Research Articles (Mathematical Sciences)https://scholar.sun.ac.za/handle/10019.1/48322024-07-15T07:42:07Z2024-07-15T07:42:07Z1151On the minimal Hamming weight of a multi-base representationKrenn, DanielSuppakitpaisarn, VorapongWagner, Stephanhttps://scholar.sun.ac.za/handle/10019.1/1262492023-05-27T23:26:54Z2020-01-01T00:00:00Zdc.title: On the minimal Hamming weight of a multi-base representation
dc.contributor.author: Krenn, Daniel; Suppakitpaisarn, Vorapong; Wagner, Stephan
dc.description.abstract: 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.
dc.description: CITATION: Krenn, D., Suppakitpaisarn, V. & Wagner, S. 2020. On the minimal Hamming weight of a multi-base representation. Journal of Number Theory, 208:168–179, doi:10.1016/j.jnt.2019.07.023.; The original publication is available at https://www.sciencedirect.com
2020-01-01T00:00:00ZSome combinatorial matrices and their LU-decompositionProdinger, Helmuthttps://scholar.sun.ac.za/handle/10019.1/1262282023-06-26T10:40:30Z2020-02-01T00:00:00Zdc.title: Some combinatorial matrices and their LU-decomposition
dc.contributor.author: Prodinger, Helmut
dc.description.abstract: 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.
dc.description: CITATION: Prodinger, H. 2020. Some combinatorial matrices and their LU-decomposition. Special Matrices, 8:61–67, doi:10.1515/spma-2020-0007.; The original publication is available at https://www.degruyter.com
2020-02-01T00:00:00ZA conceptual map of invasion biology : integrating hypotheses into a consensus networkEnders, MartinHavemann, FrankRuland, FlorianBernard-Verdier, MaudCatford, Jane A.Gomez-Aparicio, LorenaHaider, SylviaHeger, TinaKueffer, ChristophKuh, IngolfMeyerson, Laura A.Musseau, CamilleNovoa, AnaRicciardi, AnthonySagouis, AlbanSchittko, ConradStrayer, David L.Vilà, MontserratEssl, FranzHulme, Philip E.Van Kleunen, MarkKumschick, SabrinaLockwood, Julie L.Mabey, Abigail L.McGeoch, Melodie A.Estibaliz, PalmaPysek, PetrSaul, Wolf-ChristianYannelli, Florencia A.Jeschke, Jonathan M.https://scholar.sun.ac.za/handle/10019.1/1255862024-04-23T07:42:20Z2020-03-25T00:00:00Zdc.title: A conceptual map of invasion biology : integrating hypotheses into a consensus network
dc.contributor.author: 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.
dc.description.abstract: 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.
dc.description: CITATION: Enders, M. et al. 2020. A conceptual map of invasion biology: Integrating hypotheses into a consensus network. Global Ecology and Biogeography, 29(6): 978– 991. doi:10.1111/geb.13082; The original publication is available at https://onlinelibrary.wiley.com/journal/14668238
2020-03-25T00:00:00ZThe number of distinct adjacent pairs in geometrically distributed wordsArchibald, MargaretBlecher, AubreyBrennan, CharlotteKnopfmacher, ArnoldWagner, StephanWard, Mark Danielhttps://scholar.sun.ac.za/handle/10019.1/1254462024-04-23T07:42:22Z2021-01-28T00:00:00Zdc.title: The number of distinct adjacent pairs in geometrically distributed words
dc.contributor.author: Archibald, Margaret; Blecher, Aubrey; Brennan, Charlotte; Knopfmacher, Arnold; Wagner, Stephan; Ward, Mark Daniel
dc.description.abstract: 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→∞.
dc.description: CITATION: Archibald, M. et al. 2021. The number of distinct adjacent pairs in geometrically distributed words. Discrete Mathematics & Theoretical Computer Science, 22(4) doi:10.23638/DMTCS-22-4-10; The original publication is available at https://dmtcs.episciences.org/
2021-01-28T00:00:00ZInducibility of d-ary treesCzabarka, EvaDossou-Olory, Audace A. V.Szekely, Laszlo A.Wagner, Stephanhttps://scholar.sun.ac.za/handle/10019.1/1254432024-04-23T07:42:27Z2020-01-01T00:00:00Zdc.title: Inducibility of d-ary trees
dc.contributor.author: Czabarka, Eva; Dossou-Olory, Audace A. V.; Szekely, Laszlo A.; Wagner, Stephan
dc.description.abstract: 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.
