Verónica Becher

An example of a computable absolutely normal number

The first example of an absolutely normal number was given by Sierpinski in 1916, twenty years before the concept of computability was formalized. In this note we give a recursive reformulation of Sierpinskis construction which produces a computable absolutely normal number.

Turing's unpublished algorithm for normal numbers

In an unpublished manuscript, Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor errors. While doing this, we recreate Turings ideas as accurately as possible. One of his original lemmas remained unproved, but we have replaced it with a weaker lemma that still allows us to maintain Turings proof idea and obtain his result.

Efficient computation of all perfect repeats in genomic sequences of up to half a gigabyte, with a case study on the human genome

MOTIVATION There is a significant ongoing research to identify the number and types of repetitive DNA sequences. As more genomes are sequenced, efficiency and scalability in computational tools become mandatory. Existing tools fail to find distant repeats because they cannot accommodate whole chromosomes, but segments. Also, a quantitative framework for repetitive elements inside a genome or across genomes is still missing. RESULTS We present a new efficient algorithm and its implementation as a software tool to compute all perfect repeats in inputs of up to 500 million nucleotide bases, possibly containing many genomes. Our algorithm is based on a suffix array construction and a novel procedure to extract all perfect repeats in the entire input, that can be arbitrarily distant, and with no bound on the repeat length. We tested the software on the Homo sapiens DNA genome NCBI 36.49. We computed all perfect repeats of at least 40 bases occurring in any two chromosomes with exact matching. We found that each H.sapiens chromosome shares approximately 10% of its full sequence with every other human chromosome, distributed more or less evenly among the chromosome surfaces. We give statistics including a quantification of repeats by diversity, length and number of occurrences. We compared the computed repeats against all biological repeats currently obtainable from Ensembl enlarged with the output of the dust program and all elements identified by TRF and RepeatMasker (ftp://ftp.ebi.ac.uk/pub/databases/ensembl/jherrero/.repeats/all_repeats.txt.bz2). We report novel repeats as well as new occurrences of repeats matching with known biological elements. AVAILABILITY The source code, results and visualization of some statistics are accessible from http://kapow.dc.uba.ar/patterns/.

A polynomial-time algorithm for computing absolutely normal numbers

We give an algorithm to compute an absolutely normal number so that the first n digits in its binary expansion are obtained in time polynomial in n; in fact, just above quadratic. The algorithm uses combinatorial tools to control divergence from normality. Speed of computation is achieved at the sacrifice of speed of convergence to normality.

Iterable AGM Functions

The issue of iterated theory change is indeed interesting. Legal codes are under constant modification, new discoveries shape scientific theories, and robots ought to update their representation of the world each time a sensor gains new data. A pertinent criticism to the AGM formalism of theory change [Alchourron et al.,1985] is its lack of definition with respect to iterated change. Let’s start by introducing the basic elements in the AGM framework

On extending de Bruijn sequences

We give a complete proof of the following theorem: Every de Bruijn sequence of order n in at least three symbols can be extended to a de Bruijn sequence of order n+1. Every de Bruijn sequence of order n in two symbols can not be extended to order n+1, but it can be extended to order n+2.

Normal numbers and finite automata

We give an elementary and direct proof of the following theorem: A real number is normal to a given integer base if, and only if, its expansion in that base is incompressible by lossless finite-state compressors (these are finite automata augmented with an output transition function such that the automata input-output behaviour is injective; they are also known as injective finite-state transducers). As a corollary we obtain V.N. Agafonovs theorem on the preservation of normality on subsequences selected by finite automata.

A computable absolutely normal Liouville number

Fil: Becher, Veronica Andrea. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computacion; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

On the normality of numbers to different bases

We demonstrate the full logical independence of normality to multiplicatively independent bases. This establishes that the set of bases to which a real number can be normal is not tied to any arithmetical properties other than multiplicative dependence. It also establishes that the set of real numbers which are normal to at least one base is properly at the fourth level of the Borel hierarchy, which was conjectured by A. Ditzen 20 years ago. We further show that the discrepancy functions for multiplicatively independent bases are pairwise independent. In addition, for any given set of bases closed under multiplicative dependence, there are real numbers that are normal to each base in the given set, but not simply normal to any base in its complement. This answers a question first raised by Brown, Moran and Pearce.

Neutral Theory Predicts the Relative Abundance and Diversity of Genetic Elements in a Broad Array of Eukaryotic Genomes

It is universally true in ecological communities, terrestrial or aquatic, temperate or tropical, that some species are very abundant, others are moderately common, and the majority are rare. Likewise, eukaryotic genomes also contain classes or “species” of genetic elements that vary greatly in abundance: DNA transposons, retrotransposons, satellite sequences, simple repeats and their less abundant functional sequences such as RNA or genes. Are the patterns of relative species abundance and diversity similar among ecological communities and genomes? Previous dynamical models of genomic diversity have focused on the selective forces shaping the abundance and diversity of transposable elements (TEs). However, ideally, models of genome dynamics should consider not only TEs, but also the diversity of all genetic classes or “species” populating eukaryotic genomes. Here, in an analysis of the diversity and abundance of genetic elements in >500 eukaryotic chromosomes, we show that the patterns are consistent with a neutral hypothesis of genome assembly in virtually all chromosomes tested. The distributions of relative abundance of genetic elements are quite precisely predicted by the dynamics of an ecological model for which the principle of functional equivalence is the main assumption. We hypothesize that at large temporal scales an overarching neutral or nearly neutral process governs the evolution of abundance and diversity of genetic elements in eukaryotic genomes.