Ergodic Jacobi matrices and conformal maps

We study structural properties of the Lyapunov exponent $\gamma$ and the density of states $k$ for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function $w=-\gamma+i\pi k$ as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem

Empirical properties of large covariance matrices

The salient properties of large empirical covariance and correlation matrices are studied for three datasets of size 54, 55 and 330. The covariance is defined as a simple cross product of the returns, with weights that decay logarithmically slowly. The key general properties of the covariance matrices are the following. The spectrum of the covariance is very static, except for the top three to 10 eigenvalues, and decay exponentially fast toward zero. The mean spectrum and spectral density show no particular feature that would separate 'meaningful' from 'noisy' eigenvalues. The spectrum of the correlation is more static, with three to five eigenvalues that have distinct dynamics. The mean projector of rank k on the leading subspace shows that a large part of the dynamics occurs in the eigenvectors. Together, this implies that the reduction of the covariance to a few leading static eigenmodes misses most of the dynamics. Finally, all the analysed properties of the dynamics of the covariance and correlation are similar. This indicates that a covariance estimator correctly evaluates both volatilities and correlations, and separate estimators are not required.Covariance matrix, Correlation, Long memory, Spectrum, Spectral density,

Brain songs framework used for discovering the relevant timescale of the human brain

A key unresolved problem in neuroscience is to determine the relevant timescale for understanding spatiotemporal dynamics across the whole brain. While resting state fMRI reveals networks at an ultraslow timescale (below 0.1 Hz), other neuroimaging modalities such as MEG and EEG suggest that much faster timescales may be equally or more relevant for discovering spatiotemporal structure. Here, we introduce a novel way to generate wholebrain neural dynamical activity at the millisecond scale from fMRI signals. This method allows us to study the different timescales through binning the output of the model. These timescales can then be investigated using a method (poetically named brain songs) to extract the spacetime motifs at a given timescale. Using independent measures of entropy and hierarchy to characterize the richness of the dynamical repertoire, we show that both methods find a similar optimum at a timescale of around 200 ms in resting state and in task data.G.D. is supported by the Spanish Research Project PSI2016-75688-P (AEI/FEDER, EU), by the European Union’s Horizon 2020 Research and Innovation Programme under Grant Agreement Nos. 720270 (HBP SGA1) and 785907 (HBP SGA2), and by the Catalan AGAUR Programme 2017 SGR 1545. M.L.K. is supported by the ERC Consolidator Grant: CAREGIVING (No. 615539), and Center for Music in the Brain, funded by the Danish National Research Foundation (DNRF117)