On the existence of normal maximal subgroups in division rings

AbstractLet D be a division ring with centre F. Denote by D∗ the multiplicative group of D. The relation between valuations on D and maximal subgroups of D∗ is investigated. In the finite dimensional case, it is shown that F∗ has a maximal subgroup if Br(F) is non-trivial provided that the characteristic of F is zero. It is also proved that if F is a local or an algebraic number field, then D∗ contains a maximal subgroup that is normal in D∗. It should be observed that every maximal subgroup of D∗ contains either D′ or F∗, and normal maximal subgroups of D∗ contain D′, whereas maximal subgroups of D∗ do not necessarily contain F∗. It is then conjectured that the multiplicative group of any noncommutative division ring has a maximal subgroup

On subgroups in division rings of type 22

Let $D$ be a division ring with center $F$. We say that $D$ is a {\em division ring of type $2$} if for every two elements $x, y\in D,$ the division subring $F(x, y)$ is a finite dimensional vector space over $F$. In this paper we investigate multiplicative subgroups in such a ring.Comment: 10 pages, 0 figure

The AIQ Meta-Testbed: Pragmatically Bridging Academic AI Testing and Industrial Q Needs

AI solutions seem to appear in any and all application domains. As AI becomes more pervasive, the importance of quality assurance increases. Unfortunately, there is no consensus on what artificial intelligence means and interpretations range from simple statistical analysis to sentient humanoid robots. On top of that, quality is a notoriously hard concept to pinpoint. What does this mean for AI quality? In this paper, we share our working definition and a pragmatic approach to address the corresponding quality assurance with a focus on testing. Finally, we present our ongoing work on establishing the AIQ Meta-Testbed.Comment: Accepted for publication in the Proc. of the Software Quality Days 2021, Vienna, Austri

Variability in quality attributes of service-based software systems: A systematic literature review

Context and problem: Variability in software systems is generally understood as the ability of a software artifact to be changed for a specific context, in a preplanned manner. Even though variability is primarily studied in the software product line (SPL) domain, variability can occur in any software system. Moreover, variability not only affects functionality or features, but also quality attributes (QA). Considering QA throughout software development is crucial to ensure systems that meet quality requirements. It is complex to handle variability due to the growing number of constraints and also different possible configurations. However, how to handle variability in QA and in particular in service-based software systems has not received enough attention from researchers and still causes problems in software and service engineering practice. Thesis objective: Before we thoroughly address the problem of variability in QA in service-based systems, an understanding of this topic is needed. Thus, the objective of this research is to systematically study variability of QA in service-based systems and to get an insight into the current status of research issues. In detail, the goals of the thesis are a) to assess methods for variability in quality attributes b) to collect evidence about current research that suggests implications for practice, and c) to identify research trends, open problems and areas for improvement. Methods: We apply empirical research and conduct a systematic literature review (SLR). The research questions of our review are: RQ1: What methods to handle variability in quality attributes of service-based systems exist? RQ2: How much evidence is available to adopt proposed methods? RQ3: What are the limitations of current methods? The SLR includes an automatic search, rather than a manual search of software engineering venues. Results: The results of our systematic review consists of a list of methods to handle variability in QA, including evidence for the validity of those methods (this list can be used by practitioners to select a specific method in a particular context). Moreover, based on these results we identify the current status of the research and open areas and propose guidelines for further research in this domain. In detail, our results suggest that design-time quality attributes are almost non-existent in current approaches available for practitioners, and product line engineering as the traditional discipline for variability management has almost no influence how we deal with variability in quality attributes of service-based systems. Furthermore, current approaches proposed by the research community do not provide enough evidence for practitioners to adopt these approaches. Also, variability has mainly been studied in laboratory settings, leaving many unsolved challenges for practitioners.

Multiplicative groups of division rings

Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras have not completely been understood. This article is a long survey of the recent developments on the multiplicative group structure of division rings