Papers by Christoph Bandt
arXiv (Cornell University), Oct 21, 2019
Order patterns and permutation entropy have become useful tools for studying biomedical, geophysi... more Order patterns and permutation entropy have become useful tools for studying biomedical, geophysical or climate time series. Here we study day-today market data, and Brownian motion which is a good model for their order patterns. A crucial point is that for small lags (1 up to 6 days), pattern frequencies in financial data remain essentially constant. The two most important order parameters of a time series are turning rate and up-down balance. For change points in EEG brain data, turning rate is excellent while for financial data, up-down balance seems the best. The fit of Brownian motion with respect to these parameters is tested, providing a new version of a forgotten test by Bienaymé.
International Conference and Workshop on Fractals and Wavelets
Fractals and wavelets are emerging areas of mathematics with many common factors which can be use... more Fractals and wavelets are emerging areas of mathematics with many common factors which can be used to develop new technologies. This volume contains the selected contributions from the lectures and plenary and invited talks given at the International Workshop and Conference on Fractals and Wavelets held at Rajagiri School of Engineering and Technology, India from November 9-12, 2013. Written by experts, the contributions hope to inspire and motivate researchers working in this area. They provide more insight into the areas of fractals, self similarity, iterated function systems, wavelets and the applications of both fractals and wavelets. This volume will be useful for the beginners as well as experts in the fields of fractals and wavelets
Critical Care Medicine, Feb 1, 2017
Deutschland GmbH and Becton, Dickinson and Company during the conduct of the study. He disclosed ... more Deutschland GmbH and Becton, Dickinson and Company during the conduct of the study. He disclosed government work. His institution received funding from grant from BMBF and HICARE; from grants from the BMBF, Germany; and personal fees and nonfinancial support from Dräger Medical Deutschland GmbH, Pfizer Deutschland GmbH and Becton, Dickinson and Company during the conduct of the study. The remaining authors have disclosed that they do not have any potential conflicts of interest.
Journal of Combinatorial Theory, Series A, Nov 1, 1976
An (n, k)-permutation design is a collection of k! permutations on n elements such that for any c... more An (n, k)-permutation design is a collection of k! permutations on n elements such that for any choice of k elements all k! permutations of these elements appear as subpermutations in the collection. For every n we construct an (n, n-1)-permutation design. There is a rather unique (6,4)-design but no (5,3)-and (7,4)-design. Permutation designs and ordered triple systems are special cases of ordered designs. For all n $; 2 (mod 3) we have an ordered triple system containing no triples twice. 3241 3214 3214 4132 3412 3412 423 1 4231 4213 384
Permutation Entropy and Order Patterns in Long Time Series
Contributions to statistics, 2016
While ordinal techniques are commonplace in statistics, they have been introduced to time series ... more While ordinal techniques are commonplace in statistics, they have been introduced to time series fairly recently by Hallin and coauthors. Permutation entropy, an average of frequencies of order patterns, was suggested by Bandt and Pompe in 2002 and used by many authors as a complexity measure in physics, medicine, engineering, and economy. Here a modified version is introduced, the “distance to white noise.” For datasets with tens of thousands or even millions of values, which are becoming standard in many fields, it is possible to study order patterns separately, determine certain differences of their frequencies, and define corresponding autocorrelation type functions. In contrast to classical autocorrelation, these functions are invariant with respect to nonlinear monotonic transformations of the data. For order three patterns, a variance-analytic “Pythagoras formula” combines the different autocorrelation functions with our new version of permutation entropy. We demonstrate the use of such correlation type functions in sliding window analysis of biomedical and environmental data.
Springer eBooks, Dec 19, 2007
Two communication lines in a 3 × 3 matrix speller
Journal of Neural Engineering, May 10, 2011
In a matrix speller both eye fixation and attention are reflected in the event-related potentials... more In a matrix speller both eye fixation and attention are reflected in the event-related potentials to highlighting of characters. We consider the question of whether this can be used to open up two parallel channels by dissociating the attended character from the fixated one. Complementary classifiers for both characters are established and their relationship is investigated.
