On the resonance eigenstates of an open quantum baker map

We study the resonance eigenstates of a particular quantization of the open baker map. For any admissible value of Planck's constant, the corresponding quantum map is a subunitary matrix, and the nonzero component of its spectrum is contained inside an annulus in the complex plane, $|z_{min}|\leq |z|\leq |z_{max}|$. We consider semiclassical sequences of eigenstates, such that the moduli of their eigenvalues converge to a fixed radius $r$. We prove that, if the moduli converge to $r=|z_{max}|$, then the sequence of eigenstates converges to a fixed phase space measure $\rho_{max}$. The same holds for sequences with eigenvalue moduli converging to $|z_{min}|$, with a different limit measure $\rho_{min}$. Both these limiting measures are supported on fractal sets, which are trapped sets of the classical dynamics. For a general radius $|z_{min}|< r < |z_{max}|$, we identify families of eigenstates with precise self-similar properties.Comment: 32 pages, 2 figure

Spectral problems in open quantum chaos

This review article will present some recent results and methods in the study of 1-particle quantum or wave scattering systems, in the semiclassical/high frequency limit, in cases where the corresponding classical/ray dynamics is chaotic. We will focus on the distribution of quantum resonances, and the structure of the corresponding metastable states. Our study includes the toy model of open quantum maps, as well as the recent quantum monodromy operator method.Comment: Compared with the previous version, misprints and typos have been corrected, and the bibliography update

Training deep neural density estimators to identify mechanistic models of neural dynamics

Mechanistic modeling in neuroscience aims to explain observed phenomena in terms of underlying causes. However, determining which model parameters agree with complex and stochastic neural data presents a significant challenge. We address this challenge with a machine learning tool which uses deep neural density estimators-- trained using model simulations-- to carry out Bayesian inference and retrieve the full space of parameters compatible with raw data or selected data features. Our method is scalable in parameters and data features, and can rapidly analyze new data after initial training. We demonstrate the power and flexibility of our approach on receptive fields, ion channels, and Hodgkin-Huxley models. We also characterize the space of circuit configurations giving rise to rhythmic activity in the crustacean stomatogastric ganglion, and use these results to derive hypotheses for underlying compensation mechanisms. Our approach will help close the gap between data-driven and theory-driven models of neural dynamics

Extremum feedback with partial knowledge

A scalable feedback mechanism to solicit feedback from a potentially very large group of networked nodes is an important building block for many network protocols. Multicast transport protocols use it for negative acknowledgements and for delay and packet loss determination. Grid computing and peer-to-peer applications can use similar approaches to find nodes that are, at a given moment in time, best suited to serve a request. In sensor networks, such mechanisms allow to report extreme values in a resource efficient way. In this paper we analyze several extensions to the exponential feedback algorithm [5,6] that provide an optimal way to collect extreme values from a potentially very large group of networked nodes. In contrast to prior work, we focus on how knowledge about the value distribution in the group can be used to optimize the feedback process. We describe the trade-offs that have to be decided upon when using these extensions and provide additional insight into their performance by means of simulation. Furthermore, we briefly illustrate how sample applications can benefit from the proposed mechanisms

Fractal Weyl law for Linux Kernel Architecture

We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be $\nu \approx 0.63$ that corresponds to the fractal dimension of the network $d \approx 1.2$. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension $d<2$.Comment: RevTex 6 pages, 7 figs, linked to arXiv:1003.5455[cs.SE]. Research at http://www.quantware.ups-tlse.fr/, Improved version, changed forma

Probabilistic Weyl laws for quantized tori

For the Toeplitz quantization of complex-valued functions on a $2n$-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law. In numerical experiments the same Weyl law also holds for ``false'' eigenvalues created by pseudospectral effects.Comment: 33 pages, 3 figures, v2 corrected listed titl

Convergence of random zeros on complex manifolds

We show that the zeros of random sequences of Gaussian systems of polynomials of increasing degree almost surely converge to the expected limit distribution under very general hypotheses. In particular, the normalized distribution of zeros of systems of m polynomials of degree N, orthonormalized on a regular compact subset K of C^m, almost surely converge to the equilibrium measure on K as the degree N goes to infinity.Comment: 16 page

Rates of convergence of nonextensive statistical distributions to Levy distributions in full and half spaces

The Levy-type distributions are derived using the principle of maximum Tsallis nonextensive entropy both in the full and half spaces. The rates of convergence to the exact Levy stable distributions are determined by taking the N-fold convolutions of these distributions. The marked difference between the problems in the full and half spaces is elucidated analytically. It is found that the rates of convergence depend on the ranges of the Levy indices. An important result emerging from the present analysis is deduced if interpreted in terms of random walks, implying the dependence of the asymptotic long-time behaviors of the walks on the ranges of the Levy indices if N is identified with the total time of the walks.Comment: 20 page

Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives

The classical fields with fractional derivatives are investigated by using the fractional Lagrangian formulation.The fractional Euler-Lagrange equations were obtained and two examples were studied.Comment: 9 page

Fractional differentiability of nowhere differentiable functions and dimensions

Weierstrass's everywhere continuous but nowhere differentiable function is shown to be locally continuously fractionally differentiable everywhere for all orders below the `critical order' 2-s and not so for orders between 2-s and 1, where s, 1<s<2 is the box dimension of the graph of the function. This observation is consolidated in the general result showing a direct connection between local fractional differentiability and the box dimension/ local Holder exponent. Levy index for one dimensional Levy flights is shown to be the critical order of its characteristic function. Local fractional derivatives of multifractal signals (non-random functions) are shown to provide the local Holder exponent. It is argued that Local fractional derivatives provide a powerful tool to analyze pointwise behavior of irregular signals.Comment: minor changes, 19 pages, Late