dc.description: CITATION: Czabarka, E. et al. 2020. Inducibility of d-ary trees. Discrete Mathematics, 343(2). doi:10.1016/j.disc.2019.111671.; The original publication is available at https://www.sciencedirect.com/journal/discrete-mathematics
2020-01-01T00:00:00ZA wide class of Combinatorial matrices related with Reciprocal Pascal and Super Catalan matricesKilic, EmrahProdinger, Helmuthttps://scholar.sun.ac.za/handle/10019.1/1251732024-04-23T07:54:19Z2019-01-01T00:00:00Zdc.title: A wide class of Combinatorial matrices related with Reciprocal Pascal and Super Catalan matrices
dc.contributor.author: Kilic, Emrah; Prodinger, Helmut
dc.description.abstract: Abstract. In this paper, we present a number of combinatorial matrices
that are generalizations or variants of the super Catalan matrix
and the reciprocal Pascal matrix. We present explicit formul for LUdecompositions
of all the matrices and their inverses. Alternative derivations
using hypergeometric functions are also given.
dc.description: CITATION:Prodinger, H. (2019). A wide class of Combinatorial matrices related with Reciprocal Pascal and Super Catalan matrices. Contributions Discret. Math., 14.
2019-01-01T00:00:00ZMAVSCOT : a fuzzy logic-based HIV diagnostic system with indigenous multi-lingual interfaces for rural AfricaOluwagbemi, Olugbenga OluseunOluwagbem, Folakemi EtseoghenaJatto, AbdulwahabHui, Canghttps://scholar.sun.ac.za/handle/10019.1/1241392024-01-19T12:49:15Z2020-11-06T00:00:00Zdc.title: MAVSCOT : a fuzzy logic-based HIV diagnostic system with indigenous multi-lingual interfaces for rural Africa
dc.contributor.author: Oluwagbemi, Olugbenga Oluseun; Oluwagbem, Folakemi Etseoghena; Jatto, Abdulwahab; Hui, Cang
dc.description.abstract: HIV still constitutes a major public health problem in Africa, where the highest incidence and prevalence of the disease can be found in many rural areas, with multiple indigenous languages being used for communication by locals. In many rural areas of the KwaZulu-Natal (KZN) in South Africa, for instance, the most widely used languages include Zulu and Xhosa, with only limited comprehension in English and Afrikaans. Health care practitioners for HIV diagnosis and treatment, often, cannot communicate efficiently with their indigenous ethnic patients. An informatics tool is urgently needed to facilitate these health care professionals for better communication with their patients during HIV diagnosis. Here, we apply fuzzy logic and speech technology and develop a fuzzy logic HIV diagnostic system with indigenous multi-lingual interfaces, named Multi-linguAl HIV indigenouS fuzzy logiC-based diagnOstic sysTem (MAVSCOT). This HIV multilingual informatics software can facilitate the diagnosis in underprivileged rural African communities. We provide examples on how MAVSCOT can be applied towards HIV diagnosis by using existing data from the literature. Compared to other similar tools, MAVSCOT can perform better due to its implementation of the fuzzy logic. We hope MAVSCOT would help health care practitioners working in indigenous communities of many African countries, to efficiently diagnose HIV and ultimately control its transmission.
