Department of Mathematical Sciences
http://hdl.handle.net/10019.1/223
2020-08-11T01:25:16ZNevanlinna Theory and Rational Values of Meromorphic Functions
http://hdl.handle.net/10019.1/108683
Nevanlinna Theory and Rational Values of Meromorphic Functions
Chalebgwa, Taboka Prince
ENGLISH ABSTRACT: In this thesis, we are concerned with the problem of counting algebraic
points of bounded height and degree on graphs of certain transcendental
holomorphic and meromorphic functions. Adopting a Nevanlinna theoretic
approach for the latter, we attain bounds of the form C(d)(log H)b for the
number of algebraic points of height at most H and degree at most d on the
restrictions to compact subsets of domains of holomorphy of meromorphic
functions with certain growth/decay conditions. In the second half of the
thesis, we turn our attention to counting points on graphs of certain analytic
functions with growth behaviour stricter than finite order and positive
lower order. For these functions, we are able to relax the need to restrict
them to compact subsets of C, and indeed, to count points either on the
whole graph or nearly all of it. For these functions we also attain a bound
of the form C(d)(log H)h. We end this work with several pointers towards
possible extensions of our results. The results in this thesis can be seen as
extensions of the work of Boxall and Jones on algebraic values of certain
analytic functions.; AFRIKAANSE OPSOMMING: In hierdie tesis is die probleem om die algebraïese punte van begrensde
hoogte en graad op grafieke van sekere transendentale holomorfiese en meromorfiese
funksies te tel, van belang. Met behulp van die Nevanlinnateoretiese
benadering vir laasgenoemde, verkry ons grense van die vorm
C(d)(logH)b vir die getal algebraïese punte waarvan die hoogte op die
meeste H en die graad op die meeste d is, met die beperkings tot kompakte
deelversamelings van domeine van holomorfie van meromorfiese funksies
met sekere groei-/verval-voorwaardes. In die tweede helfte van die tesis
vestig ons ons aandag op die tel van punte op grafieke van sekere analitiese
funksies met groei-gedrag strenger as eindige orde en positiewe onderorde.
Vir hierdie funksies kan ons die beperking tot kompakte deelversamelings
van C ophef en, inderdaad, die punte op óf die hele grafiek, óf byna
die hele grafiek, tel. Vir hierdie funksies verkry ons ook a grens van die
vorm C(d)(logH)h. Ons sluit hierdie werk af met verkeie aanduidings van
moontlike uitbreidings van ons resultate. Die resultate in hierdie tesis kan
as uitbreidings van die werk van Boxall en Jones oor algebraïese waardes
van sekere analitiese funksies, beskou word.
2019-04-01T00:00:00ZA simple ecological model captures the transmission pattern of COVID-19 outbreak in China
http://hdl.handle.net/10019.1/108584
A simple ecological model captures the transmission pattern of COVID-19 outbreak in China
Zhang, Feng; Zhang, Jinmei; Cao, Menglan; Hui, Cang
Background The rapid spread of the 2019 novel coronavirus disease (COVID-19), initially reported in the city of
Wuhan in China, and quickly transmitted to the entire nation and beyond, has become an international public
health emergency. Estimating the final number of infection cases and the turning point (time with the fastest
spreading rate) is crucial to assessing and improving the national and international control measures currently
being applied.
Methods We develop a simple model based on infectious growth with a time-varying infection rate, and
estimate the final number of infections and the turning point using data updated daily from 3 February 2020,
when China escalated its initial public health measures, to 10 February.
Results Our model provides an extremely good fit to the existing data and therefore a reasonable estimate of
the time-varying infection rate that has largely captured the transmission pattern of this epidemic outbreak. Our
estimation suggests that (i) the final number of infections in China could reach 78,000 with an upper 95%
confidence limit of 88,880; (ii) the turning point of the fastest spread was on the 4th or the 5th of February; and
(iii) the projected period for the end of the outbreak (i.e., when 95% of the final predicted number of infection is
reached) will be the 24th of February, with an upper 95% confidence limit on the 19th of March.
Conclusions Our results suggest that the current control measures in China are excellent, and more than
sufficient to contain the spread of this highly infectious novel coronavirus, and that the application of such
measures could be considered internationally for the global control of this outbreak.
