<i>Fractals Everywhere</i>

Barnsley, Michael F.; Hurd, Alan J.

doi:10.1119/1.15823

Cited by 206 publications

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“…which is a consequence of the box counting theorem (Barnsley, 1988;Sandau, 1996). The use of the maximal number N max implies several advantageous properties.…”

Section: The Extended Counting Methods (Xcm)mentioning

confidence: 99%

“…Replacing N q X by Q q X in Eq. (1) does not change the fractal dimension (Barnsley, 1988). This is why this dimension has often been denoted as the box dimension.…”

Section: Fractal Dimension and Fractal Setsmentioning

confidence: 99%

“…Are the borders of carbon particles or the cracks in rocks (Flook, 1978;Pape et al, 1987) really images of self-similar sets or at least good approximations? Are fractals really everywhere (Barnsley, 1988;Murray, 1995)? In the case of blood vessels, it is clear that self-similarity may not be assumed if the terminal arterial or venous branches together with the capillaries are represented in the image.…”

Section: Theoretical Considerations and Practical Aspectsmentioning

confidence: 99%

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Measuring fractal dimension and complexity — an alternative approach with an application

Sandau¹,

Kurz

1997

Journal of Microscopy

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SummaryFractal dimension has often been applied as a parameter of complexity, related to, for example, surface roughness, or for classifying textures or line patterns. Fractal dimension can be estimated statistically, if the pattern is known to be self-similar. However, the fractal dimension of more general patterns cannot be estimated, even though the concept may be retained to characterize complexity. We here show that the usual statistical methods, e.g. the box counting method, are not appropriate to measure complexity. A recently developed approach, the extended counting method, whose properties are closer to what fractal dimension means, is considered here in more detail. The methods are applied to geometric and to blood vessel patterns. The weak assumptions about the structure, and the lower variance of the estimate, suggest that the extended counting method has beneficial properties for comparing complexity of naturally occurring patterns.

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“…which is a consequence of the box counting theorem (Barnsley, 1988;Sandau, 1996). The use of the maximal number N max implies several advantageous properties.…”

Section: The Extended Counting Methods (Xcm)mentioning

confidence: 99%

“…Replacing N q X by Q q X in Eq. (1) does not change the fractal dimension (Barnsley, 1988). This is why this dimension has often been denoted as the box dimension.…”

Section: Fractal Dimension and Fractal Setsmentioning

confidence: 99%

Section: Theoretical Considerations and Practical Aspectsmentioning

confidence: 99%

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Measuring fractal dimension and complexity — an alternative approach with an application

Sandau¹,

Kurz

1997

Journal of Microscopy

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“…An especially highly prized result is a "bifurcation," a "line in the parameter space" that separates systems that behave differently (for example, Nowak et al (1994b); Nowak Martin & May Robert (1993); Nowak et al (1994a)). The study of bifurcation is closely tied to the study of fractals, complicated geometrical designs that can evolve from simple mathematical expressions (Mandelbrot, 1983;Barnsley, 1993;Wolfram, 2002 ). Also highly prized, of course, is the opposite result which indicates that a system tends to evolve in a particular direction, regardless of where it starts or how it might be exogenously shocked.…”

Section: Simulation Modeling and Hypothesis Constructionmentioning

confidence: 99%

Monte Carlo Analysis in Academic Research

Johnson

2013

Oxford Handbooks Online

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Monte Carlo analysis is a research strategy that incorporates randomness into the design, implementation or evaluation of theoretical models. It began in the 1940s, when the development of computer hardware and mathematical models made it possible to generate streams of random numbers. These random number streams are combined with mathematical models in order to create models and evaluate theories of random processes. This chapter attempts to tame this diverse, unmanageable collection of concepts and methods by dividing simulation projects into three types. The first, commonly called "Monte Carlo simulation," is used to evaluate statistical estimators. When an estimation procedure is proposed, it is standard procedure to test it against a variety of simulated research problems. A second type of project, referred to as "Markov chain Monte Carlo" (or MCMC), helps researchers to draw conclusions about complicated probability models for which conventional research strategies do not yield insights. The third 1 type of project arises in the study of complex systems, which are characterized by a large number of loosely interconnected, autonomous elements. Commonly known as agent-based models, these simulations have found enthusiastic advocates in environmental and social sciences.Keywords: Monte Carlo, Markov chain Monte Carlo (MCMC), pseudo random number generation (PRNG), Bayesian statistics, agent-based modeling Monte Carlo (MC) analysis is a general term that refers to research that employs random numbers, usually in the form of a computer model (or simulation). Although this research began in the natural sciences, computer science, and mathematics, it is now widely applied in social science as well. This essay attempts to explain the fundamental ideas that spurred the creation of these new procedures as well as their eventual adaptation for use in social science research.This chapter is not a "how to" guide for simulation, but rather it is a "what for" or "why you might want to" guide. Some of the difficulties that arise in MC research projects are considered as well. It begins with some background information on the development of computers and algorithms for random numbers. After that, the chapter takes up applications in the evaluation of proposed statistical estimators, the practice of Bayesian statistics via computer simulation, and investigation of complex systems through agent-based models. Some conclusions about the challenges that face the field are presented, along with a conclusion. A significant part of the presentation is about the exciting developments that have occurred since 1990. Rapid improvements in hardware and software have opened up opportunities for scholars to work with models that were previously prohibited by conceptual and technical barriers. At the current time, we are able to conceptualize and implement models that were simply impossible just 10 years ago. The extremely rapid progress has been driven by a fruitful interaction of substantive researchers in the natural and social science...

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“…If c n is the number of laps (pieces of monotonicity) of f n , then (2) h(f ) = lim n→∞ 1 n log c n (see, e.g., [1]). Since in each lap of f n there can be at most one element of f −n (x), we get (3) h(f ) ≥ lim sup n→∞ 1 n log Card(f −n (x)).…”

Section: Introduction Motivation and Questionsmentioning

confidence: 99%

Counting preimages

Misiurewicz¹,

Rodrigues²

2017

Ergod. Th. Dynam. Sys.

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Abstract. For noninvertible maps, mainly subshifts of finite type and piecewise monotone interval maps, we investigate what happens if we follow backward trajectories, random in the sense that at each step every preimage can be chosen with equal probability. In particular, we ask what happens if we try to compute the entropy this way. It tuns out that instead of the topological entropy we get the metric entropy of a special measure, which we call the fair measure. In general this entropy (the fair entropy) is smaller than the topological entropy. In such a way, for the systems that we consider, we get a new natural measure and a new invariant of topological conjugacy.

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Fractals Everywhere

Cited by 206 publications

References 0 publications

Measuring fractal dimension and complexity — an alternative approach with an application

Measuring fractal dimension and complexity — an alternative approach with an application

Monte Carlo Analysis in Academic Research

Counting preimages

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