dc.description: CITATION: Oluwagbemi, O. O., et al. 2020. MAVSCOT : a fuzzy logic-based HIV diagnostic system with indigenous multi-lingual interfaces for rural Africa. PLoS ONE 15(11): e0241864, doi:10.1371/journal.pone.0241864.; The original publication is available at https://journals.plos.org/plosone/; Publication of this article was funded by the Stellenbosch University Open Access Fund
2020-11-06T00:00:00ZPrejudice, privilege, and power : conflicts and cooperation between recognizable groupsBingham, JeremyLandi, PietroHui, Canghttps://scholar.sun.ac.za/handle/10019.1/1235172024-01-19T12:43:29Z2019-01-01T00:00:00Zdc.title: Prejudice, privilege, and power : conflicts and cooperation between recognizable groups
dc.contributor.author: Bingham, Jeremy; Landi, Pietro; Hui, Cang
dc.description.abstract: The problem of cooperation remains one of the fundamental questions in the fields of biology, sociology, and economics. The emergence and maintenance of cooperation are naturally affected by group dynamics, since individuals are likely to behave differently based on shared group membership. We here formulate a model of socio-economic power between two prejudiced groups, and explore the conditions for their cooperative coexistence under two social scenarios in a well-mixed environment. Each scenario corresponds to an asymmetrical increase in the payoffs for mutual cooperation in either cross-group or within-group interactions. In the 'inter-dependence' scenario payoffs of cross-group cooperation are enhanced, while in the 'group-cohesion' scenario payoffs of within-group cooperation are enhanced. We find that stable cooperative coexistence is possible only in the inter-dependence scenario. The conditions for such coexistence are highly sensitive to prejudice, defined as the reduction in probability for cross-group cooperation, and less sensitive to privilege, defined as the enhancements to payoffs of cross-group cooperation.
dc.description: CITATION: Bingham, J., Landi, P. & Hui, C. 2019. Prejudice, privilege, and power : conflicts and cooperation between recognizable groups.
Mathematical Biosciences and Engineering, 16(5):4092-4106, doi:10.3934/mbe.2019203.; The original publication is available at https://www.aimspress.com
2019-01-01T00:00:00ZA complete study of the ramification for any separable cubic global function fieldMarques, SophieWard, Jacobhttps://scholar.sun.ac.za/handle/10019.1/1233602024-04-23T07:42:20Z2019-01-01T00:00:00Zdc.title: A complete study of the ramification for any separable cubic global function field
dc.contributor.author: Marques, Sophie; Ward, Jacob
dc.description.abstract: We explicitly describe the ramified places in any separable cubic extension of a cubic global function field in terms of a unique given parameter. This is all done using the uniqueness of the purely cubic closure, which is a useful new tool for the study of cubic function fields. We give a notion of local standard forms, that is useful for many purposes, including classifying and computing of integral bases. We then determine explicitly the genus of any separable cubic extension of any global function field given the minimal polynomial of the extension. The formulae we obtain is particularly useful for further study owing to the well-understood and straightforward close relation between the parameter we define and ramification within the extension.
dc.description: CITATION: Marques, S. & Ward, J. 2019. A complete study of the ramification for any separable cubic global function field. Research in Number Theory, 5(36). doi:10.1007/s40993-019-0173-y; The original publication is available at https://www.springer.com/journal/40993
2019-01-01T00:00:00ZLarge deviations of random walks on random graphsCoghi, FrancescoMorand, JulesTouchette, Hugohttps://scholar.sun.ac.za/handle/10019.1/1233102024-04-23T07:42:22Z2019-02-26T00:00:00Zdc.title: Large deviations of random walks on random graphs
dc.contributor.author: Coghi, Francesco; Morand, Jules; Touchette, Hugo
dc.description.abstract: We study the rare fluctuations or large deviations of time-integrated functionals or observables of an unbiased random walk evolving on Erdös-Rényi random graphs, and construct a modified, biased random walk that explains how these fluctuations arise in the long-time limit. Two observables are considered: the sum of the degrees visited by the random walk and the sum of their logarithm, related to the trajectory entropy. The modified random walk is used for both quantities to explain how sudden changes in degree fluctuations, similar to dynamical phase transitions, are related to localization transitions. For the second quantity, we also establish links between the large deviations of the trajectory entropy and the maximum entropy random walk.
dc.description: CITATION: Coghi, F.; Morand, J. & Touchette, H. 2019. Large deviations of random walks on random graphs. Physical Review E, 99. doi:10.1103/PhysRevE.99.022137; The original publication is available at https://journals.aps.org/pre/abstract/10.1103/PhysRevE.99.022137
2019-02-26T00:00:00Z