Reconstructing Rational Functions with FireFly\texttt{FireFly}

We present the open-source $\texttt{C++}$ library $\texttt{FireFly}$ for the reconstruction of multivariate rational functions over finite fields. We discuss the involved algorithms and their implementation. As an application, we use $\texttt{FireFly}$ in the context of integration-by-parts reductions and compare runtime and memory consumption to a fully algebraic approach with the program $\texttt{Kira}$.Comment: 46 pages, 3 figures, 6 tables; v2: matches published versio

Efficient adaptive sampling applied to multivariate, multiple output rational interpolation models, with applications in electromagnetics-based device modelling

Thesis (PhD) -- Stellenbosch University, 2001.ENGLISH ABSTRACT: A robust and efficient adaptive sampling algorithm for multivariate, multiple output rational interpolation models, based on convergents of Thiele-type branched continued fractions, is presented. A variation of the standard branched continued fraction method is proposed that uses approximation to establish a non-rectangular grid of support points. Starting with a low order interpolant, the technique systematically increases the order by optimally choosing new support points in the areas of highest error, until the desired accuracy is achieved. In this way, accurate surrogate models are established by a small number of support points, without assuming any a priori knowledge of the microwave structure under study. The technique is illustrated and evaluated on several passive microwave structures, however it is general enough to be applied to many modelling problems.AFRIKAANSE OPSOMMING: 'n Robuuste en effektiewe aanpasbare monsternemingsalgoritme vir multi-veranderlike, multi-uittree rasionale interpolasiemodelle, gegrond op konvergente van Thiele vertakte volgehoue breukuitbreidings, word beskryf. 'n Variasie op die konvensionele breukuitbreidingsmetode word voorgestel, wat 'n nie-reghoekige rooster van ondersteuningspunte gebruik in die funksiebenadering. Met 'n lae orde interpolant as beginpunt, verhoog die algoritme stelselmatig die orde van die interpolant deur optimaal verbeterde ondersteuningspunte te kies waar die grootste fout voorkom, totdat die gewensde akuraatheid bereik word. Hierdeur word akkurate surrogaat modelle opgebou ten spyte van min inisiele ondersteuningspunte, asook sonder voorkennis van die mikrogolfstruktuur ter sprake. Die algoritme word gedemonstreer en geevalueer op verskeie passiewe mikrogolfstrukture, maar is veelsydig genoeg om toepassing te vind in meer algemene modelleringsprobleme

Theoretical Computer Science and Discrete Mathematics

This book includes 15 articles published in the Special Issue "Theoretical Computer Science and Discrete Mathematics" of Symmetry (ISSN 2073-8994). This Special Issue is devoted to original and significant contributions to theoretical computer science and discrete mathematics. The aim was to bring together research papers linking different areas of discrete mathematics and theoretical computer science, as well as applications of discrete mathematics to other areas of science and technology. The Special Issue covers topics in discrete mathematics including (but not limited to) graph theory, cryptography, numerical semigroups, discrete optimization, algorithms, and complexity

(Acido)bacterial diversity in space and time

Recent technological achievements enabled microbiologists to fully grasp the vast diversity of microbial life that is resident in soils, highly complex matrices of alternating micro-habitats on very small scales. Since then, microbial community composition has been catalogued for many different terrestrial habitats. This triggered the investigation and definition of processes which shape these communities. In most cases, the environment determines community composition, and similar habitats may feature similar microbial communities despite being far apart. However, some habitats have been described as subjected to pronounced neutral processes, which are dispersal, ecological drift or speciation. The balance between these process types is now the subject of many studies looking at microbial communities. It is also clear that these processes need to be monitored on both temporal and spatial scales, as the two dimensions are inseparably interlinked. However, most microbial studies deal with only one aspect, but do not control for the other. In this work, the outcome of a highly sophisticated plot scale experiment is presented encompassing 358 sampling locations distributed between six intra-annual sampling points on a 10 m x 10 m unfertilized grassland site in the Swabian Alb. RNA was extracted from the A-horizon of each soil and the hypervariable region 3 of the ribosomal small subunit was amplified and sequenced with barcoded Illumina sequencing. Roughly 400 million eubacterial reads were obtained. The dataset was used to assess the population dynamics of Acidobacteria, as well as the spatio-temporal co-occurenze of functionally depending microorganism. Additionally, preliminary results motivated the assessment of common methods for the examination of rhizospheric communities. In combination, the diversity of bacterial communities in space and time was tested from different angles, reflecting different research question, and they all revealed a far more complex reality than previously thought