Research Articles (Mathematical Sciences)
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- ItemUse of Language By generative AI Tools in Mathematical Problem Solving: The Case of ChatGPT(Taylor & Francis, 2024-08-18) Daher, Wajeeh; Gierdien, FaaizTexts generated by artificial intelligence agents have been suggested as tools supporting students’ learning. The present research analyses the language of texts generated by ChatGPT when solving mathematical problems related to the quadratic equation. We use the functional grammar theoretical framework that includes three meta-functions: the ideational meta-function, the interpersonal meta-function and the textual meta-function. The results indicated that in at least one of six problem-solving tasks ChatGPT provided a mathematically incorrect answer. The processes appearing in ChatGPT texts, aiming at developing students’ understanding of mathematical concepts, included verbal, mental, existential, relational and behavioural processes but no material processes. Specifically, ChatGPT performed a mathematically incorrect existential process. ChatGPT generally used the first plural pronoun ‘we’ when describing the processes of solving mathematical problems, while it generally used the first-person singular pronoun when taking responsibility for a specific mistake or when expressing happiness for the actions of the user. Moreover, generally the text of the solution did not include direct imperatives but used ‘let us do’. The advancement of the ChatGPT textual solution was made usually through steps like ‘first’, ‘second’, etc. The research results indicated that the way ChatGPT responded to the mathematical problems would be useful in supporting learners’ understanding of ways to solve quadratic equations, but only if the teacher critically accompanies the student in the problem-solving process. Self-study with ChatGPT could lead to or confirm students’ mathematical misconceptions.
- ItemOn the minimal Hamming weight of a multi-base representation(Elsevier, 2020) Krenn, Daniel; Suppakitpaisarn, Vorapong; Wagner, StephanGiven a finite set of bases b1, b2, ..., br (integers greater than 1), a multi-base representation of an integer n is a sum with summands dbα1 1 b α2 2 ··· bαr r , where the αj are nonnegative integers and the digits d are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our main result states that the order of magnitude of the minimal Hamming weight of an integer n, i.e., the minimal number of nonzero summands in a representation of n, is log n/(log log n). This is independent of the number of bases, the bases themselves, and the digit set. For the proof, the existing upper bound for prime bases is generalized to multiplicatively independent bases; for the required analysis of the natural greedy algorithm, an auxiliary result in Diophantine approximation is derived. The lower bound follows by a counting argument and alternatively by using communication complexity; thereby improving the existing bounds and closing the gap in the order of magnitude.
- ItemSome combinatorial matrices and their LU-decomposition(De Gruyter, 2020-02) Prodinger, HelmutThree combinatorial matrices were considered and their LU-decompositions were found. This is typically done by (creative) guessing, and the proofs are more or less routine calculations.
- ItemA conceptual map of invasion biology : integrating hypotheses into a consensus network(Wiley, 2020-03-25) Enders, Martin; Havemann, Frank; Ruland, Florian; Bernard-Verdier, Maud; Catford, Jane A.; Gomez-Aparicio, Lorena; Haider, Sylvia; Heger, Tina; Kueffer, Christoph; Kuh, Ingolf; Meyerson, Laura A.; Musseau, Camille; Novoa, Ana; Ricciardi, Anthony; Sagouis, Alban; Schittko, Conrad; Strayer, David L.; Vilà, Montserrat; Essl, Franz; Hulme, Philip E.; Van Kleunen, Mark; Kumschick, Sabrina; Lockwood, Julie L.; Mabey, Abigail L.; McGeoch, Melodie A.; Estibaliz, Palma; Pysek, Petr; Saul, Wolf-Christian; Yannelli, Florencia A.; Jeschke, Jonathan M.Background and aims: Since its emergence in the mid-20th century, invasion biology has matured into a productive research field addressing questions of fundamental and applied importance. Not only has the number of empirical studies increased through time, but also has the number of competing, overlapping and, in some cases, contradictory hypotheses about biological invasions. To make these contradictions and redundancies explicit, and to gain insight into the field’s current theoretical structure, we developed and applied a Delphi approach to create a consensus network of 39 existing invasion hypotheses. Results: The resulting network was analysed with a link-clustering algorithm that revealed five concept clusters (resource availability, biotic interaction, propagule, trait and Darwin’s clusters) representing complementary areas in the theory of invasion biology. The network also displays hypotheses that link two or more clusters, called connecting hypotheses, which are important in determining network structure. The network indicates hypotheses that are logically linked either positively (77 connections of support) or negatively (that is, they contradict each other; 6 connections). Significance: The network visually synthesizes how invasion biology’s predominant hypotheses are conceptually related to each other, and thus, reveals an emergent structure – a conceptual map – that can serve as a navigation tool for scholars, practitioners and students, both inside and outside of the field of invasion biology, and guide the development of a more coherent foundation of theory. Additionally, the outlined approach can be more widely applied to create a conceptual map for the larger fields of ecology and biogeography.
- ItemThe number of distinct adjacent pairs in geometrically distributed words(Episciences.org, 2021-01-28) Archibald, Margaret; Blecher, Aubrey; Brennan, Charlotte; Knopfmacher, Arnold; Wagner, Stephan; Ward, Mark DanielA sequence of geometric random variables of length n is a sequence of n independent and identically distributed geometric random variables (Γ1,Γ2,…,Γn) where P(Γj=i)=pqi−1 for 1 ≤ j ≤ n with p+q=1. We study the number of distinct adjacent two letter patterns in such sequences. Initially we directly count the number of distinct pairs in words of short length. Because of the rapid growth of the number of word patterns we change our approach to this problem by obtaining an expression for the expected number of distinct pairs in words of length n. We also obtain the asymptotics for the expected number as n→∞.