423 research outputs found
Non-Markovian Levy diffusion in nonhomogeneous media
We study the diffusion equation with a position-dependent, power-law
diffusion coefficient. The equation possesses the Riesz-Weyl fractional
operator and includes a memory kernel. It is solved in the diffusion limit of
small wave numbers. Two kernels are considered in detail: the exponential
kernel, for which the problem resolves itself to the telegrapher's equation,
and the power-law one. The resulting distributions have the form of the L\'evy
process for any kernel. The renormalized fractional moment is introduced to
compare different cases with respect to the diffusion properties of the system.Comment: 7 pages, 2 figure
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
Dissipation time and decay of correlations
We consider the effect of noise on the dynamics generated by
volume-preserving maps on a d-dimensional torus. The quantity we use to measure
the irreversibility of the dynamics is the dissipation time. We focus on the
asymptotic behaviour of this time in the limit of small noise. We derive
universal lower and upper bounds for the dissipation time in terms of various
properties of the map and its associated propagators: spectral properties,
local expansivity, and global mixing properties. We show that the dissipation
is slow for a general class of non-weakly-mixing maps; on the opposite, it is
fast for a large class of exponentially mixing systems which include uniformly
expanding maps and Anosov diffeomorphisms.Comment: 26 Pages, LaTex. Submitted to Nonlinearit
Fractal Weyl law for chaotic microcavities: Fresnel's laws imply multifractal scattering
We demonstrate that the harmonic inversion technique is a powerful tool to
analyze the spectral properties of optical microcavities. As an interesting
example we study the statistical properties of complex frequencies of the fully
chaotic microstadium. We show that the conjectured fractal Weyl law for open
chaotic systems [W. T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91,
154101 (2003)] is valid for dielectric microcavities only if the concept of the
chaotic repeller is extended to a multifractal by incorporating Fresnel's laws.Comment: 8 pages, 12 figure
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 that
corresponds to the fractal dimension of the network . 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 .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
Stability of Coalescence Hidden variable Fractal Interpolation Surfaces
In the present paper, the stability of Coalescence Hidden variable Fractal
Interpolation Surfaces(CHFIS) is established. The estimates on error in
approximation of the data generating function by CHFIS are found when there is
a perturbation in independent, dependent and hidden variables. It is proved
that any small perturbation in any of the variables of generalized
interpolation data results in only small perturbation of CHFIS. Our results are
likely to be useful in investigations of texture of surfaces arising from the
simulation of surfaces of rocks, sea surfaces, clouds and similar natural
objects wherein the generating function depends on more than one variable
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
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
Delocalization of slowly damped eigenmodes on Anosov manifolds
We look at the properties of high frequency eigenmodes for the damped wave
equation on a compact manifold with an Anosov geodesic flow. We study
eigenmodes with spectral parameters which are asymptotically close enough to
the real axis. We prove that such modes cannot be completely localized on
subsets satisfying a condition of negative topological pressure. As an
application, one can deduce the existence of a "strip" of logarithmic size
without eigenvalues below the real axis under this dynamical assumption on the
set of undamped trajectories.Comment: 28 pages; compared with version 1, minor modifications, add two
reference
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
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