3-Adic Cantor function on local fields and its p-adic derivative

The iterative elimination of the middle spacing in the random division of intervals with two points ldquoat randomrdquo - in the narrow sense of uniformly distributed - generates a random middle Cantor set. We compute the Hausdorff dimension (which intuitively evaluates how ldquodenserdquo a set is) of the random middle third Cantor set, and we verify that although the deterministic middle third Cantor set is the expectation of the random middle third Cantor set, it is more dense than its stochastic counterpart. This can be explained by the dependence of order statistics.

Advanced prerequisite for E-infinity theory

Golden mean relevance for chaos inhibition in a system of two coupled modified van der Pol oscillators

Starting from the random extension of the Cantor middle set in [0,1], by iteratively removing the central uniform spacing from the intervals remaining in the previous step, we define random Beta(p,1)Cantor sets, and compute their Hausdorff dimension. Next we define a deterministic counterpart, by iteratively removing the expected value of the spacing defined by the appropriate Beta(p,1) order statistics. We investigate the reasons why the Hausdorff dimension of this deterministic fractal is greater than the Hausdorff dimension of the corresponding random fractals.

Stochastic self-similar and fractal universe

Roughness is, among human sensations, just as fundamental as color or pitch, or as heaviness or hotness. But its study had remained in a more primitive state, by far. The reason was that both geometry and science were first drawn to smooth shapes. Thus, color and pitch came to be measured in cycles per seconds, that is, were reduced to sinusoids, in other words to uniform motions around a circle - the epitome of a smooth shape. A study of roughness had necessarily to wait until specific mathematical tools had been discovered and, much later, suitably interpreted. Fractal geometry began when I reinterpreted the flight from nature that had led mathematicians to conceive of notions like the Holder exponent, the Cantor set, or the Hausdorff dimension. They boasted of these notions being ‘monstrous’ but in fact I turned them over into everyday tools of science. I also added further tools that - taken together - made roughness quantitatively measurable for the first time. Acquiring a quantitative measure is the step that moves a field into maturity. And this move instantly led to a striking conjecture. In 1984, ‘Nature’ published an article I wrote with D. A. Passoja and A. J. Paullay on metal fractures. We found that the traditional measures of their roughness range all over. To the contrary, their fractal roughness varies very little not only between samples but also between materials. Last time I checked the “universality” had been extended but not explained. The new intrinsic measure created a major intellectual mystery. The first major new tool that I added to those contributed by the likes of Holder, Cantor, and Hausdorff was multifractality, for both measures and functions. I was motivated by the urge to model the intermittence of turbulence but my first full paper (in 1972) also noted that the same techniques ought to apply to the intermittence in the variation of financial prices. An ancient adage claimed that the City of London is as unpredictable as the weather. I found unexpectedly quantitative truth to this adage by showing that both phenomena can be tackled with essentially the same tools. Roughness is everywhere therefore fractal geometry has little fear of running out of problems. This address will sketch the fractal geometry of roughness and explore some new developments relevant to this Congress.

Abstract. The predictability of the monthly North Atlantic Oscillation, NAO, index is analysed from the point of view of different fractal concepts and dynamic system theory such as lacunarity, rescaled analysis (Hurst exponent) and reconstruction theorem (embedding and correlation dimensions, Kolmogorov entropy and Lyapunov exponents). The main results point out evident signs of randomness and the necessity of stochastic models to represent time evolution of the NAO index. The results also show that the monthly NAO index behaves as a white-noise Gaussian process. The high minimum number of nonlinear equations needed to describe the physical process governing the NAO index fluctuations is evidence of its complexity. A notable predictive instability is indicated by the positive Lyapunov exponents. Besides corroborating the complex time behaviour of the NAO index, present results suggest that random Cantor sets would be an interesting tool to model lacunarity and time evolution of the NAO index.

At its most basic level physics starts with space-time topology and geometry. On the other hand topology’s and geometry’s simplest and most basic elements are random Cantor sets. It follows then that nonlinear dynamics i.e. deterministic chaos and fractal geometry is the best mathematical theory to apply to the problems of high energy particle physics and cosmology. In the present work we give a short survey of some recent achievements of applying nonlinear dynamics to notoriously difficult subjects such as quantum entanglement as well as the origin and true nature of dark energy, negative absolute temperature and the fractal meaning of the constancy of the speed of light.

Part I Foundations: mathematical background Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractals products of fractals intersections of fractals. Part II Applications and examples: fractals defined by transformations examples from number theory graphs of functions examples from pure mathematics dynamical systems iteration of complex functions-Julia sets random fractals Brownian motion and Brownian surfaces multifractal measures physical applications.

