Fractals

Consider a family of bounded domains Ω_t in the plane (or more generally any Euclidean space) that depend analytically on the parameter t, and consider the ordinary Neumann Laplacian Δ_t on each of them. Then we can organize all the eigenfunctions into continuous families with eigenvalues also varying continuously with t, although the relative sizes of the eigenvalues will change with t at crossings where . We call these families homotopies of eigenfunctions. We study two explicit examples. The first example has Ω₀ equal to a square and Ω₁ equal to a circle; in both cases the eigenfunctions are known explicitly, so our homotopies connect these two explicit families. In the second example we approximate the Sierpinski carpet starting with a square, and we continuously delete subsquares of varying sizes. (Data available in full at ).

Automatic generation of colored patterns with discrete planar symmetries is considered from a dynamical system's point of view. Invariant mappings with such symmetries are constructed to serve as the density functions for the generation of colorful images.

This paper researches the dynamic behavior of a general form of the Fibonacci function, which is a quasi-sine Fibonacci function. It analyses the fixed points of the quasi-sine Fibonacci function on the real axis and the complex plane, and then constructs the Julia set of it using the escape-time method, discovering that the Julia set is fractal and it is on the x-axis symmetry. Using the conception of critical point, the quasi-sine Fibonacci function is generalized. Later the paper examines the dynamic behavior of the generalized quasi-sine Fibonacci function on critical points, and finds that the Mandelbrot set is also on the x-axis symmetry. Finally, it is discovered that there is a jumping phenomenon on the critical points.

The size distributions of solid particles and pores in porous media are approximately hierarchical and statistically fractals. In this paper, a model for single-phase fractal media is constructed, and the analytical expressions for area, fractal dimension and distribution function for solid particles are derived. The distribution function of solid particles obtained from the proposed model is in good agreement with available experimental data. Then, a model for approximate two-phase fractal media is developed. Good agreement is found between the predicted fractal dimensions for pore space from the two-phase fractal medium model and the existing measured data. A model for approximate three-phase fractal media is also presented by extending the obtained two-phase model.

We prove that if a recurrent double sequence coincide in a sufficiently large starting square with a double sequence produced by context-free substitutions, then they must coincide everywhere. We apply this result for some examples.

We perform a Lévy index analysis of relativistic shower particles produced in the interactions of ³²S nuclei at 200A GeV with nuclear emulsion using the results of factorial moment F_q in two-dimensional anisotropic (η - ϕ) space. We carry out the same investigation for target fragments produced in ³²S-AgBr interactions at 200A GeV energy for both events in emission angle space and azimuthal angle space. The analysis reveals that for pions the value of Lévy index μ is 1.491 ± 0.025 for ring-like events and μ ~ 2.004 ± 0.054 for jet-like events which indicates different degree of multifractality. In case of target fragments jet-like events show a value of μ ~ 1.871 ± 0.010 whereas ring-like events yields an unphysical value (μ > 2) the implication of which need to be explored.

In this paper, we detected the fractal/multifractal and asymmetric properties in a simple financial market model which is an analog of the Ising model. We introduced the virtual market with heterogeneous agents characterized by agents with bounded rationality, by which we mean that agents only have local information, and a market maker who is responsible for market liquidity. To investigate the heterogeneity and psychological factors in real financial market, we designed the parameters of individual expectations of agents to this model. Applying fractal/multifractal and Zipf techniques, we conducted many simulations under different scenarios and then analyzed the generated time series of this virtual market. We acquired some nontrivial findings: first, the virtual price returns generated by our model display fractal and multifractal features; secondly, we found that the price have the asymmetric behaviors; finally, our findings have qualitative similarities with many empirical results, which imply that although our toy model is seemingly simple, it can generate complex dynamics and thus can be a useful tool to investigate complex market behaviors and phenomena.

