30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)

Optimized ancestral state reconstruction using Sankoff parsimony

<p>Abstract</p> <p>Background</p> <p>Parsimony methods are widely used in molecular evolution to estimate the most plausible phylogeny for a set of characters. Sankoff parsimony determines the minimum number of changes required in a given phylogeny when a cost is associated to transitions between character states. Although optimizations exist to reduce the computations in the number of taxa, the original algorithm takes time <it>O</it>(<it>n</it>²) in the number of states, making it impractical for large values of <it>n</it>.</p> <p>Results</p> <p>In this study we introduce an optimization of Sankoff parsimony for the reconstruction of ancestral states when ultrametric or additive cost matrices are used. We analyzed its performance for randomly generated matrices, Jukes-Cantor and Kimura's two-parameter models of DNA evolution, and in the reconstruction of elongation factor-1<it>α </it>and ancestral metabolic states of a group of eukaryotes, showing that in all cases the execution time is significantly less than with the original implementation.</p> <p>Conclusion</p> <p>The algorithms here presented provide a fast computation of Sankoff parsimony for a given phylogeny. Problems where the number of states is large, such as reconstruction of ancestral metabolism, are particularly adequate for this optimization. Since we are reducing the computations required to calculate the parsimony cost of a single tree, our method can be combined with optimizations in the number of taxa that aim at finding the most parsimonious tree.</p

Comparative Bacterial Proteomics: Analysis of the Core Genome Concept

While comparative bacterial genomic studies commonly predict a set of genes indicative of common ancestry, experimental validation of the existence of this core genome requires extensive measurement and is typically not undertaken. Enabled by an extensive proteome database developed over six years, we have experimentally verified the expression of proteins predicted from genomic ortholog comparisons among 17 environmental and pathogenic bacteria. More exclusive relationships were observed among the expressed protein content of phenotypically related bacteria, which is indicative of the specific lifestyles associated with these organisms. Although genomic studies can establish relative orthologous relationships among a set of bacteria and propose a set of ancestral genes, our proteomics study establishes expressed lifestyle differences among conserved genes and proposes a set of expressed ancestral traits

Evaluation schemes in the ring of quaternionic polynomials

In this paper we focus on computational aspects associated with polynomial problems in the ring of one-sided quaternionic polynomials. The complexity and error bounds of quaternion arithmetic are considered and several evaluation schemes are analyzed from their complexity point of view. The numerical stability of generalized Horner’s and Goertzel’s algorithms to evaluate polynomials with quaternion floating-point coefficients is addressed. Numerical tests illustrate the behavior of the algorithms from the point of view of performance and accuracy.Research at CMAT was financed by Portuguese Funds through FCT - Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013. Research at NIPE was carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683 (UID/ECO/03182/2013), with the FCT/MEC’s (Fundação para a Ciência e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on “Competitiveness and Internationalization - COMPETE 2020” under the PT2020 Partnership Agreement.info:eu-repo/semantics/publishedVersio