Let F1 and F2 be independent copies of one-dimensional correlated fractal percolation, with almost sure Hausdorff dimensions dimH(F1) and dimH(F2). Consider the following question: does dimH(F1) + dimH(F2) > 1 imply that their algebraic difference F1 ? F2 will contain an interval? The well known Palis conjecture states that ‘generically’ this should be true. Recent work by Kuijvenhoven and the first author (Dekking and Kuijvenhoven in J. Eur. Math. Soc., to appear) on random Cantor sets cannot answer this question as their condition on the joint survival distributions of the generating process is not satisfied by correlated fractal percolation. We develop a new condition which permits us to solve the problem, and we prove that the condition of Dekking and Kuijvenhoven (J. Eur. Math. Soc., to appear) implies our condition. Independently of this we give a solution to the critical case, yielding that a strong version of the Palis conjecture holds for fractal percolation and correlated fractal percolation: the algebraic difference contains an interval almost surely if and only if the sum of the Hausdorff dimensions of the random Cantor sets exceeds one.

The problems associated with the visible set have been explored by various scholars. In this paper, we investigate the Hausdorff dimension and the topological properties of the visible set in relation to the products of homogeneous Cantor sets. To address these issues and establish our results, we employ beta expansion theory, numerical calculations and several technical results from fractal geometry. Our research reveals that the case of the homogeneous Cantor set differs from those of the middle Cantor sets. Furthermore, we identify a critical number that is linked to both the Hausdorff dimension and the topological properties of the visible set.

We establish various new results on a problem proposed by Mahler [Some suggestions for further research. Bull. Aust. Math. Soc.29 (1984), 101–108] concerning rational approximation to fractal sets by rational numbers inside and outside the set in question. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich, and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover, we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle-third set $C$ to the set $C$, in terms of the denominator $q$. We further discuss patterns of rational numbers in fractal sets. We highlight two of them: firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets; and secondly, we find properties of the denominator structure of rational points in ‘missing-digit’ Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

Exponential demand for broadband applications has elicited renewed interest in characterizing noise in power line communication (PLC) networks for optimal error performance analysis. PLC is economical and ubiquitous, however due to multipath phenomena and nonstationary impulsive noise, the PLC channel is very harsh to signal transmission. Recent studies reveal that PLC noise is cyclostationary and modelling its characteristic should exploit the spectral and temporal redundancy. In addition, preliminary research shows that PLC noise exhibits self-similarity and can be predicted by fractal theory. In modelling noise employing fractal theory, the Hausdorff dimension (D) needs to be approximated. D is related to Hurst parameter (H), which is a measure of self-similarity. Hence, it is important to estimate H for accurate noise prediction using fractal theory. H parameter estimators assume stationarity in time series data. Therefore, preprocessing is required for the case of PLC noise before estimating H to make it stationary as periodicity and nonstatioanrity affects the H estimators. In this study, we remove the periodicity hidden in the PLC noise obtained from measurement campaign within several university offices, workshops and laboratories. Then long-range dependence is estimated from this pre-processed noise data by employing three estimation algorithms: R/S (rescaled adjusted range) method, aggregated variance method and absolute values of aggregated series method. Results from the said estimation methods though not the same, show that the Hurst parameters obtained from various noise measurements are greater than 0.5, implying presence of self-similarity with long-term memory. Results from this study will find valuable applications in PLC noise modelling using fractal and chaos theory and in error performance analysis in both narrowband and broadband PLC systems.

We prove that the temporal autocorrelation function C(t) for quantum systems with Cantor spectra has an algebraic decay C(t)\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$, where \ensuremath{\delta} equals the generalized dimension ${\mathit{D}}_{2}$ of the spectral measure and is bounded by the Hausdorff dimension ${\mathit{D}}_{0}$\ensuremath{\ge}\ensuremath{\delta}. We study various incommensurate systems with singular continuous and absolutely continuous Cantor spectra and find extremely slow correlation decays in singular continuous cases (\ensuremath{\delta}=0.14 for the critical Harper model and 0\ensuremath{\le}0.84 for the Fibonacci chains). In the kicked Harper model we deomonstrate that the quantum mechanical decay is unrelated to the existence of classical chaos.

In this paper, we consider a family of random Cantor sets on the line and consider the question of whether the condition that the sum of the Hausdorff dimensions is larger than one implies the existence of interior points in the difference set of two independent copies. We give a new and complete proof that this is the case for the random Cantor sets introduced by Per Larsson.