1,240 research outputs found
Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences
We consider infinite sequences of superstable orbits (cascades) generated by
systematic substitutions of letters in the symbolic dynamics of one-dimensional
nonlinear systems in the logistic map universality class. We identify the
conditions under which the topological entropy of successive words converges as
a double exponential onto the accumulation point, and find the convergence
rates analytically for selected cascades. Numerical tests of the convergence of
the control parameter reveal a tendency to quantitatively universal
double-exponential convergence. Taking a specific physical example, we consider
cascades of stable orbits described by symbolic sequences with the symmetries
of quasilattices. We show that all quasilattices can be realised as stable
trajectories in nonlinear dynamical systems, extending previous results in
which two were identified.Comment: This version: updated figures and added discussion of generalised
time quasilattices. 17 pages, 4 figure
Kinematic Diffraction from a Mathematical Viewpoint
Mathematical diffraction theory is concerned with the analysis of the
diffraction image of a given structure and the corresponding inverse problem of
structure determination. In recent years, the understanding of systems with
continuous and mixed spectra has improved considerably. Simultaneously, their
relevance has grown in practice as well. In this context, the phenomenon of
homometry shows various unexpected new facets. This is particularly so for
systems with stochastic components. After the introduction to the mathematical
tools, we briefly discuss pure point spectra, based on the Poisson summation
formula for lattice Dirac combs. This provides an elegant approach to the
diffraction formulas of infinite crystals and quasicrystals. We continue by
considering classic deterministic examples with singular or absolutely
continuous diffraction spectra. In particular, we recall an isospectral family
of structures with continuously varying entropy. We close with a summary of
more recent results on the diffraction of dynamical systems of algebraic or
stochastic origin.Comment: 30 pages, invited revie
Chaos in computer performance
Modern computer microprocessors are composed of hundreds of millions of
transistors that interact through intricate protocols. Their performance during
program execution may be highly variable and present aperiodic oscillations. In
this paper, we apply current nonlinear time series analysis techniques to the
performances of modern microprocessors during the execution of prototypical
programs. Our results present pieces of evidence strongly supporting that the
high variability of the performance dynamics during the execution of several
programs display low-dimensional deterministic chaos, with sensitivity to
initial conditions comparable to textbook models. Taken together, these results
show that the instantaneous performances of modern microprocessors constitute a
complex (or at least complicated) system and would benefit from analysis with
modern tools of nonlinear and complexity science
The noncommutative Kubo Formula: Applications to Transport in Disordered Topological Insulators with and without Magnetic Fields
The non-commutative theory of charge transport in mesoscopic aperiodic
systems under magnetic fields, developed by Bellissard, Shulz-Baldes and
collaborators in the 90's, is complemented with a practical numerical
implementation. The scheme, which is developed within a -algebraic
framework, enable efficient evaluations of the non-commutative Kubo formula,
with errors that vanish exponentially fast in the thermodynamic limit.
Applications to a model of a 2-dimensional Quantum spin-Hall insulator are
given. The conductivity tensor is mapped as function of Fermi level, disorder
strength and temperature and the phase diagram in the plane of Fermi level and
disorder strength is quantitatively derived from the transport simulations.
Simulations at finite magnetic field strength are also presented.Comment: 10 figure
Data-Adaptive Wavelets and Multi-Scale Singular Spectrum Analysis
Using multi-scale ideas from wavelet analysis, we extend singular-spectrum
analysis (SSA) to the study of nonstationary time series of length whose
intermittency can give rise to the divergence of their variance. SSA relies on
the construction of the lag-covariance matrix C on M lagged copies of the time
series over a fixed window width W to detect the regular part of the
variability in that window in terms of the minimal number of oscillatory
components; here W = M Dt, with Dt the time step. The proposed multi-scale SSA
is a local SSA analysis within a moving window of width M <= W <= N.
Multi-scale SSA varies W, while keeping a fixed W/M ratio, and uses the
eigenvectors of the corresponding lag-covariance matrix C_M as a data-adaptive
wavelets; successive eigenvectors of C_M correspond approximately to successive
derivatives of the first mother wavelet in standard wavelet analysis.
Multi-scale SSA thus solves objectively the delicate problem of optimizing the
analyzing wavelet in the time-frequency domain, by a suitable localization of
the signal's covariance matrix. We present several examples of application to
synthetic signals with fractal or power-law behavior which mimic selected
features of certain climatic and geophysical time series. A real application is
to the Southern Oscillation index (SOI) monthly values for 1933-1996. Our
methodology highlights an abrupt periodicity shift in the SOI near 1960. This
abrupt shift between 4 and 3 years supports the Devil's staircase scenario for
the El Nino/Southern Oscillation phenomenon.Comment: 24 pages, 19 figure
Estimation of the control parameter from symbolic sequences: Unimodal maps with variable critical point
The work described in this paper can be interpreted as an application of the
order patterns of symbolic dynamics when dealing with unimodal maps.
Specifically, it is shown how Gray codes can be used to estimate the
probability distribution functions (PDFs) of the order patterns of parametric
unimodal maps. Furthermore, these PDFs depend on the value of the parameter,
what eventually provides a handle to estimate the parameter value from symbolic
sequences (in form of Gray codes), even when the critical point depends on the
parameter.Comment: 10 pages, 14 figure
Switching Regression Models and Causal Inference in the Presence of Discrete Latent Variables
Given a response and a vector of predictors,
we investigate the problem of inferring direct causes of among the vector
. Models for that use all of its causal covariates as predictors enjoy
the property of being invariant across different environments or interventional
settings. Given data from such environments, this property has been exploited
for causal discovery. Here, we extend this inference principle to situations in
which some (discrete-valued) direct causes of are unobserved. Such cases
naturally give rise to switching regression models. We provide sufficient
conditions for the existence, consistency and asymptotic normality of the MLE
in linear switching regression models with Gaussian noise, and construct a test
for the equality of such models. These results allow us to prove that the
proposed causal discovery method obtains asymptotic false discovery control
under mild conditions. We provide an algorithm, make available code, and test
our method on simulated data. It is robust against model violations and
outperforms state-of-the-art approaches. We further apply our method to a real
data set, where we show that it does not only output causal predictors, but
also a process-based clustering of data points, which could be of additional
interest to practitioners.Comment: 46 pages, 14 figures; real-world application added in Section 5.2;
additional numerical experiments added in the Appendix
Generalized Fast-Convolution-based Filtered-OFDM: Techniques and Application to 5G New Radio
This paper proposes a generalized model and methods for fast-convolution
(FC)-based waveform generation and processing with specific applications to
fifth generation new radio (5G-NR). Following the progress of 5G-NR
standardization in 3rd generation partnership project (3GPP), the main focus is
on subband-filtered cyclic prefix (CP) orthogonal frequency-division
multiplexing (OFDM) processing with specific emphasis on spectrally well
localized transmitter processing. Subband filtering is able to suppress the
interference leakage between adjacent subbands, thus supporting different
numerologies for so-called bandwidth parts as well as asynchronous multiple
access. The proposed generalized FC scheme effectively combines overlapped
block processing with time- and frequency-domain windowing to provide highly
selective subband filtering with very low intrinsic interference level. Jointly
optimized multi-window designs with different allocation sizes and design
parameters are compared in terms of interference levels and implementation
complexity. The proposed methods are shown to clearly outperform the existing
state-of-the-art windowing and filtering-based methods.Comment: To appear in IEEE Transactions on Signal Processin
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