Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations

This paper is concerned with the initial-boundary value problem for the Einstein equations in a first-order generalized harmonic formulation. We impose boundary conditions that preserve the constraints and control the incoming gravitational radiation by prescribing data for the incoming fields of the Weyl tensor. High-frequency perturbations about any given spacetime (including a shift vector with subluminal normal component) are analyzed using the Fourier-Laplace technique. We show that the system is boundary-stable. In addition, we develop a criterion that can be used to detect weak instabilities with polynomial time dependence, and we show that our system does not suffer from such instabilities. A numerical robust stability test supports our claim that the initial-boundary value problem is most likely to be well-posed even if nonzero initial and source data are included.Comment: 27 pages, 4 figures; more numerical results and references added, several minor amendments; version accepted for publication in Class. Quantum Gra

Multivariate Granger Causality and Generalized Variance

Granger causality analysis is a popular method for inference on directed interactions in complex systems of many variables. A shortcoming of the standard framework for Granger causality is that it only allows for examination of interactions between single (univariate) variables within a system, perhaps conditioned on other variables. However, interactions do not necessarily take place between single variables, but may occur among groups, or "ensembles", of variables. In this study we establish a principled framework for Granger causality in the context of causal interactions among two or more multivariate sets of variables. Building on Geweke's seminal 1982 work, we offer new justifications for one particular form of multivariate Granger causality based on the generalized variances of residual errors. Taken together, our results support a comprehensive and theoretically consistent extension of Granger causality to the multivariate case. Treated individually, they highlight several specific advantages of the generalized variance measure, which we illustrate using applications in neuroscience as an example. We further show how the measure can be used to define "partial" Granger causality in the multivariate context and we also motivate reformulations of "causal density" and "Granger autonomy". Our results are directly applicable to experimental data and promise to reveal new types of functional relations in complex systems, neural and otherwise.Comment: added 1 reference, minor change to discussion, typos corrected; 28 pages, 3 figures, 1 table, LaTe

Whitham Averaged Equations and Modulational Stability of Periodic Traveling Waves of a Hyperbolic-Parabolic Balance Law

In this note, we report on recent findings concerning the spectral and nonlinear stability of periodic traveling wave solutions of hyperbolic-parabolic systems of balance laws, as applied to the St. Venant equations of shallow water flow down an incline. We begin by introducing a natural set of spectral stability assumptions, motivated by considerations from the Whitham averaged equations, and outline the recent proof yielding nonlinear stability under these conditions. We then turn to an analytical and numerical investigation of the verification of these spectral stability assumptions. While spectral instability is shown analytically to hold in both the Hopf and homoclinic limits, our numerical studies indicates spectrally stable periodic solutions of intermediate period. A mechanism for this moderate-amplitude stabilization is proposed in terms of numerically observed "metastability" of the the limiting homoclinic orbits.Comment: 27 pages, 5 figures. Minor changes throughou

A modified sequence domain impedance definition and its equivalence to the dq-domain impedance definition for the stability analysis of AC power electronic systems

Representations of AC power systems by frequency dependent impedance equivalents is an emerging technique in the dynamic analysis of power systems including power electronic converters. The technique has been applied for decades in DC-power systems, and it was recently adopted to map the impedances in AC systems. Most of the work on AC systems can be categorized in two approaches. One is the analysis of the system in the \textit{dq}-domain, whereas the other applies harmonic linearization in the phase domain through symmetric components. Impedance models based on analytical calculations, numerical simulation and experimental studies have been previously developed and verified in both domains independently. The authors of previous studies discuss the advantages and disadvantages of each domain separately, but neither a rigorous comparison nor an attempt to bridge them has been conducted. The present paper attempts to close this gap by deriving the mathematical formulation that shows the equivalence between the \textit{dq}-domain and the sequence domain impedances. A modified form of the sequence domain impedance matrix is proposed, and with this definition the stability estimates obtained with the Generalized Nyquist Criterion (GNC) become equivalent in both domains. The second contribution of the paper is the definition of a \textit{Mirror Frequency Decoupled} (MFD) system. The analysis of MFD systems is less complex than that of non-MFD systems because the positive and negative sequences are decoupled. This paper shows that if a system is incorrectly assumed to be MFD, this will lead to an erroneous or ambiguous estimation of the equivalent impedance