CITATION: Zhang, F. et al. 2020. A simple ecological model captures the transmission pattern of COVID-19 outbreak in China. BMC Infectious Diseases, doi:10.21203/rs.3.rs-27784/v1.; The original publication is available at https://bmcinfectdis.biomedcentral.com
2020-01-01T00:00:00ZNear-vector spaces constructed from near domains
http://hdl.handle.net/10019.1/108510
Near-vector spaces constructed from near domains
Howell, Karin-Therese; Sanon, Sogo Pierre
In this paper we prove some new results on near-vector spaces and near domains and give a first application of the nearring of quotients with respect to a multiplicative set, namely we construct a new class of near-vector spaces from near domains.
CITATION: Howell, K. T. & Sanon, S. P. 2018. Near-vector spaces constructed from near domains. Miskolc Mathematical Notes, 19(2):883-897, doi:10.18514/MMN.2018.2186.; The original publication is available at http://mat76.mat.uni-miskolc.hu/mnotes/contents/19/2
2018-01-01T00:00:00ZWhy does not the leaf weight-area allometry of bamboos follow the 3/2-power law?
http://hdl.handle.net/10019.1/107672
Why does not the leaf weight-area allometry of bamboos follow the 3/2-power law?
Lin, Shuyan; Shao, Lijuan; Hui, Cang; Song, Yu; Reddy, Gadi V. P.; Gielis, Johan; Li, Fang; Ding, Yulong; Wei, Qiang; Shi, Peijian
The principle of similarity (Thompson, 1917) states that the weight of an organism follows the 3/2-power law of its surface area and is proportional to its volume on the condition that the density is constant. However, the allometric relationship between leaf weight and leaf area has been reported to greatly deviate from the 3/2-power law, with the irregularity of leaf density largely ignored for explaining this deviation. Here, we choose 11 bamboo species to explore the allometric relationships among leaf area (A), density (ρ), length (L), thickness (T), and weight (W). Because the edge of a bamboo leaf follows a simplified two-parameter Gielis equation, we could show that A ∝ L2 and that A ∝ T2. This then allowed us to derive the density-thickness allometry ρ ∝ Tb and the weight-area allometry W ∝ A(b+3)/2 ≈ A9/8, where b approximates −3/4. Leaf density is strikingly negatively associated with leaf thickness, and it is this inverse relationship that results in the weight-area allometry to deviate from the 3/2-power law. In conclusion, although plants are prone to invest less dry mass and thus produce thinner leaves when the leaf area is sufficient for photosynthesis, such leaf thinning needs to be accompanied with elevated density to ensure structural stability. The findings provide the insights on the evolutionary clue about the biomass investment and output of photosynthetic organs of plants. Because of the importance of leaves, plants could have enhanced the ratio of dry material per unit area of leaf in order to increase the efficiency of photosynthesis, relative the other parts of plants. Although the conclusion is drawn only based on 11 bamboo species, it should also be applicable to the other plants, especially considering previous works on the exponent of the weight-area relationship being less than 3/2 in plants.
CITATION: Lin, S., et al. 2018. Why does not the leaf weight-area allometry of bamboos follow the 3/2-power law?. Frontiers in Plant Science, 9:583, doi:10.3389/fpls.2018.00583; The original publication is available at https://www.frontiersin.org
2018-01-01T00:00:00ZSpatial segregation facilitates the coexistence of tree species in temperate forests
http://hdl.handle.net/10019.1/107661
Spatial segregation facilitates the coexistence of tree species in temperate forests
Shi, Peijian; Gao, Jie; Song, Zhaopeng; Liu, Yanhong; Hui, Cang
Competition between plants has an important role during the natural succession of forest communities. Niche separation between plants can reduce such interspecific competition and enable multispecies plant to achieve coexistence, although this proposition has rarely been supported in experiments. Plant competition can be captured by spatial segregation of the competing species to avoid fierce direct conflicts for nutrients and light. We investigated a site of 400 m × 1000 m in Beijing Pine Mountain National Nature Reserve that was established for protecting Chinese pine and some rare fungi. Six dominant tree species (Fraxinus chinensis Roxb., Syringa reticulata (Blume) H. Hara var. amurensis (Rupr.) J. S. Pringle, Quercus mongolica Fisch. ex Ledeb., Armeniaca sibirica (L.) Lam., Pinus tabuliformis Carrière, and Ulmus pumila L.) were individually marked. Metrics of spatial segregation, based on the theory of spatial point process, were calculated to detect spatial competition. The corresponding type (species)-specific probabilities and the p-values from a spatially implicit test revealed significant overall spatial segregation between the six tree species. We further used the cross-type L-function to check the spatial correlation between Chinese pine and the other tree species, and detected a significant spatial repulsion relationship with four other tree species. Our study shows that each of the six dominant tree species occupies a different subarea in the landscape to effectively reduce direct spatial competition. We thus argue that patchy distributions of different tree species could be common in late forest community succession, and the coexistence of plants could be maintained over a large spatial scale. Management intervention, such as thinning the densities of dominant tree species, could be used to foster species coexistence and ensure the productivity of commercial stands.
