Isospectral Compression and Other Useful Isospectral Transformations of Dynamical Networks

It is common knowledge that a key dynamical characteristic of a network is its spectrum (the collection of all eigenvalues of the network's weighted adjacency matrix). In \cite{BW10} we demonstrated that it is possible to reduce a network, considered as a graph, to a smaller network with fewer vertices and edges while preserving the spectrum (or spectral information) of the original network. This procedure allows for the introduction of new equivalence relations between networks, where two networks are spectrally equivalent if they can be reduced to the same network. Additionally, using this theory it is possible to establish whether a network, modeled as a dynamical system, has a globally attracting fixed point (is strongly synchronizing). In this paper we further develop this theory of isospectral network transformations and demonstrate that our procedures are applicable to families of parameterized networks and networks of arbitrary size.Comment: 26 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1010.327

Lower bounds for the approximation with variation-diminishing splines

We prove lower bounds for the approximation error of the variation-diminishing Schoenberg operator on the interval [0, 1] in terms of classical moduli of smoothness depending on the degree of the spline basis. For this purpose we use a functional analysis framework in order to characterize the spectrum of the Schoenberg operator and investigate the asymptotic behavior of its iterates

Markovian evolution of quantum coherence under symmetric dynamics

Both conservation laws and practical restrictions impose symmetry constraints on the dynamics of open quantum systems. In the case of time-translation symmetry, which arises naturally in many physically relevant scenarios, the quantum coherence between energy eigenstates becomes a valuable resource for quantum information processing. In this work we identify the minimum amount of decoherence compatible with this symmetry for a given population dynamics. This yields a generalisation to higher-dimensional systems of the relation T2 2T1 for qubit decoherence and relaxation times. It also enables us to witness and assess the role of non-Markovianity as a resource for coherence preservation and transfer. Moreover, we discuss the relationship between ergodicity and the ability of Markovian dynamics to indenitely sustain a superposition of diferent energy states. Finally, we establish a formal connection between the resource-theoretic and the master equation approaches to thermodynamics, with the former being a non-Markovian generalisation of the latter. Our work thus brings the abstract study of quantum coherence as a resource towards the realm of actual physical applications

Improving model fidelity and sensitivity for complex systems through empirical information theory

In many situations in contemporary science and engineering, the analysis and prediction of crucial phenomena occur often through complex dynamical equations that have significant model errors compared with the true signal in nature. Here, a systematic information theoretic framework is developed to improve model fidelity and sensitivity for complex systems including perturbation formulas and multimodel ensembles that can be utilized to improve both aspects of model error simultaneously. A suite of unambiguous test models is utilized to demonstrate facets of the proposed framework. These results include simple examples of imperfect models with perfect equilibrium statistical fidelity where there are intrinsic natural barriers to improving imperfect model sensitivity. Linear stochastic models with multiple spatiotemporal scales are utilized to demonstrate this information theoretic approach to equilibrium sensitivity, the role of increasing spatial resolution in the information metric for model error, and the ability of imperfect models to capture the true sensitivity. Finally, an instructive statistically nonlinear model with many degrees of freedom, mimicking the observed non-Gaussian statistical behavior of tracers in the atmosphere, with corresponding imperfect eddy-diffusivity parameterization models are utilized here. They demonstrate the important role of additional stochastic forcing of imperfect models in order to systematically improve the information theoretic measures of fidelity and sensitivity developed here