19 citations · 75 across the 15 of their papers we have counts for
19 papers
Elementary fractal geometry. 3. Complex Pisot factors imply finite type
Christoph Bandt
Self-similar sets require a separation condition to admit a nice mathematical structure. The classical open set condition (OSC) is difficult to verify. Zerner proved that there is…
Two new parameters for the ordinal analysis of images
Christoph Bandt, Katharina Wittfeld
Local patterns play an important role in statistical physics as well as in image processing. Two-dimensional ordinal patterns were studied by Ribeiro et al. who determined permutat…
Statistics and modelling of order patterns in univariate time series
Christoph Bandt
Order patterns apply well to many fields, because of minimal stationarity assumptions. Here we fix the methodology of patterns of length 3 by introducing an orthogonal system of fo…
A reproduction rate which perfectly fits Covid-19
Christoph Bandt
We present a simple technique to compare the development of the Covid-19 epidemic in different regions, based only on the time series of confirmed cases. Weekly new infections, tak…
Transparent Covid-19 prediction
Christoph Bandt
We present a very simple and transparent method to interpret time series of confirmed Covid-19 cases. Roughly speaking, the analysis of weekly new infections for each day is a tool…
Elementary fractal geometry. Networks and carpets involving irrational rotations
Christoph Bandt, Dmitry Mekhontsev
Self-similar sets with open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in…
Order patterns, their variation and change points in financial time series and Brownian motion
Christoph Bandt
Order patterns and permutation entropy have become useful tools for studying biomedical, geophysical or climate time series. Here we study day-to-day market data, and Brownian moti…
Computer geometry: Rep-tiles with a hole
Christoph Bandt, Dmitry Mekhontsev
A cube is an 8-rep-tile: it is the union of eight smaller copies of itself. Is there a set with a hole which has this property? The computer found an interesting and complicated so…
New relatives of the Sierpinski gasket
Christoph Bandt, Dmitry Mekhontsev
By slight modification of the data of the Sierpinski gasket, keeping the open set condition fulfilled, we obtain self-similar sets with very dense parts, similar to fractals in nat…
Crude EEG parameter provides sleep medicine with well-defined continuous hypnograms
Christoph Bandt
To evaluate EEG data, one can count local maxima and minima on a fine scale, in a sliding window analysis. This straightforward calculation, which simplifies and improves previous…
A single fractal pinwheel tile
Christoph Bandt, Dmitry Mekhontsev, Andrei Tetenov
The pinwheel triangle of Conway and Radin is a standard example for tilings with self-similarity and statistical circular symmetry. Many modifications were constructed, all based o…
The two-dimensional density of Bernoulli Convolutions
Christoph Bandt
Bernoulli convolutions form a one-parameter family of self-similar measures on the unit interval. We suggest to study their two-dimensional density which has an intricate combinato…
Polynomial approximation of self-similar measures and the spectrum of the transfer operator
Christoph Bandt, Helena Peña
We consider self-similar measures on The Hutchinson operator acts on measures and is the dual of the transfer operator which acts on continuous functions. We d…
Finite orbits in multivalued maps and Bernoulli convolutions
Christoph Bandt
Bernoulli convolutions are certain measures on the unit interval depending on a parameter between 1 and 2. In spite of their simple definition, they are not yet well understood…
Autocorrelation type functions for big and dirty data series
Christoph Bandt
One form of big data are signals - time series of consecutive values. In physical experiments, billions of values can now be measured within a second. Signals of heart and brain in…
Conjugacies provided by fractal transformations I : Conjugate measures, Hilbert spaces, orthogonal expansions, and flows, on self-referential spaces
Christoph Bandt, Michael Barnsley, Markus Hegland +1
Theorems and explicit examples are used to show how transformations between self-similar sets (general sense) may be continuous almost everywhere with respect to stationary measure…
Local structure of self-affine sets
Christoph Bandt, Antti Käenmäki
The structure of a self-similar set with open set condition does not change under magnification. For self-affine sets the situation is completely different. We consider planar self…
Differentiability of fractal curves
Christoph Bandt, Alexey Kravchenko
While self-similar sets have no tangents at any single point, self-affine curves can be smooth. We consider plane self-affine curves without double points and with two pieces. Ther…
Combinatorial topology of three-dimensional self-affine tiles
Christoph Bandt
We develop tools to study the topology and geometry of self-affine fractals in dimension three and higher. We use the self-affine structure and obtain rather detailed information a…