We introduce two-dimensional periodic arrays that are a variant of the de Bruijn tori. We call them nested perfect arrays. Instead of asking that every array of a given size has exactly one occurrence, we partition the positions in congruence classes and we ask exactly one occurrence in each congruence class. We also ask that this property applies recursively to each of the subarrays. We give a method to construct nested perfect arrays based on Pascal triangle matrix modulo 2. For the two-symbol alphabet, and for n being a power of 2, we partition the positions of the arrays in$n^{2}$many congruence classes by taking the row number modulo n and the column number modulo n. We construct arrays where each possible$n\times n$array occurs$n^{2}$times, once in each congruence class. Our method yields exponentially many (in$n^{2}$) different nested perfect arrays.

What parts of the classical descriptive set theory done in Polish spaces still hold for more general topological spaces, possibly T0 or T1, but not T2 (i.e. not Hausdorff)? This question has been addressed by Selivanov in a series of papers centred on algebraic domains. And recently it has been considered by de Brecht for quasi-Polish spaces, a framework that contains both countably based continuous domains and Polish spaces. In this paper, we present alternative unifying topological spaces, that we call approximation spaces. They are exactly the spaces for which player Nonempty has a stationary strategy in the Choquet game. A natural proper subclass of approximation spaces coincides with the class of quasi-Polish spaces. We study the Borel and Hausdorff difference hierarchies in approximation spaces, revisiting the work done for the other topological spaces. We also consider the problem of effectivization of these results.

We prove some results on the Wadge order on the space of sets of natural numbers endowed with Scott topology, and more generally, on omega-continuous domains. Using alternating decreasing chains we characterize the property of Wadge hardness for the classes of the Hausdorff difference hierarchy (iterated differences of open sets). A similar characterization holds for Wadge one-to-one and finite-to-one completeness. We consider the same questions for the effectivization of the Wadge relation. We also show that for the space of sets of natural numbers endowed with the Scott topology, in each class of the Hausdorff difference hierarchy there are two strictly increasing chains of Wadge degrees of sets properly in that class. The length of these chains is the rank of the considered class, and each element in one chain is incomparable with all the elements in the other chain.

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/.

Abstract We obtain a large class of significant examples ofn-random reals (i.e., Martin-Löf random in oracle ∅(n−1)) à la Chaitin. Any such real is defined as the probability that a universal monotone Turing machine performing possibly infinite computations on infinite (resp. finite large enough, resp. finite self-delimited) inputs produces an output in a given set . In particular, we develop methods to transfer many-one completeness results of index sets ton-randomness of associated probabilities.

Abstract We consider the question of randomness of the probability ΩU[X] that an optimal Turing machine U halts and outputs a string in a fixed set X. The main results are as follows: • ΩU[X] is random whenever X is Σn0-complete or Πn0-complete for some n ≥ 2. • However, for n ≥ 2, ΩU[X] is not n-random when X is Σn0 or Πn0. Nevertheless, there exists Δn+10 sets such that ΩU[X] is n-random. • There are Δ20 sets X such that ΩU[X] is rational. Also, for every n ≥ 1, there exists a set X which is Δn+10 and Σn0-hard such that ΩU[X] is not random. We also look at the range of ΩU as an operator. We prove that the set {ΩU[X]: X ⊆ 2≤ω} is a finite union of closed intervals. It follows that for any optimal machine U and any sufficiently small real r, there is a set X ⊆ 2≤ω recursive in ∅′ ⊕ r, such that ΩU[X] = r. The same questions are also considered in the context of infinite computations, and lead to similar results.

Abstract Using possibly infinite computations on universal monotone Turing machines, we prove Martin-Löf randomness in ∅′ of the probability that the output be in some set under complexity assumptions about .