to ~ 5% and for the ell = 2, 4 modes up to ~ 25%. We argue that, in the MDR1 cosmology, positivity of the ell = 0 mode gives a firm upper limit of k ~ 0.74 h/Mpc for the validity of the one-loop EFT prediction in redshift space using only the lowest-order counterterm. We show that replacing the one-loop growth factors by their Einstein-de Sitter counterparts is a good approximation for the ell = 0 mode, but can induce deviations as large as 2% for the ell = 2, 4 modes. An accompanying software bundle, distributed under open source licenses, includes Mathematica notebooks describing the calculation, together with parallel pipelines capable of computing both the necessary one-loop SPT integrals and the effective field theory counterterms

Proceedings of AUTOMATA 2011 : 17th International Workshop on Cellular Automata and Discrete Complex Systems

International audienceThe proceedings contain full (reviewed) papers and short (non reviewed) papers that were presented at the workshop

Numerical Relativity and Critical Black Holes.

PhD ThesesIn this thesis we have studied static black holes in direct product spacetimes. In this setting, there exists a universal sector describing black holes which are uniformly extended along the internal space. This sector naturally exhibits a hierarchy of length scales and dynamical instabilities of the Gregory-La amme type appear when one of those lengths is much larger than the other. At the onset of the instability there exists a 0-mode that hints on the existence of a new branch of black holes that are non-uniformly extended along the internal space. Typically, this new family of black holes cannot be found analytically and one has to rely on numerical methods. In particular, we have numerically constructed these `non-uniform black holes' in Kaluza-Klein theory and Anti-de Sitter spaces times a sphere. Moving along the space of solutions, eventually the branches of non-uniform black holes merge with another branch of black holes which have di erent horizon topology. These are also constructed in the Kaluza-Klein case. In this thesis we have focused on a detailed study of the extreme black holes very near the critical (or merger) point. It had been predicted that the physical properties of black holes near the critical solution are controlled by a local Ricci- at cone that governs, locally, the singularity at the merger. We verify this prediction by extracting the critical exponents of various physical quantities in the Kaluza-Klein setting in D = 10. In this particular case, properties of black holes can be computed by solving a dual super Yang-Mills theory on the lattice. In another study, we consider critical non-uniform black holes in AdSp Sq for (p; q) = (5; 5) and (p; q) = (4; 7), which are the relevant cases for the gauge/gravity duality, and compute, for the rst time, the critical exponents. Remarkably, in these two cases our study suggests a non-Ricci- at cone, which is consistent with the presence of nontrivial uxes in the setting. Our results are new and non-trivial predictions of the gauge/gravity duality without supersymmetry

Vortex Filament Equation for some Regular Polygonal Curves

One of the most interesting phenomena in fluid literature is the occurrence and evolution of vortex filaments. Some of their examples in the real world are smoke rings, whirlpools, and tornadoes. For an ideal fluid, there have been several models and governing equations to describe this evolution; however, due to its simplicity and geometric properties, the vortex filament equation (VFE) has recently gained attention. As an approximation of the dynamics of a vortex filament, the equation first appeared in the work of Da Rios at the beginning of the twentieth century. This model is usually known as the local induction approximation (LIA). In this work, we examine the evolution of VFE for regular polygonal curves both from a numerical and theoretical point of view in the Euclidean as well as hyperbolic geometry. In the first part of the thesis, we observe the evolution of the Vortex Filament equation taking $M$-sided regular polygons with nonzero torsion as initial data in the Euclidean space. Using algebraic techniques, backed by numerical simulations, we show that the solutions are polygons at rational times, as in the zero-torsion case. However, unlike in that case, the evolution is not periodic in time; moreover, the multifractal trajectory of the point X(0, t) is not planar and appears to be a helix for large times. These new solutions of VFE can be used to illustrate numerically that the smooth solutions of VFE given by helices and straight lines share the same instability as the one already established for circles. This is accomplished by showing the existence of variants of the so-called Riemann’s non-differentiable function that are as close to smooth curves as desired when measured in the right topology. This topology is motivated by some recent results on the well-posedness of VFE, which prove that the self-similar solutions of VFE have finite renormalized energy. In the rest of the work, we delve into the hyperbolic setting and examine the evolution of VFE for a planar $l$-polygon, i.e., a regular planar polygon in the Minkowski 3-space $\mathbb{R}^{1,2}$. Unlike in the Euclidean case, a planar $l$-polygon is open which makes the problem more challenging from a numerical point of view. After trying several numerical methods, we conclude that a finite-difference discretization in space combined with an explicit Runge--Kutta method in time, gives the best numerical results both in terms of efficiency and accuracy. On the other hand, using theoretical arguments, we recover the evolution algebraically, and thus, we show the agreement between the two approaches. During the numerical evolution, it has been observed that the trajectory of a corner is multifractal and as the parameter $l$ goes to zero, it converges to the Riemann’s non-differentiable function. Furthermore, as in the Euclidean case, we provide strong numerical evidence to show that at infinitesimal times, the evolution of VFE for a planar $l$-polygon as an initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only we can compute the speed of the center of mass of the planar $l$-polygon theoretically, the relationship also reveals important properties of the trajectory of its corners which we compare with its equivalent in the Euclidean case. Finally, a nonzero torsion in the hyperbolic case, yields two different kinds of helical polygonal curves, however, with the numerical and theoretical techniques developed so far, we are able to address them as well. This remains part of the future work of the thesis