Automaton Semigroups and Groups: On the Undecidability of Problems Related to Freeness and Finiteness

In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. In the course of this study, we also exhibit some connections between the algebraic structure of automaton (semi)groups and their dynamics on the boundary. First, we show that it is undecidable to check whether the group generated by a given invertible automaton has a positive relation, i.e. a relation p = 1 such that p only contains positive generators. Besides its obvious relation to the freeness of the group, the absence of positive relations has previously been studied and is connected to the triviality of some stabilizers of the boundary. We show that the emptiness of the set of positive relations is equivalent to the dynamical property that all (directed positive) orbital graphs centered at non-singular points are acyclic. Gillibert showed that the finiteness problem for automaton semigroups is undecidable. In the second part of the paper, we show that this undecidability result also holds if the input is restricted to be bi-reversible and invertible (but, in general, not complete). As an immediate consequence, we obtain that the finiteness problem for automaton subsemigroups of semigroups generated by invertible, yet partial automata, so called automaton-inverse semigroups, is also undecidable. Erratum: Contrary to a statement in a previous version of the paper, our approach does not show that that the freeness problem for automaton semigroups is undecidable. We discuss this in an erratum at the end of the paper

Towards a Simplified Dynamic Wake Model using POD Analysis

We apply the proper orthogonal decomposition (POD) to large eddy simulation data of a wind turbine wake in a turbulent atmospheric boundary layer. The turbine is modeled as an actuator disk. Our analyis mainly focuses on the question whether POD could be a useful tool to develop a simplified dynamic wake model. The extracted POD modes are used to obtain approximate descriptions of the velocity field. To assess the quality of these POD reconstructions, we define simple measures which are believed to be relevant for a sequential turbine in the wake such as the energy flux through a disk in the wake. It is shown that only a few modes are necessary to capture basic dynamical aspects of these measures even though only a small part of the turbulent kinetic energy is restored. Furthermore, we show that the importance of the individual modes depends on the measure chosen. Therefore, the optimal choice of modes for a possible model could in principle depend on the application of interest. We additionally present a possible interpretation of the POD modes relating them to specific properties of the wake. For example the first mode is related to the horizontal large scale movement. Besides yielding a deeper understanding, this also enables us to view our results in comparison to existing dynamic wake models

On the Complexity of the Word Problem for Automaton Semigroups and Automaton Groups

In this paper, we study the word problem for automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups, which are generated by partial, yet invertible automata. We show that there is an automaton-inverse semigroup and, thus, an automaton semigroup with a PSPACE-complete word problem. We also show that there is an automaton group for which the word problem with a single rational constraint is PSPACE-complete. Additionally, we provide simpler constructions for the uniform word problems of these classes. For the uniform word problem for automaton groups (without rational constraints), we show NL-hardness. Finally, we investigate a question asked by Cain about a better upper bound for the length of a word on which two distinct elements of an automaton semigroup must act differently

Productivity convergence and economic growth: A frontier production function approach

The empirical growth literature has focused on capital accumulation but largely ignored productivity growth. To address this imbalance, we propose a methodology for analyzing productivity convergence based on frontier production functions. We examine whether departures from the frontier are cointegrated, determine the extent and speed of catch-up, and assess the importance of efficiency changes for economic growth. Using a sample of 26 OECD countries from 1965-90, we find convergence and catch-up is fairly strong among EU countries but not among the G-7. Overall, the ability to absorb new technology is an important source of economic growth. --cointegration,convergence,growth,productivity,unit root

Charging of a quantum dot coupled to Luttinger liquid leads

Luttinger liquid behavior of one-dimensional correlated electron systems is characterized by power-law scaling of a variety of physical observables with exponents determined by a single interaction dependent parameter K. We suggest a setup to study Luttinger liquid behavior in quantum wires which allows to determine K from two independent measurements: resonant transport through a quantum dot embedded in the wire and the charge on the dot. Consistency of the two measured values of K for a single probe would provide strong experimental evidence for the Luttinger liquid paradigm.Comment: 4 pages, 4 figures included, version accepted for publication in PR

Comovement and catch-up in productivity across sectors: Evidence from the OECD

A method for analyzing productivity convergence based on frontier production functions is proposed. It is examined whether departures from the frontier|country{level ineciencies|exhibit long{run relationships and convergence. The method is applied to 1-digit industries of 14 OECD countries from 1970{90. Results suggests that comovements in eciency are concentrated between the EU{countries. Catch{up is found in all but one sector. Even manufacturing, which previous studies have found not to display convergence, shows signs of catch-up. It is examined whether departures from the frontier|country{level ineciencies|exhibit long{run relationships and convergence. The method is applied to 1-digit industries of 14 OECD countries from 1970{90. Results suggests that comovements in eciency are concentrated between the EU{countries. Catch{up is found in all but one sector. Even manufacturing, which previous studies have found not to display convergence, shows signs of catch{up. --cointegration,convergence,economic growth,efficiency,productivity,technological diffusion,unit root