Pacific Journal of Mathematics, 1986
Chaos, Jun 1, 2018
By slight modification of the data of the Sierpi nski gasket, keeping the open set condition fulf... more By slight modification of the data of the Sierpi nski gasket, keeping the open set condition fulfilled, we obtain self-similar sets with very dense parts, similar to fractals in nature and in random models. This is caused by a complicated structure of the open set and is revealed only under magnification. Thus, the family of self-similar sets with separation condition is much richer and has higher modelling potential than usually expected. An interactive computer search for such examples and new properties for their classification are discussed.
Nonlinearity, May 1, 1993
By the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly con... more By the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling map h on the circle T. In particular, a connected and locally connected Julia set can be considered as a topological factor T/ R« of T with respect to a special h-invariant equivalence relation « o n l , which is called Julia equivalence by Keller. Following an idea of Thurston, Bandt and Keller have investigated a map a-> & a from T onto the set of all Julia equivalences, which gives a natural abstract description of the Mandelbrot set. By the use of a symbol sequence called the kneading sequence of the point a, they gave a topological classification of the abstract Julia sets T/ & a. It turns out that T/ «" contains simple closed curves iff the point a has a periodic kneading sequence. The present article characterizes the set of points possessing a periodic kneading sequence and discusses this set in relation to Julia sets and to the Mandelbrot set.
Aequationes Mathematicae, Feb 1, 1997
In this paper we solve the functional equation x(u + v)e(u-v) = f,(u)g~ (v) +A(u)g2(v) under the ... more In this paper we solve the functional equation x(u + v)e(u-v) = f,(u)g~ (v) +A(u)g2(v) under the assumption that Z, q~,fJ,f2,gl,g2 are complex-valued functions on R", n e N arbitrary, and X ~ 0 and q~ ~ 0 are continuous. Our main result shows that, apart from degeneracy and some obvious modifications, theta functions of one complex variable are the only continuous solutions of this functional equation.
Fractals, Wavelets, and their Applications : Contributions from the International Conference and Workshop on Fractals and Wavelets
Springer eBooks, 2014
Part I: Fractal Theory.- Introduction to Fractals.- Geometry of self similar sets.- An introducti... more Part I: Fractal Theory.- Introduction to Fractals.- Geometry of self similar sets.- An introduction to Julia and Fatou set.- Crazy topology in complex dynamics.- Measure preserving fractal homeomorphisms.- The dimension theory of almost self affine sets and measures.- Countable alphabet non autonomous self affine sets.- On transverse hyperplanes to self similar Jordan arcs.- Fractals in product fuzzy metric space.- Some properties on Koch curve.- Projections of Mandelbrot percolation in higher dimensions.- Some examples of finite type fractals in three dimensional space.- Fractals in partial metric spaces.- Part II: Wavelet Theory.- Frames and extension problems I.- Frames and extension problems II.- Local fractal functions and function spaces.- Some historical precedents of the fractal functions.- A new class of rational quadratic fractal function with positive shape preservation.- Interval wavelets determined by points on the circle.- Construction of multi scaling functions using matrix polynomials.- A remark on reconstruction of splines from their local weighted average samples.- C1rational cubic fractal interpolation surface using functional values.- On fractal rational functions.- Part III: Applications of Fractals and Wavelets.- Innovation on the tortuous path: Fractal Electronics.- Permutation entropy analysis of EEG of mild cognitive impairment patients during memory activation task.- A multifractal based image analysis for cervical dysplasia classification.- Self similar network traffic modeling using fractal point process Markovian approach.- Validation of variance based fitting for self similar network traffic.- Self similar network traffic modeling using circulant Markov modulated poisson process.- Investigation of priority based optical packet switch under self similar variable length input traffic using matrix queuing theory.- Computationally efficient wavelet domain solver for fluorescence diffuse optical tomography.- Implementation of wavelet based and discrete cosine based algorithms on panchromatic image.- Trend, time series and wavelet analysis of river water dynamics.- An efficient wavelet based approximation method to film - pore diffusion model arising in chemical engineering.- A new wavelet based hybrid method for Fisher type equations.
