Amalgamation of Transition Sequences in the PEPA Formalism

This report presents a proposed formal approach towards reduction of sequences in PEPA components. By performing the described amalgamation procedure we may remove, from the Markov chain underlying an initial PEPA model, those states for which detailed local balance equations cannot be formulated. This transformation may lead to a simpler model with product form solution. Some classes of reduced models preserve those performance measures which we are interested in and, moreover, the steady state solution vector is much easier to find from the computational point of view

A note on insensitivity in stochastic networks

We give a simple and direct treatment of insensitivity in stochastic networks which is quite general and which provides probabilistic insight into the phenomenon. In the case of multi-class networks, the results generalise those of Bonald and Proutiere (2002, 2003).Comment: 12 pages, to appear in J. Appl. Probab., 44, No 1 (March 2007

On the concept of Bell's local causality in local classical and quantum theory

The aim of this paper is to give a sharp definition of Bell's notion of local causality. To this end, first we unfold a framework, called local physical theory, integrating probabilistic and spatiotemporal concepts. Formulating local causality within this framework and classifying local physical theories by whether they obey local primitive causality --- a property rendering the dynamics of the theory causal, we then investigate what is needed for a local physical theory, with or without local primitive causality, to be locally causal. Finally, comparing Bell's local causality with the Common Cause Principles and relating both to the Bell inequalities we find a nice parallelism: Bell inequalities cannot be derived neither from local causality nor from a common cause unless the local physical theory is classical or the common cause is commuting, respectively.Comment: 24 pages, 5 figure

Chaos and Complexity of quantum motion

The problem of characterizing complexity of quantum dynamics - in particular of locally interacting chains of quantum particles - will be reviewed and discussed from several different perspectives: (i) stability of motion against external perturbations and decoherence, (ii) efficiency of quantum simulation in terms of classical computation and entanglement production in operator spaces, (iii) quantum transport, relaxation to equilibrium and quantum mixing, and (iv) computation of quantum dynamical entropies. Discussions of all these criteria will be confronted with the established criteria of integrability or quantum chaos, and sometimes quite surprising conclusions are found. Some conjectures and interesting open problems in ergodic theory of the quantum many problem are suggested.Comment: 45 pages, 22 figures, final version, at press in J. Phys. A, special issue on Quantum Informatio

A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems

We address the numerical solution of infinite-dimensional inverse problems in the framework of Bayesian inference. In the Part I companion to this paper (arXiv.org:1308.1313), we considered the linearized infinite-dimensional inverse problem. Here in Part II, we relax the linearization assumption and consider the fully nonlinear infinite-dimensional inverse problem using a Markov chain Monte Carlo (MCMC) sampling method. To address the challenges of sampling high-dimensional pdfs arising from Bayesian inverse problems governed by PDEs, we build on the stochastic Newton MCMC method. This method exploits problem structure by taking as a proposal density a local Gaussian approximation of the posterior pdf, whose construction is made tractable by invoking a low-rank approximation of its data misfit component of the Hessian. Here we introduce an approximation of the stochastic Newton proposal in which we compute the low-rank-based Hessian at just the MAP point, and then reuse this Hessian at each MCMC step. We compare the performance of the proposed method to the original stochastic Newton MCMC method and to an independence sampler. The comparison of the three methods is conducted on a synthetic ice sheet inverse problem. For this problem, the stochastic Newton MCMC method with a MAP-based Hessian converges at least as rapidly as the original stochastic Newton MCMC method, but is far cheaper since it avoids recomputing the Hessian at each step. On the other hand, it is more expensive per sample than the independence sampler; however, its convergence is significantly more rapid, and thus overall it is much cheaper. Finally, we present extensive analysis and interpretation of the posterior distribution, and classify directions in parameter space based on the extent to which they are informed by the prior or the observations.Comment: 31 page

Directed abelian algebras and their applications to stochastic models

To each directed acyclic graph (this includes some D-dimensional lattices) one can associate some abelian algebras that we call directed abelian algebras (DAA). On each site of the graph one attaches a generator of the algebra. These algebras depend on several parameters and are semisimple. Using any DAA one can define a family of Hamiltonians which give the continuous time evolution of a stochastic process. The calculation of the spectra and ground state wavefunctions (stationary states probability distributions) is an easy algebraic exercise. If one considers D-dimensional lattices and choose Hamiltonians linear in the generators, in the finite-size scaling the Hamiltonian spectrum is gapless with a critical dynamic exponent $z = D$. One possible application of the DAA is to sandpile models. In the paper we present this application considering one and two dimensional lattices. In the one dimensional case, when the DAA conserves the number of particles, the avalanches belong to the random walker universality class (critical exponent $\sigma_{\tau} = 3/2$). We study the local densityof particles inside large avalanches showing a depletion of particles at the source of the avalanche and an enrichment at its end. In two dimensions we did extensive Monte-Carlo simulations and found $\sigma_{\tau} = 1.782 \pm 0.005$.Comment: 14 pages, 9 figure

Constructing Hamiltonian quantum theories from path integrals in a diffeomorphism invariant context

Osterwalder and Schrader introduced a procedure to obtain a (Lorentzian) Hamiltonian quantum theory starting from a measure on the space of (Euclidean) histories of a scalar quantum field. In this paper, we extend that construction to more general theories which do not refer to any background, space-time metric (and in which the space of histories does not admit a natural linear structure). Examples include certain gauge theories, topological field theories and relativistic gravitational theories. The treatment is self-contained in the sense that an a priori knowledge of the Osterwalder-Schrader theorem is not assumed.Comment: Plain Latex, 25 p., references added, abstract and title changed (originally :``Osterwalder Schrader Reconstruction and Diffeomorphism Invariance''), introduction extended, one appendix with illustrative model added, accepted by Class. Quantum Gra