The stochastic nature of oil price fluctuations is investigated over a twelve-year period, borrowing feedback from an existing database (USA Energy Information Administration database, available online). We evaluate the scaling exponents of the fluctuations by employing different statistical analysis methods, namely rescaled range analysis (R/S), scale windowed variance analysis (SWV) and the generalized Hurst exponent (GH) method. Relying on the scaling exponents obtained, we apply a rescaling procedure to investigate the complex characteristics of the probability density functions (PDFs) dominating oil price fluctuations. It is found that PDFs exhibit scale invariance, and in fact collapse onto a single curve when increments are measured over microscales (typically less than 30 days). The time evolution of the distributions is well fitted by a Lévy-type stable distribution. The relevance of a Lévy distribution is made plausible by a simple model of nonlinear transfer. Our results also exhibit a degree of multifractality as the PDFs change and converge toward to a Gaussian distribution at the macroscales.

The framework of a new scale invariant analysis on a Cantor set C ⊂ I = [0,1], presented recently¹ is clarified and extended further. For an arbitrarily small ε > 0, elements in I\C satisfying , x ∈ C together with an inversion rule are called relative infinitesimals relative to the scale ε. A non-archimedean absolute value , ε → 0 is assigned to each such infinitesimal which is then shown to induce a non-archimedean structure in the full Cantor set C. A valued measure constructed using the new absolute value is shown to give rise to the finite Hausdorff measure of the set. The definition of differentiability on C in the non-archimedean sense is introduced. The associated Cantor function is shown to relate to the valuation on C which is then reinterpreated as a locally constant function in the extended non-archimedean space. The definitions and the constructions are verified explicitly on a Cantor set which is defined recursively from I deleting q number of open intervals each of length leaving out p numbers of closed intervals so that p + q = r.

In this paper, we introduce fractal interpolation functions (FIFs) and linear FIFs on a post critically finite (p.c.f. for short) self-similar set K. We present a sufficient condition such that linear FIFs have finite energy and prove that the solution of Dirichlet problem -Δ_μ u = f,u|_∂K = 0 is a linear FIF on K if f is a linear FIF.

Multifractal box counting analysis has been widely applied to study the scaling characteristics of grayscale images. Since bit depth is an important property of such images it is desirable know the impact of varying bit depths on the estimation of the generalized dimensions (D_q). We generated random geometrical multifractal grayscale fields, which were then transformed from double precision to 16, 13, and 8 bit depths. Digitized grayscale images of soil thin sections at 13 bit depth were also selected for study and transformed to 8 bit depth. The moment based box counting method was applied to evaluate the bit depth effects on D_q. The partition functions for the multifractal fields became noticeably nonlinear on a log-log scale when q ≪ 0 as the bit depth decreased. This trend can be attributed to loss of grayscale details, changes in the local mass distribution, and the occurrence of zeros due to the bit depth transformation and data normalization processes. These effects were most pronounced for positively skewed multifractal fields, with a high proportion of extremely small mass fractions. As a result, the generalized dimensions estimated by linear regression were not always accurate, and an alternative method based on numerical derivatives was explored. The numerical method significantly improved the accuracy of the multifractal analyses; the maximum absolute difference between the analytical and numerically-derived estimates of D_q was only 9.62 × 10^-3. However, when applied to situations in which the box counting scale factor did not match the scale factor used to generate the multifractal field, the numerically-derived estimates of D_q were severely biased. In this case, the linear regression method is preferable even though some error may occur due to limited bit depths. All of the soil grayscale images exhibited multifractal scaling characteristics, although there was little effect of bit depth on the resulting D_q values. Because of random fluctuations in the partition functions, the linear regression method proved to be more robust than the numerical derivative method for estimating the generalized dimensions of natural grayscale images.

In 1969, Kannan¹ gave the definition of a new mapping which had presented a condition which is more lenient than contraction condition. The purpose of this note is to introduce K-Iterated Function System using Kannan mapping which will cover a larger range of mappings. We also prove the Collage theorem for the K-Iterated Function System.