Fractals from genomes: exact solutions of a biology-inspired problem

This is a review of a set of recent papers with some new data added. After a brief biological introduction a visualization scheme of the string composition of long DNA sequences, in particular, of bacterial complete genomes, will be described. This scheme leads to a class of self-similar and self-overlapping fractals in the limit of infinitely long constotuent strings. The calculation of their exact dimensions and the counting of true and redundant avoided strings at different string lengths turn out to be one and the same problem. We give exact solution of the problem using two independent methods: the Goulden-Jackson cluster method in combinatorics and the method of formal language theory.Comment: 24 pages, LaTeX, 5 PostScript figures (two in color), psfi

Fractal structure in the Chinese yuan/US dollar rate

Price changes of the Chinese yuan/US dollar rate are found to display a Sierpinski triangle in an Iterative Function System clumpiness test. This fractal structure commonly emerges in “the chaos gameâ€, where randomness coexists with deterministic rules. We show that a threshold model with four states, two deterministic and two stochastic is able to replicate the properties of the yuan/dollar returns in general, and the Sierpinski triangle in particular.

The Chinese Chaos Game

The yuan-dollar returns prior to the 2005 revaluation show a Sierpinski triangle in an iterated function system clumpiness test. Yet the fractal vanishes after the revaluation. The Sierpinski commonly emerges in the chaos game, where randomness coexists with deterministic rules [2, 3]. Here it is explained by the yuan’s pegs to the US dollar, which made more than half of the data points close to zero. Extra data from the Brazilian and Argentine experiences do confirm that the fractal emerges whenever exchange rate pegs are kept for too long.

Celulární automat a CML systémy

The main aim of this thesis is the study of cellular automata and discrete dynamical systems on a lattice. Both tools, cellular automata as well as dynamical systems on a lattice are introduced and elementary properties described. The relation between cellular automata and dynamical system on lattice is derived. The main goal of the thesis is also the use of the cellular automata as that mathematical tool of evolution visualization of discrete dynamical systems. The theory of cellular automata is applied to the discrete dynamical systems on a lattice Laplacian type and implemented in Java language.Hlavním cílem práce je studium vztahu celulárních automatů a diskrétních dynamických systémů na mřížce. Oba nástroje, jak celulární automat tak dynamický systém na mřížce, jsou zavedeny a jejich základní vlastnosti popsány. Vztah mezi celulárními automaty a dynamickými systémy na mřížce je podrobně popsán. Hlavním cílem práce je dále použití nástroje celulárního automatu jako matematického vizualizačního prostředku evoluce diskrétních dynamických systémů. Teorie celulárních automatů je použita na dynamické systémy na mřížce Lamplaceova typu a implementována v prostředí Java.470 - Katedra aplikované matematikyvelmi dobř

Universal sequence map (USM) of arbitrary discrete sequences

BACKGROUND: For over a decade the idea of representing biological sequences in a continuous coordinate space has maintained its appeal but not been fully realized. The basic idea is that any sequence of symbols may define trajectories in the continuous space conserving all its statistical properties. Ideally, such a representation would allow scale independent sequence analysis – without the context of fixed memory length. A simple example would consist on being able to infer the homology between two sequences solely by comparing the coordinates of any two homologous units. RESULTS: We have successfully identified such an iterative function for bijective mappingψ of discrete sequences into objects of continuous state space that enable scale-independent sequence analysis. The technique, named Universal Sequence Mapping (USM), is applicable to sequences with an arbitrary length and arbitrary number of unique units and generates a representation where map distance estimates sequence similarity. The novel USM procedure is based on earlier work by these and other authors on the properties of Chaos Game Representation (CGR). The latter enables the representation of 4 unit type sequences (like DNA) as an order free Markov Chain transition table. The properties of USM are illustrated with test data and can be verified for other data by using the accompanying web-based tool:http://bioinformatics.musc.edu/~jonas/usm/. CONCLUSIONS: USM is shown to enable a statistical mechanics approach to sequence analysis. The scale independent representation frees sequence analysis from the need to assume a memory length in the investigation of syntactic rules

Qubism: self-similar visualization of many-body wavefunctions

A visualization scheme for quantum many-body wavefunctions is described, which we have termed qubism. Its main property is its recursivity: increasing the number of qubits reflects in an increase in the image resolution. Thus, the plots are typically fractal. As examples, we provide images for the ground states of commonly used Hamiltonians in condensed matter and cold atom physics, such as Heisenberg or ITF. Many features of the wavefunction, such as magnetization, correlations and criticality, can be visualized as properties of the images. In particular, factorizability can be easily spotted, and a way to estimate the entanglement entropy from the image is provided