The \mu-Calculus Alternation Hierarchy Collapses over Structures with Restricted Connectivity

It is known that the alternation hierarchy of least and greatest fixpoint operators in the mu-calculus is strict. However, the strictness of the alternation hierarchy does not necessarily carry over when considering restricted classes of structures. A prominent instance is the class of infinite words over which the alternation-free fragment is already as expressive as the full mu-calculus. Our current understanding of when and why the mu-calculus alternation hierarchy is not strict is limited. This paper makes progress in answering these questions by showing that the alternation hierarchy of the mu-calculus collapses to the alternation-free fragment over some classes of structures, including infinite nested words and finite graphs with feedback vertex sets of a bounded size. Common to these classes is that the connectivity between the components in a structure from such a class is restricted in the sense that the removal of certain vertices from the structure's graph decomposes it into graphs in which all paths are of finite length. Our collapse results are obtained in an automata-theoretic setting. They subsume, generalize, and strengthen several prior results on the expressivity of the mu-calculus over restricted classes of structures.Comment: In Proceedings GandALF 2012, arXiv:1210.202

N-body methods for relativistic cosmology

We present a framework for general relativistic N-body simulations in the regime of weak gravitational fields. In this approach, Einstein's equations are expanded in terms of metric perturbations about a Friedmann-Lema\^itre background, which are assumed to remain small. The metric perturbations themselves are only kept to linear order, but we keep their first spatial derivatives to second order and treat their second spatial derivatives as well as sources of stress-energy fully non-perturbatively. The evolution of matter is modelled by an N-body ensemble which can consist of free-streaming nonrelativistic (e.g. cold dark matter) or relativistic particle species (e.g. cosmic neutrinos), but the framework is fully general and also allows for other sources of stress-energy, in particular additional relativistic sources like modified-gravity models or topological defects. We compare our method with the traditional Newtonian approach and argue that relativistic methods are conceptually more robust and flexible, at the cost of a moderate increase of numerical difficulty. However, for a LambdaCDM cosmology, where nonrelativistic matter is the only source of perturbations, the relativistic corrections are expected to be small. We quantify this statement by extracting post-Newtonian estimates from Newtonian N-body simulations.Comment: 30 pages, 3 figures. Invited contribution to a Classical and Quantum Gravity focus issue on "Relativistic Effects in Cosmology", edited by Kazuya Koyam

An exact approach for evaluating the benefits from technological change

It is commonly believed that taxing agricultural commodities in developing countries, and subsidizing agricultural commodities in industrial countries, reduces incentives in the developing countries for both current production and longer-term investments in capital, knowledge, technology, and infrastructure. It is argued that distortions in agricultural markets have kept investments in research and development, and productivity rates low in agriculture in developing countries. Martin and Alston lay the theoretical foundation for empirical studies of how such distortions affect returns to agricultural research and development in developing countries. Earlier studies of the benefits from technological change have typically used partial equilibrium models with Marshallian welfare measures. Such models have not allowed for a general set of market distortions and market interactions. Techniques recently developed for evaluating welfare in the context of general equilibrium models better measure the implications of trade distorting policies. Martin and Alston describe how to harness these approaches to evaluate the benefits and costs of technological changes. They show that a modified trade expenditure function can be used to measure welfare changes exactly, with a model consistent with the optimizing behavior of both producers and consumers. They do so in a general setting that allows for multiple market distortions and multiple paths of general equilibrium feedback. They illustrate this approach using a quadratic form for a profit function that is a component of the trade expenditure function. They spell out, in principle, how to apply this approach with minimal requirements for additional information, using the results from a computable general equilibrium model. They provide a diagram to illustrate the application of the technique.Environmental Economics&Policies,Economic Theory&Research,Access to Markets,Markets and Market Access,Consumption

gevolution: a cosmological N-body code based on General Relativity

We present a new N-body code, gevolution, for the evolution of large scale structure in the Universe. Our code is based on a weak field expansion of General Relativity and calculates all six metric degrees of freedom in Poisson gauge. N-body particles are evolved by solving the geodesic equation which we write in terms of a canonical momentum such that it remains valid also for relativistic particles. We validate the code by considering the Schwarzschild solution and, in the Newtonian limit, by comparing with the Newtonian N-body codes Gadget-2 and RAMSES. We then proceed with a simulation of large scale structure in a Universe with massive neutrinos where we study the gravitational slip induced by the neutrino shear stress. The code can be extended to include different kinds of dark energy or modified gravity models and going beyond the usually adopted quasi-static approximation. Our code is publicly available.Comment: 28 pages + appendix, 10 figures. v2: revised and extended version accepted by JCAP; code available at https://github.com/gevolution-cod