9,138 research outputs found
Dimension Theory of Graphs and Networks
Starting from the working hypothesis that both physics and the corresponding
mathematics have to be described by means of discrete concepts on the
Planck-scale, one of the many problems one has to face in this enterprise is to
find the discrete protoforms of the building blocks of continuum physics and
mathematics. A core concept is the notion of dimension. In the following we
develop such a notion for irregular structures like (large) graphs and networks
and derive a number of its properties. Among other things we show its stability
under a wide class of perturbations which is important if one has 'dimensional
phase transitions' in mind. Furthermore we systematically construct graphs with
almost arbitrary 'fractal dimension' which may be of some use in the context of
'dimensional renormalization' or statistical mechanics on irregular sets.Comment: 20 pages, 7 figures, LaTex2e, uses amsmath, amsfonts, amssymb,
latexsym, epsfi
Trigonometric series and self-similar sets
Let be a self-similar set on associated to contractions
, , for some finite ,
such that is not a singleton. We prove that if is
irrational for some , then is a set of multiplicity, that is,
trigonometric series are not in general unique in the complement of . No
separation conditions are assumed on . We establish our result by showing
that every self-similar measure on is a Rajchman measure: the Fourier
transform as . The rate of
is also shown to be logarithmic if is diophantine for some . The proof is based on quantitative
renewal theorems for random walks on .Comment: 18 pages, v2: improved the main theore
Wavelet analysis of the multivariate fractional Brownian motion
The work developed in the paper concerns the multivariate fractional Brownian
motion (mfBm) viewed through the lens of the wavelet transform. After recalling
some basic properties on the mfBm, we calculate the correlation structure of
its wavelet transform. We particularly study the asymptotic behavior of the
correlation, showing that if the analyzing wavelet has a sufficient number of
null first order moments, the decomposition eliminates any possible long-range
(inter)dependence. The cross-spectral density is also considered in a second
part. Its existence is proved and its evaluation is performed using a von
Bahr-Essen like representation of the function \sign(t) |t|^\alpha. The
behavior of the cross-spectral density of the wavelet field at the zero
frequency is also developed and confirms the results provided by the asymptotic
analysis of the correlation
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
Special issue: Fractal functions and applications
This volume gathers some important advances in the fields of fractional calculus and fractal curves and functions. Fractional derivatives and integrals play an increasingly important role in applied science, and these types of models are ubiquitous in the current scientific literature. The references [1, 2] are devoted to fractional calculus and an application of it to a coronavirus spreading model. The first one studies three procedures of inverse Laplace Transforms: A Sinc–Thiele approximation, a Sinc and a Sinc–Gaussian (SG) method. Both Sinc versions are exact methods of inverse Laplace Transforms. The author proves that SG-based transformations present some advantages over the pure Sinc version regarding stability and convergence properties. The convergence is of exponential type. All the methods presented are applied to Mittag-Leffler functions depending on one, two and three parameters, and the author proves that the representation of this kind of functions is very effective. The author concludes that even for variable-order fractional differential or integral equations, the Sinc–Gaussian method is a powerful procedure..
Singularity and similarity detection for signals using the wavelet transform
The wavelet transform and related techniques are used to analyze singular and fractal signals. The normalized wavelet scalogram is introduced to detect singularities including jumps, cusps and other sharply changing points. The wavelet auto-covariance is applied to estimate the self-similarity exponent for statistical self-affine signals
Dimensions, Maximal Growth Sites and Optimization in the Dielectric Breakdown Model
We study the growth of fractal clusters in the Dielectric Breakdown Model
(DBM) by means of iterated conformal mappings. In particular we investigate the
fractal dimension and the maximal growth site (measured by the Hoelder exponent
) as a function of the growth exponent of the DBM model.
We do not find evidence for a phase transition from fractal to non-fractal
growth for a finite -value. Simultaneously, we observe that the limit of
non-fractal growth () is consistent with .
Finally, using an optimization principle, we give a recipe on how to estimate
the effective value of from temporal growth data of fractal aggregates.Comment: 5 pages, 7 figures; v2: extra figures and new data adde
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