Fractals

Applied blindly, the formula for the dimension of the intersection can give negative results. Extending Minkowski's definition of the dimension by ∊-neighborhoods to ∊-pseudo-neighborhoods, that is, replacing (A ∩ B)_∊ with A_∊ ∩ B_∊, we introduce the notion of negative dimensions through several examples of random fractal constructions.

A two-phase phenomenon in three financial exchange prices is studied. To understand the underlying mechanism for the formation of market prices, we perform the multifractal analysis and the detrended fluctuation analysis in terms of time series of market prices. We also examine higher order temporal correlations for the market price. Although the multifractal properties of market prices are obtained, it cannot be reproduced the binomial multiplicative process through that was used to understand fully developed turbulence.

The aim of this paper is twofold. First, with reference to Refs. 1 and 2 we show maximal and minimal box and Hausdorff dimensions over all continuous real functions compactly supported on ℝⁿ, with integrability 0 < p ≤ ∞ and exact smoothness s > 0. We estimate also the box dimensions of trigonometrical Weierstrass-type functions. Secondly, we investigate some criteria in order to decide when the graph of a function is not a d-set. Moreover, for each d between n and n + 1 we construct a function on [0,1]ⁿ the graph of which is a d-set.

We provide a solution to the conjecture of Ref. 1 on the global minimal fractal dimension of Sierpiński petal triangles. We show that the dimension, ln 3/ln 2, of the Sierpiński triangle is the strict minimal value of the dimensions among all Sierpiński pedal triangles associated with acute triangles.

The formulation of a new analysis on a zero measure Cantor set C(⊂I = [0,1]) is presented. A non-Archimedean absolute value is introduced in C exploiting the concept of relative infinitesimals and a scale invariant ultrametric valuation of the form log_ε^-1 (ε/x) for a given scale ε > 0 and infinitesimals 0 < x < ε, x ∈ I\C. Using this new absolute value, a valued (metric) measure is defined on C and is shown to be equal to the finite Hausdorff measure of the set, if it exists. The formulation of a scale invariant real analysis is also outlined, when the singleton {0} of the real line R is replaced by a zero measure Cantor set. The Cantor function is realized as a locally constant function in this setting. The ordinary derivative dx/dt in R is replaced by the scale invariant logarithmic derivative d log x/d log t on the set of valued infinitesimals. As a result, the ordinary real valued functions are expected to enjoy some novel asymptotic properties, which might have important applications in number theory and in other areas of mathematics.

A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order α, 0 < α ≤ 1, called F^α-integral, is defined, which is suitable to integrate functions with fractal support F of dimension α. Further, a derivative of order α, 0 < α ≤ 1, called F^α-derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, "changing" only on a fractal set. The F^α-derivative is local unlike the classical fractional derivative. The F^α-calculus retains much of the simplicity of ordinary calculus. Several results including analogues of fundamental theorems of calculus are proved.

The integral staircase function, which is a generalization of the functions like the Cantor staircase function, plays a key role in this formulation. Further, it gives rise to a new definition of dimension, the γ-dimension.

Spaces of F^α-differentiable and F^α-integrable functions are analyzed. Analogues of Sobolev Spaces are constructed on F and F^α-differentiability is generalized using Sobolev-like construction.

F^α-differential equations are equations involving F^α-derivatives. They can be used to model sublinear dynamical systems and fractal time processes, since sublinear behaviors are associated with staircase-like functions which occur naturally as their solutions. As examples, we discuss a fractal-time diffusion equation, and one-dimensional motion of a particle undergoing friction in a fractal medium.

This manuscript describes three activities connecting the Tower of Hanoi puzzle to three familiar fractal forms. The first connects coin flipping, paper folding, and the Tower of Hanoi to the Dragon Curve. The second illustrates mathematician Ian Stewart's method for showing how the relationships between possible states of the Tower of Hanoi are related to stages in Sierpinski's Gasket. The final activity compares right-left moves of Tower of Hanoi disks to iterations of the Von Koch Curve.

Two current conflicts, in Afghanistan and in Iraq were studied to identify a possibility of a critical behavior in an asymmetric conflict. The analysis was performed using temporal dependence of fatalities. The data included daily fatality information from the beginning of each of the conflicts until 31 December 2007. The results suggest that these examples of asymmetric, counter-insurgency warfare can possibly be characterized in terms of self-organized criticality (SOC). While SOC could be an attractor for such conflicts in general, not all asymmetric conflicts are actually at the point of criticality. The conflict in Afghanistan is an example of such a subcritical conflict. The conflict in Iraq, on the other hand is an example of an already critical system. The two types of conflict have in common the power-law dependence between the numbers of fatalities and the frequency of occurrences. However, while for a critical system the numbers of fatalities are correlated over time, for a subcritical system they are anti-correlated. From the point of view of counter-insurgency, the subcritical state is a preferred option. However, from the system point of view the two cases are just two different phases of the same type of a dynamical system.

We examine a dynamic spatial system of self-aggregating individuals that migrate under mutual gravitational attraction, forming clusters of varying sizes and distributions. To that end, we develop a computational model in which a frictional factor limits agent mobility, which in turn leads to the concentration of population in clusters. Our numerical results reveal that the clustering behavior of the system generates two distinct phases broken down by a phase transition. In the sub-critical phase, the spatial size-distribution of clusters follows the well-known exponential or power-law distribution. By contrast, in the super-critical phase the entire population converges into either a single giant cluster (a "Black-hole"), or a few large clusters ("Oligarchs"). In between these two phases, there exists a threshold critical level corresponding to a phase transition in which clusters abruptly enter a period of rapid growth. The key contribution of this study is to solve for the phase transition regime analytically. We also discuss the application of our results to study the development of cities, the evolution of languages, the growth of asteroids, and the emergence of monopoly and oligopoly.

This paper proposes a fractal image encoding algorithm based on matching error threshold: first the authors set up two kick-out conditions to reduce the capacity of the codebook, and then set up a matching threshold when searching the best matching blocks, which can shorten its runtime greatly. Meanwhile, the authors discard the isometric transformations that are mentioned in most literatures, because the usage of the isometric transformations only increase the computational complexity, the same or even better reconstructed image can achieve through reducing the sliding step of producing domain blocks. Experimental results indicate that the proposed algorithm can both shorten the encoding time greatly and achieve the same or better reconstructed image quality as compared with the basic fractal encoding algorithm with full search.

In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman's lemma and find that Hausdorff dimension can be computed through a counting argument rather than by taking the infimum of a sum of certain covers. This formulation is then applied to obtain a simple proof of the doubling of the dimension of certain sets under a Brownian motion.

When we distribute particles and bonds onto a cubic lattice, we assume that a plaquette is formed, if four corner sites of a square are occupied or four bonds of a square are occupied. When these plaquettes are produced in a cubic lattice and occupy a certain number of square surfaces in the cubic lattice, then a macroscopic surface is formed following a chain construction. This is the second phase transition added to usual site and bond percolation problems where a spreading chain is formed.