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

NCUWM Talk Abstracts 2010

Dr. Bryna Kra, Northwestern University “From Ramsey Theory to Dynamical Systems and Back” Dr. Karen Vogtmann, Cornell University “Ping-Pong in Outer Space” Lindsay Baun, College of St. Benedict Danica Belanus, University of North Dakota Hayley Belli, University of Oregon Tiffany Bradford, Saint Francis University Kathryn Bryant, Northern Arizona University Laura Buggy, College of St. Benedict Katharina Carella, Ithaca College Kathleen Carroll, Wheaton College Elizabeth Collins-Wildman, Carleton College Rebecca Dorff, Brigham Young University Melisa Emory, University of Nebraska at Omaha Avis Foster, George Mason University Xiaojing Fu, Clarkson University Jennifer Garbett, Kenyon College Nicki Gaswick, University of Nebraska-Lincoln Rita Gnizak, Fort Hays State University Kailee Gray, University of South Dakota Samantha Hilker, Sam Houston State University Ruthi Hortsch, University of Michigan Jennifer Iglesias, Harvey Mudd College Laura Janssen, University of Nebraska-Lincoln Laney Kuenzel, Stanford University Ellen Le, Pomona College Thu Le, University of the South Shauna Leonard, Arkansas State University Tova Lindberg, Bethany Lutheran College Lisa Moats, Concordia College Kaitlyn McConville, Westminster College Jillian Neeley, Ithaca College Marlene Ouayoro, George Mason University Kelsey Quarton, Bradley University Brooke Quisenberry, Hope College Hannah Ross, Kenyon College Karla Schommer, College of St. Benedict Rebecca Scofield, University of Iowa April Scudere, Westminster College Natalie Sheils, Seattle University Kaitlin Speer, Baylor University Meredith Stevenson, Murray State University Kiri Sunde, University of North Carolina Kaylee Sutton, John Carroll University Frances Tirado, University of Florida Anna Tracy, University of the South Kelsey Uherka, Morningside College Danielle Wheeler, Coe College Lindsay Willett, Grove City College Heather Williamson, Rice University Chengcheng Yang, Rice University Jie Zeng, Michigan Technological Universit

The Einstein Relation on Metric Measure Spaces

This note is based on F. Burghart's master thesis at Stuttgart university from July 2018, supervised by Prof. Freiberg. We review the Einstein relation, which connects the Hausdorff, local walk and spectral dimensions on a space, in the abstract setting of a metric measure space equipped with a suitable operator. This requires some twists compared to the usual definitions from fractal geometry. The main result establishes the invariance of the three involved notions of fractal dimension under bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more generally, how the transport of the analytic and stochastic structure behind the Einstein relation works. While any homeomorphism suffices for this transport of structure, non-Lipschitz maps distort the Hausdorff and the local walk dimension in different ways. To illustrate this, we take a look at H\"older regular transformations and how they influence the local walk dimension and prove some partial results concerning the Einstein relation on graphs of fractional Brownian motions. We conclude by giving a short list of further questions that may help building a general theory of the Einstein relation.Comment: 28 pages, 3 figure

Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension

A fundamental problem in the dimension theory of self-affine sets is the construction of high-dimensional measures which yield sharp lower bounds for the Hausdorff dimension of the set. A natural strategy for the construction of such high-dimensional measures is to investigate measures of maximal Lyapunov dimension; these measures can be alternatively interpreted as equilibrium states of the singular value function introduced by Falconer. Whilst the existence of these equilibrium states has been well-known for some years their structure has remained elusive, particularly in dimensions higher than two. In this article we give a complete description of the equilibrium states of the singular value function in the three-dimensional case, showing in particular that all such equilibrium states must be fully supported. In higher dimensions we also give a new sufficient condition for the uniqueness of these equilibrium states. As a corollary, giving a solution to a folklore open question in dimension three, we prove that for a typical self-affine set in $\mathbb{R}^3$, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension

Fractal dimension of a random invariant set

AbstractIn recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in Debussche's paper and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2d Navier–Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations

Practical implementation of nonlinear time series methods: The TISEAN package

Nonlinear time series analysis is becoming a more and more reliable tool for the study of complicated dynamics from measurements. The concept of low-dimensional chaos has proven to be fruitful in the understanding of many complex phenomena despite the fact that very few natural systems have actually been found to be low dimensional deterministic in the sense of the theory. In order to evaluate the long term usefulness of the nonlinear time series approach as inspired by chaos theory, it will be important that the corresponding methods become more widely accessible. This paper, while not a proper review on nonlinear time series analysis, tries to make a contribution to this process by describing the actual implementation of the algorithms, and their proper usage. Most of the methods require the choice of certain parameters for each specific time series application. We will try to give guidance in this respect. The scope and selection of topics in this article, as well as the implementational choices that have been made, correspond to the contents of the software package TISEAN which is publicly available from http://www.mpipks-dresden.mpg.de/~tisean . In fact, this paper can be seen as an extended manual for the TISEAN programs. It fills the gap between the technical documentation and the existing literature, providing the necessary entry points for a more thorough study of the theoretical background.Comment: 27 pages, 21 figures, downloadable software at http://www.mpipks-dresden.mpg.de/~tisea

An investigation into Quadtree fractal image and video compression

Digital imaging is the representation of drawings, photographs and pictures in a format that can be displayed and manipulated using a conventional computer. Digital imaging has enjoyed increasing popularity over recent years, with the explosion of digital photography, the Internet and graphics-intensive applications and games. Digitised images, like other digital media, require a relatively large amount of storage space. These storage requirements can become problematic as demands for higher resolution images increases and the resolution capabilities of digital cameras improve. It is not uncommon for a personal computer user to have a collection of thousands of digital images, mainly photographs, whilst the Internet’s Web pages present a practically infinite source. These two factors 一 image size and abundance 一 inevitably lead to a storage problem. As with other large files, data compression can help reduce these storage requirements. Data compression aims to reduce the overall storage requirements for a file by minimising redundancy. The most popular image compression method, JPEG, can reduce the storage requirements for a photographic image by a factor of ten whilst maintaining the appearance of the original image 一 or can deliver much greater levels of compression with a slight loss of quality as a trade-off. Whilst JPEG's efficiency has made it the definitive image compression algorithm, there is always a demand for even greater levels of compression and as a result new image compression techniques are constantly being explored. One such technique utilises the unique properties of Fractals. Fractals are relatively small mathematical formulae that can be used to generate abstract and often colourful images with infinite levels of detail. This property is of interest in the area of image compression because a detailed, high-resolution image can be represented by a few thousand bytes of formulae and coefficients rather than the more typical multi-megabyte filesizes. The real challenge associated with Fractal image compression is to determine the correct set of formulae and coefficients to represent the image a user is trying to compress; it is trivial to produce an image from a given formula but it is much, much harder to produce a formula from a given image. เท theory, Fractal compression can outperform JPEG for a given image and quality level, if the appropiate formulae can be determined. Fractal image compression can also be applied to digital video sequences, which are typically represented by a long series of digital images 一 or 'frames'