Renormalization of cellular automata and self-similarity

We study self-similarity in one-dimensional probabilistic cellular automata (PCA) using the renormalization technique. We introduce a general framework for algebraic construction of renormalization groups (RG) on cellular automata and apply it to exhaustively search the rule space for automata displaying dynamic criticality. Previous studies have shown that there exists several exactly renormalizable deterministic automata. We show that the RG fixed points for such self-similar CA are unstable in all directions under renormalization. This implies that the large scale structure of self-similar deterministic elementary cellular automata is destroyed by any finite error probability. As a second result we show that the only non-trivial critical PCA are the different versions of the well-studied phenomenon of directed percolation. We discuss how the second result supports a conjecture regarding the universality class for dynamic criticality defined by directed percolation.Comment: 14 pages, 4 figure

A model-independent approach to infer hierarchical codon substitution dynamics

<p>Abstract</p> <p>Background</p> <p>Codon substitution constitutes a fundamental process in molecular biology that has been studied extensively. However, prior studies rely on various assumptions, e.g. regarding the relevance of specific biochemical properties, or on conservation criteria for defining substitution groups. Ideally, one would instead like to analyze the substitution process in terms of raw dynamics, independently of underlying system specifics. In this paper we propose a method for doing this by identifying groups of codons and amino acids such that these groups imply closed dynamics. The approach relies on recently developed spectral and agglomerative techniques for identifying hierarchical organization in dynamical systems.</p> <p>Results</p> <p>We have applied the techniques on an empirically derived Markov model of the codon substitution process that is provided in the literature. Without system specific knowledge of the substitution process, the techniques manage to "blindly" identify multiple levels of dynamics; from amino acid substitutions (via the standard genetic code) to higher order dynamics on the level of amino acid groups. We hypothesize that the acquired groups reflect earlier versions of the genetic code.</p> <p>Conclusions</p> <p>The results demonstrate the applicability of the techniques. Due to their generality, we believe that they can be used to coarse grain and identify hierarchical organization in a broad range of other biological systems and processes, such as protein interaction networks, genetic regulatory networks and food webs.</p

Characterization of exact lumpability for vector fields on smooth manifolds

We characterize the exact lumpability of smooth vector fields on smooth manifolds. We derive necessary and sufficient conditions for lumpability and express them from four different perspectives, thus simplifying and generalizing various results from the literature that exist for Euclidean spaces. We introduce a partial connection on the pullback bundle that is related to the Bott connection and behaves like a Lie derivative. The lumping conditions are formulated in terms of the differential of the lumping map, its covariant derivative with respect to the connection and their respective kernels. Some examples are discussed to illustrate the theory. © 2016 Published by Elsevier B.V

A SPECTRAL METHOD FOR AGGREGATING VARIABLES IN LINEAR DYNAMICAL SYSTEMS WITH APPLICATION TO CELLULAR AUTOMATA RENORMALIZATION

We present a method for identifying coarse-grained dynamics through aggregation of variables or states in linear dynamical systems. The condition for aggregation is expressed as a permutation symmetry of a set of dual eigenvectors of the matrix that defines the dynamics. The applicability of the condition is illustrated in examples from three different generic classes of reducible Markov chains: systems consisting of independent subsystems, dynamics with symmetries, and nearly decoupled Markov chains. Furthermore we show how the method can be used to coarse-grain cellular automata.Lumpability, aggregated Markov chains, aggregation of variables, aggregated linear dynamics, quotient processes, state space reduction, renormalization, coarse-graining, cellular automata

Reduktion der Evolutionsgleichungen in Banach-Räumen

In this thesis we analyze lumpability of infinite dimensional dynamical systems. Lumping is a method to project a dynamics by a linear reduction operator onto a smaller state space on which a self-contained dynamical description exists. We consider a well-posed dynamical system defined on a Banach space X and generated by an operator F, together with a linear and bounded map M : X → Y, where Y is another Banach space. The operator M is surjective but not an isomorphism and it represents a reduction of the state space. We investigate whether the variable y = M x also satisfies a well-posed and self-contained dynamics on Y . We work in the context of strongly continuous semigroup theory. We first discuss lumpability of linear systems in Banach spaces. We give conditions for a reduced operator to exist on Y and to describe the evolution of the new variable y . We also study lumpability of nonlinear evolution equations, focusing on dissipative operators, for which some interesting results exist, concerning the existence and uniqueness of solutions, both in the classical sense of smooth solutions and in the weaker sense of strong solutions. We also investigate the regularity properties inherited by the reduced operator from the original operator F . Finally, we describe a particular kind of lumping in the context of C*-algebras. This lumping represents a different interpretation of a restriction operator. We apply this lumping to Feller semigroups, which are important because they can be associated in a unique way to Markov processes. We show that the fundamental properties of Feller semigroups are preserved by this lumping. Using these ideas, we give a short proof of the classical Tietze extension theorem based on C*-algebras and Gelfand theory