A journey through computability, topology and analysis

This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and descriptive set theory. We will especially focus on Weihrauch reducibility, as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of the open and clopen Ramsey theorems. In particular, since there is not a canonical way to phrase these theorems as multi-valued functions, we identify 8 different multi-valued functions (5 corresponding to the open Ramsey theorem and 3 corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch and arithmetic Weihrauch reducibility. We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. These notions turn out to be extremely relevant when exploring the Weihrauch degree of the problem DS of computing descending sequences in ill-founded linear orders. They allow us to show that DS, and the Weihrauch equivalent problem BS of finding bad sequences through non-well quasi-orders, while being very "hard" to solve, are rather weak in terms of uniform computational strength. We then generalize DS and BS by considering Gamma-presented orders, where Gamma is a Borel pointclass or Delta11, Sigma11, Pi11. We study the obtained DS-hierarchy and BS-hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level. Finally, we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and of Salem sets. We first work in the hyperspace K([0,1]) of compact subsets of [0,1] and show that the closed Salem sets form a Pi03-complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in K([0,1]^d). We also generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still Pi03-complete when we endow K(R^d) with the Fell topology. A similar result holds also for the Vietoris topology. We conclude by showing how these results can be used to characterize the Weihrauch degree of the functions computing the Hausdorff and Fourier dimensions

Weighted Tree Automata -- May it be a little more?

This is a book on weighted tree automata. We present the basic definitions and some of the important results in a coherent form with full proofs. The concept of weighted tree automata is part of Automata Theory and it touches the area of Universal Algebra. It originated from two sources: weighted string automata and finite-state tree automata

Mythic Discourse Analysis

Peer reviewe

A study on cerebral activity by means of combined EEG-fMRI in neuropsychological disorders in childhood

This study is composed by two parts, both focusing on post-calcarine ventral, occipito-temporal visual pathway (“ventral stream”), and on occipito- temporal cortex, structures involved in images and in face processing. In the first part of the study I have analyzed gamma-band ERSP (event-related spectral perturbations) and fMRI BOLD activations in response to recognizable and not recognizable visual stimuli, in typical children and in children affected by "ventral type" Cerebral Visual Impairment, trying to show how the deficits in "ventral" tasks could be investigated using both a neurophysiological and a neuroimaging approach. However I was not able to reproduce preliminary, promising data on gamma-band ERSP because of excessive electrical noise during EEG recordings, most likely because of an equipment radical and unexpected change. Despite these issues, taking advantage of the peculiar features and strength points of the new equipment (a dense-array EEG machine), I continued my work on visual perception and the occipito- temporal visual network using ERPs recordings (part 2), that are substantially less affected from the AC electrical noise usually present in every EEG recording. In particular, I recorded high density ERP responses to neutral and emotional visual face stimuli in typical children and in children affected by Autism Spectrum Disorder, a condition in which face-processing neural networks have been often found dysfunctional in neurophysiological and functional neuroimaging studies. However, evidence regarding face processing in Autism Spectrum Disorders is still contradictory and neurophysiological methods used are heterogeneous. Therefore I designed and applied an experimental paradigm trying to control most of the known or suspected confounding variables in this kind of studies. Using neutral and emotional faces, and trees as non-face stimuli, I was able to modulate both latency and amplitude of the main face-sensitive ERPs (N170, P1, peak-to-peak N170) as a function of stimulus and group conditions. These findings support the hypothesis of an early (first 200 msec) impairment in both neutral and emotional face processing in children affected by Autism Spectrum Disorders

Genomics analyses in kelp species

Kelps are large brown algae in the order Laminariales. Kelp species have different distribution ranges along temperate to Polar rocky coastal lines. We sequenced DNA from three Laminaria species namely Laminaria digitata, Laminaria solidungula and Laminaria rodriguezii. Laminaria digitata is found in the Northern Atlantic region with a southern boundary in Brittany (France) or Massachusetts (USA) and a northern boundary in the Artic. Laminaria solidungula is endemic to the Artic and Laminaria rodriguezii is restricted to deep waters of Mediterranean Sea. Currently, not much is known about the nuclear and organellar genomes of kelp species. To initiate the analysis of sequencing data in kelp species the organellar genomes of Laminaria species were generated. The mitochondrial and chloroplast genomes of Laminaria rodriguezii and Laminaria solidungula, and chloroplast genome of Laminaria digitata were analysed and compared with phylogenetic trees derived from publicly available complete mitochondrial and chloroplast kelp genomes. All analysed kelp organellar genomes were found collinear, where large insertion, deletion (indels) or rearrangements were rare with some essential exceptions. Laminaria rodriguezii is very closely related to the North Atlantic temperate to Arctic Laminaria digitata according to the chloroplast and mitochondrial phylogeny. In the mitochondrial genome of Laminaria rodriguezii a stretch of more than 700 base pairs was found, which was not present in any other kelp sequenced so far. The translated Open Reading Frame (ORF) matches a protein coding region in the mitochondrial genome from Desmarestia viridis, a brown seaweed with a cold-temperate to Arctic distribution in the order Desmarestiales, which is closely related to the Laminariales. The high similarity of overlapping parts of two ORFs suggests that it originated through independent introduction, potentially by infection with similar mitoviruses, which is currently known in fungi and plants only. In the chloroplast genomes of Laminaria solidungula a small rearrangement at the inverted repeat regions was found. These rearrangements led to the pseudogenisation of ycf37 gene in Laminaria solidungula, a gene possibly required under high light conditions. This defunct gene might be one of the reasons why the habitat ranges of Laminaria solidungula is restricted to lowlight sublittoral sites in the incomplete lineage sorting of chloroplast genomes in kelp species. This work laid the foundation for analysis of nuclear genome (ca. 400Mb) of Laminaria digitata. The Single nucleotide polymorphism analysis yielded a first glimpse into the population diversity of this species. The draft genome analysis of Laminaria digitata will be part of the comprehensive analysis of brown algal genomes in the framework of the international Phaeoexplorer project led by the Biologigue de Roscoff in France