Research Articles (Mathematical Sciences)

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    On the minimal Hamming weight of a multi-base representation
    (Elsevier, 2020) Krenn, Daniel; Suppakitpaisarn, Vorapong; Wagner, Stephan
    Given 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.
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    Some combinatorial matrices and their LU-decomposition
    (De Gruyter, 2020-02) Prodinger, Helmut
    Three 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.
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    A 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.
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    The number of distinct adjacent pairs in geometrically distributed words
    (, 2021-01-28) Archibald, Margaret; Blecher, Aubrey; Brennan, Charlotte; Knopfmacher, Arnold; Wagner, Stephan; Ward, Mark Daniel
    A 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→∞.
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    Inducibility of d-ary trees
    (Elsevier, 2020) Czabarka, Eva; Dossou-Olory, Audace A. V.; Szekely, Laszlo A.; Wagner, Stephan
    Imitating the binary inducibility, a recently introduced invariant of binary trees (Cz- abarka et al., 2017), we initiate the study of the inducibility of d-ary trees (rooted trees whose vertex outdegrees are bounded from above by d ≥ 2). We determine the exact inducibility for stars and binary caterpillars. For T in the family of strictly d-ary trees (every vertex has 0 or d children), we prove that the difference between the maximum density of a d-ary tree D in T and the inducibility of D is of order O(|T |−1/2) compared to the general case where it is shown that the difference is O(|T |−1) which, in particular, responds positively to a conjecture on the inducibility in binary trees. We also discover that the inducibility of a binary tree in d-ary trees is independent of d. Furthermore, we establish a general lower bound on the inducibility and also provide a bound for some special trees. Moreover, we find that the maximum inducibility is attained for binary caterpillars for every d.