Conditionally Monotone Functions and Three Sperner Type Conditions
Journal of Automata, Languages and Combinatorics, 1981
Advances in Mathematics, 2018
Bernoulli convolutions are certain measures on the unit interval depending on a parameter β betwe... more Bernoulli convolutions are certain measures on the unit interval depending on a parameter β between 1 and 2. In spite of their simple definition, they are not yet well understood. We study their two-dimensional density which exists by a theorem of Solomyak. To each Bernoulli convolution, there is an interval D called the overlap region, and a map which assigns two values to each point of D and one value to all other points of [0, 1]. There are two types of finite orbits of these multivalued maps which correspond to zeros and potential singularities of the density, respectively. Orbits which do not meet D belong to an ordinary map called β-transformation and exist for all β > 1.6182. They were studied by Erdos, Joo, Komornik, Sidorov, de Vries and others as points with unique addresses, and by Jordan, Shmerkin and Solomyak as points with maximal local dimension. In the two-dimensional view, these orbits form address curves related to the Milnor-Thurston itineraries in one-dimensional dynamics. The curves depend smoothly on the parameter and represent quantiles of all corresponding Bernoulli convolutions. Finite orbits which intersect D have a network-like structure and can exist only at Perron parameters β. Their points are intersections of extended address curves, and can have finite or countable number of addresses, as found by Sidorov. For an uncountable number of parameters, the central point 1 2 has only two addresses. The intersection of periodic address curves can lead to singularities of the measures. We give examples which are not Pisot or Salem parameters.
Lecture Notes in Mathematics, 1992
By the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly con... more By the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling map h on the circle T. In particular, a connected and locally connected Julia set can be considered as a topological factor T/ R« of T with respect to a special h-invariant equivalence relation « o n l , which is called Julia equivalence by Keller. Following an idea of Thurston, Bandt and Keller have investigated a map a-> & a from T onto the set of all Julia equivalences, which gives a natural abstract description of the Mandelbrot set. By the use of a symbol sequence called the kneading sequence of the point a, they gave a topological classification of the abstract Julia sets T/ & a. It turns out that T/ «" contains simple closed curves iff the point a has a periodic kneading sequence. The present article characterizes the set of points possessing a periodic kneading sequence and discusses this set in relation to Julia sets and to the Mandelbrot set.
Journal of Statistical Physics, Aug 1, 2005
Two fundamental models of critical phenomena are connected. We show that the discrete Bak-Sneppen... more Two fundamental models of critical phenomena are connected. We show that the discrete Bak-Sneppen evolution model is conjugate to the classical contact process. This holds in discrete and continuous time, on all graphs and for random as well as for deterministic choice of neighbors. Thus the extensive theory for the contact process applies to the discrete Bak-Sneppen model, too.
Nonlinearity, Aug 12, 2011
While self-similar sets have no tangents at any single point, self-affine curves can be smooth. W... more 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. There is an open subset of parameter space for which the curve is differentiable at all points except for a countable set. For a parameter set of codimension one, the curve is continuously differentiable. However, there are no twice differentiable self-affine curves in the plane, except for parabolic arcs.
Fundamenta Mathematicae, 1994
The horseshoe or bucket handle continuum, defined as the inverse limit of the tent map, is one of... more The horseshoe or bucket handle continuum, defined as the inverse limit of the tent map, is one of the standard examples in continua theory as well as in dynamical systems. It is not arcwise connected. Its arcwise components coincide with composants, and with unstable manifolds in the dynamical setting. Knaster asked whether these composants are all homeomorphic, with the obvious exception of the zero composant. Partial results were obtained by Bellamy (1979), Dębski and Tymchatyn (1987), and Aarts and Fokkink (1991). We answer Knaster's question in the affirmative. The main tool is a very simple type of symbolic dynamics for the horseshoe and related continua.
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Papers by Christoph Bandt