Approximation of Rough Functions

For given $p\in\lbrack1,\infty]$ and $g\in L^{p}\mathbb{(R)}$, we establish the existence and uniqueness of solutions $f\in L^{p}(\mathbb{R)}$, to the equation $$f(x)-af(bx)=g(x),$$ where $a\in\mathbb{R}$, $b\in\mathbb{R} \setminus \{0\}$, and $\left\vert a\right\vert \neq\left\vert b\right\vert ^{1/p}$. Solutions include well-known nowhere differentiable functions such as those of Bolzano, Weierstrass, Hardy, and many others. Connections and consequences in the theory of fractal interpolation, approximation theory, and Fourier analysis are established.Comment: 16 pages, 3 figure

Hurst Coefficient in long time series of population size: Model for two plant populations with different reproductive strategies

Can the fractal dimension of fluctuations in population size be used to estimate extinction risk? The problem with estimating this fractal dimension is that the lengths of the time series are usually too short for conclusive results. This study answered this question with long time series data obtained from an iterative competition model. This model produces competitive extinction at different perturbation intensities for two different germination strategies: germination of all seeds vs. dormancy in half the seeds. This provided long time series of 900 years and different extinction risks. The results support the hypothesis for the effectiveness of the Hurst coefficient for estimating extinction risk

Poisson-to-Wigner crossover transition in the nearest-neighbor spacing statistics of random points on fractals

We show that the nearest-neighbor spacing distribution for a model that consists of random points uniformly distributed on a self-similar fractal is the Brody distribution of random matrix theory. In the usual context of Hamiltonian systems, the Brody parameter does not have a definite physical meaning, but in the model considered here, the Brody parameter is actually the fractal dimension. Exploiting this result, we introduce a new model for a crossover transition between Poisson and Wigner statistics: random points on a continuous family of self-similar curves with fractal dimension between 1 and 2. The implications to quantum chaos are discussed, and a connection to conservative classical chaos is introduced.Comment: Low-resolution figure is included here. Full resolution image available (upon request) from the author

Fractal Dimensions in Perceptual Color Space: A Comparison Study Using Jackson Pollock's Art

The fractal dimensions of color-specific paint patterns in various Jackson Pollock paintings are calculated using a filtering process which models perceptual response to color differences (\Lab color space). The advantage of the \Lab space filtering method over traditional RGB spaces is that the former is a perceptually-uniform (metric) space, leading to a more consistent definition of ``perceptually different'' colors. It is determined that the RGB filtering method underestimates the perceived fractal dimension of lighter colored patterns but not of darker ones, if the same selection criteria is applied to each. Implications of the findings to Fechner's 'Principle of the Aesthetic Middle' and Berlyne's work on perception of complexity are discussed.Comment: 21 pp LaTeX; two postscript figure

SPRAT: Spectrograph for the Rapid Acquisition of Transients

We describe the development of a low cost, low resolution (R ~ 350), high throughput, long slit spectrograph covering visible (4000-8000) wavelengths. The spectrograph has been developed for fully robotic operation with the Liverpool Telescope (La Palma). The primary aim is to provide rapid spectral classification of faint (V ∼ 20) transient objects detected by projects such as Gaia, iPTF (intermediate Palomar Transient Factory), LOFAR, and a variety of high energy satellites. The design employs a volume phase holographic (VPH) transmission grating as the dispersive element combined with a prism pair (grism) in a linear optical path. One of two peak spectral sensitivities are selectable by rotating the grism. The VPH and prism combination and entrance slit are deployable, and when removed from the beam allow the collimator/camera pair to re-image the target field onto the detector. This mode of operation provides automatic acquisition of the target onto the slit prior to spectrographic observation through World Coordinate System fitting. The selection and characterisation of optical components to maximise photon throughput is described together with performance predictions. © (2014) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only

Inter-Intra Molecular Dynamics as an Iterated Function System

The dynamics of units (molecules) with slowly relaxing internal states is studied as an iterated function system (IFS) for the situation common in e.g. biological systems where these units are subjected to frequent collisional interactions. It is found that an increase in the collision frequency leads to successive discrete states that can be analyzed as partial steps to form a Cantor set. By considering the interactions among the units, a self-consistent IFS is derived, which leads to the formation and stabilization of multiple such discrete states. The relevance of the results to dynamical multiple states in biomolecules in crowded conditions is discussed.Comment: 7 pages, 7 figures. submitted to Europhysics Letter

Synthetic Turbulence, Fractal Interpolation and Large-Eddy Simulation

Fractal Interpolation has been proposed in the literature as an efficient way to construct closure models for the numerical solution of coarse-grained Navier-Stokes equations. It is based on synthetically generating a scale-invariant subgrid-scale field and analytically evaluating its effects on large resolved scales. In this paper, we propose an extension of previous work by developing a multiaffine fractal interpolation scheme and demonstrate that it preserves not only the fractal dimension but also the higher-order structure functions and the non-Gaussian probability density function of the velocity increments. Extensive a-priori analyses of atmospheric boundary layer measurements further reveal that this Multiaffine closure model has the potential for satisfactory performance in large-eddy simulations. The pertinence of this newly proposed methodology in the case of passive scalars is also discussed

Fractal Image Coding as Projections Onto Convex Sets

Abstract. We show how fractal image coding can be viewed and gen-eralized in terms of the method of projections onto convex sets (POCS). In this approach, the fractal code denes a set of spatial domain sim-ilarity constraints. We also show how such a reformulation in terms of POCS allows additional contraints to be imposed during fractal image decoding. Two applications are presented: image construction with an incomplete fractal code and image denoising.

On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions

We consider the "Mandelbrot set" $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $\lambda$ in the unit disk such that the attractor $A_\lambda$ of the IFS $\{\lambda z-1, \lambda z+1\}$ is connected. We show that a non-trivial portion of $M$ near the imaginary axis is contained in the closure of its interior (it is conjectured that all non-real points of $M$ are in the closure of the set of interior points of $M$). Next we turn to the attractors $A_\lambda$ themselves and to natural measures $\nu_\lambda$ supported on them. These measures are the complex analogs of much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os and Garsia, we demonstrate how certain classes of complex algebraic integers give rise to singular and absolutely continuous measures $\nu_\lambda$. Next we investigate the Hausdorff dimension and measure of $A_\lambda$, for $\lambda$ in the set $M$, for Lebesgue-a.e. $\lambda$. We also obtain partial results on the absolute continuity of $\nu_\lambda$ for a.e. $\lambda$ of modulus greater than $\sqrt{1/2}$.Comment: 22 pages, 5 figure