Bilinear Fractal Interpolation and Box Dimension

In the context of general iterated function systems (IFSs), we introduce bilinear fractal interpolants as the fixed points of certain Read-Bajraktarevi\'{c} operators. By exhibiting a generalized "taxi-cab" metric, we show that the graph of a bilinear fractal interpolant is the attractor of an underlying contractive bilinear IFS. We present an explicit formula for the box-counting dimension of the graph of a bilinear fractal interpolant in the case of equally spaced data points

Equilibrium states and invariant measures for random dynamical systems

Random dynamical systems with countably many maps which admit countable Markov partitions on complete metric spaces such that the resulting Markov systems are uniformly continuous and contractive are considered. A non-degeneracy and a consistency conditions for such systems, which admit some proper Markov partitions of connected spaces, are introduced, and further sufficient conditions for them are provided. It is shown that every uniformly continuous Markov system associated with a continuous random dynamical system is consistent if it has a dominating Markov chain. A necessary and sufficient condition for the existence of an invariant Borel probability measure for such a non-degenerate system with a dominating Markov chain and a finite (16) is given. The condition is also sufficient if the non-degeneracy is weakened with the consistency condition. A further sufficient condition for the existence of an invariant measure for such a consistent system which involves only the properties of the dominating Markov chain is provided. In particular, it implies that every such a consistent system with a finite Markov partition and a finite (16) has an invariant Borel probability measure. A bijective map between these measures and equilibrium states associated with such a system is established in the non-degenerate case. Some properties of the map and the measures are given.Comment: The article is published in DCDS-A, but without the 3rd paragraph on page 4 (the complete removal of the paragraph became the condition for the publication in the DCDS-A after the reviewer ran out of the citation suggestions collected in the paragraph

Facilitating children's emergent literacy using shared reading: A comparison of two models

This paper investigates early home literacy practices and their influence on preschool children's literacy and reading development. In particular, two recently developed Australian home literacy interventions are reviewed that were based on a parent shared reading and dialogic reading framework. While both interventions facilitated preschool children's reading development and parent involvement, each intervention had a different focus. One of the interventions was designed for children with language delays and it concentrated on motivating book reading. The second intervention was designed for children with a family history of reading disability, and this intervention concentrated more on children's alphabetical and phonological awareness development, along with home reading. The strategies used for both interventions have the potential to be incorporated into mainstream early childhood literacy education and tuition

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

Did changing primary care delivery models change performance? A population based study using health administrative data

<p>Abstract</p> <p>Background</p> <p>Primary care reform in Ontario, Canada started with the introduction of new enrollment models, the two largest of which are Family Health Networks (FHNs), a capitation-based model, and Family Health Groups (FHGs), a blended fee-for-service model. The purpose of this study was to evaluate differences in performance between FHNs and FHGs and to compare performance before and after physicians joined these new primary care groups.</p> <p>Methods</p> <p>This study used Ontario administrative claims data to compare performance measures in FHGs and FHNs. The study population included physicians who belonged to a FHN or FHG for at least two years. Patients were included in the analyses if they enrolled with a physician in the two years after the physician joined a FHN or FHG, and also if they saw the physician in a two year period prior to the physician joining a FHN or FHG. Performance was derived from the administrative data, and included measures of preventive screening for cancer (breast, cervical, colorectal) and chronic disease management (diabetes, heart failure, asthma).</p> <p>Results</p> <p>Performance measures did not vary consistently between models. In some cases, performance approached current benchmarks (Pap smears, mammograms). In other cases it was improving in relation to previous measures (colorectal cancer screening). There were no changes in screening for cervical cancer or breast cancer after joining either a FHN or FHG. Colorectal cancer screening increased in both FHNs and FHGs. After enrolling in either a FHG or a FHN, prescribing performance measures for diabetes care improved. However, annual eye examinations decreased for younger people with diabetes after joining a FHG or FHN. There were no changes in performance measures for heart failure management or asthma care after enrolling in either a FHG or FHN.</p> <p>Conclusions</p> <p>Some improvements in preventive screening and diabetes management which were seen amongst people after they enrolled may be attributed to incentive payments offered to physicians within FHGs and FHNs. However, these primary care delivery models need to be compared with other delivery models and fee for service practices in order to describe more specifically what aspects of model delivery and incentives affect care.</p

A differential method for bounding the ground state energy

For a wide class of Hamiltonians, a novel method to obtain lower and upper bounds for the lowest energy is presented. Unlike perturbative or variational techniques, this method does not involve the computation of any integral (a normalisation factor or a matrix element). It just requires the determination of the absolute minimum and maximum in the whole configuration space of the local energy associated with a normalisable trial function (the calculation of the norm is not needed). After a general introduction, the method is applied to three non-integrable systems: the asymmetric annular billiard, the many-body spinless Coulombian problem, the hydrogen atom in a constant and uniform magnetic field. Being more sensitive than the variational methods to any local perturbation of the trial function, this method can used to systematically improve the energy bounds with a local skilled analysis; an algorithm relying on this method can therefore be constructed and an explicit example for a one-dimensional problem is given.Comment: Accepted for publication in Journal of Physics

Drip Paintings and Fractal Analysis

It has been claimed [1-6] that fractal analysis can be applied to unambiguously characterize works of art such as the drip paintings of Jackson Pollock. This academic issue has become of more general interest following the recent discovery of a cache of disputed Pollock paintings. We definitively demonstrate here, by analyzing paintings by Pollock and others, that fractal criteria provide no information about artistic authenticity. This work has also led to two new results in fractal analysis of more general scientific significance. First, the composite of two fractals is not generally scale invariant and exhibits complex multifractal scaling in the small distance asymptotic limit. Second the statistics of box-counting and related staircases provide a new way to characterize geometry and distinguish fractals from Euclidean objects

Quantum Iterated Function Systems

Iterated functions system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum iterated functions system (QIFS), where functions act randomly with prescribed probabilities in the Hilbert space. In a more general setting a QIFS consists of completely positive maps acting in the space of density operators. We present exemplary classical IFSs, the invariant measure of which exhibits fractal structure, and study properties of the corresponding QIFSs and their invariant states.Comment: 12 pages, 1 figure include