CITATION: Shi, P., et al. 2018. Spatial segregation facilitates the coexistence of tree species in temperate forests. Forests, 9(12):768, doi:10.3390/f9120768.; The original publication is available at http://www.mdpi.com
2018-01-01T00:00:00ZCalibrating a stochastic SIR-model to simulated data using different calibration methods : a tutorial & comparison of methods
http://hdl.handle.net/10019.1/108187
Calibrating a stochastic SIR-model to simulated data using different calibration methods : a tutorial & comparison of methods
Van Staden, Wynand-Junior
ENGLISH ABSTRACT: Mathematical models have helped researchers identify and quantify trends in observed
data, which is especially useful in the field of epidemiology. Fitting models to data enhances
the credibility of model results, since the underlying framework of disease, is
quantified and epidemiological drivers can be found. However, many calibration methods
exist that quantify key parameters of a model, given observed data, and choosing
which calibration method to use in a study needs justification. Also, understanding how
different calibration methods work, can improve the quality and reduce uncertainty of
estimated parameters. Four calibration methods (two optimization methods and two
sampling methods) were reviewed and compared by calibrating a simple stochastic
SIR model to model simulated data, with all four methods. With the target parameters
known and by evaluating the performance of the calibration methods by using
bias, accuracy and coverage measures, it was found that sampling methods (Bayesian
Maximum Likelihood Estimation and the Approximate Bayesian Computation rejection
algorithm) outperform optimization methods (Least Squares and Maximum Likelihood
Estimation).; AFRIKAANSE OPSOMMING: Wiskundige modelle help navorses om die neigings in waargeneemde data te identifiseer
en te kwantifiseer, wat veral nuttig in die van epidemiologie is. Deur modelle
aan data te kalibreer, word die geloofwaardigheid van model resultate verhoog, aangesien
die onderliggende raamwerk van ’n siekte gekwantifiseer word en epidemiologiese
drywers gevind kan word. Daar bestaan egter baie kalibrasiemetodes wat die
sleutel parameters van ’n model kwantifiseer, gegewe waargenome data en die keuse
van die kalibrasiemetode om in ’n studie te gebruik, moet gereverdig word. Deur om
te verstaan hoe verskillende kalibrasiemetodes werk, kan dit die kwaliteit verbeter en
onsekerheid van geskatte parameters verminder. Vier kalibrasiemetodes (twee optimeringsmetodes
en twee steekproef metodes) is hersien en vergelyk deur ’n eenvoudige
stogastiese SIR-model te kalibreer aan gesimuleerde data met al vier metodes te modelleer.
Met die teikenparameters bekend en deur die werking van die kalibrasiemetodes
te evalueer deur die berekening van vooroordeligheid, akkuraatheid en bedekking, is
daar gevind dat steekproefmetodes (Bayesian Maximum Likelihood Estimation en die
Approximate Bayesian Computation verwerpings algoritme) beter as optimeringsmetodes (Least Squares en Maximum Likelihood Estimation) vaar.
Thesis (MSc)--Stellenbosch University, 2020.
2020-04-01T00:00:00ZDrinfeld modular forms of higher rank from a lattice-oriented point of view
http://hdl.handle.net/10019.1/108242
Drinfeld modular forms of higher rank from a lattice-oriented point of view
Baker, Liam Bradwin
ENGLISH ABSTRACT: Please refer to full text for abstract; AFRIKAANSE OPSOMMING: Raadpleeg asseblief volteks vir opsomming
Thesis (PhD)--Stellenbosch University, 2020.
2020-04-01T00:00:00ZComparison of old and new algorithms for s,t -network reliability
http://hdl.handle.net/10019.1/108246
Comparison of old and new algorithms for s,t -network reliability
Zainabu, Simotwo Chepkirui Faith
ENGLISH ABSTRACT: Network reliability is the probability of an operative path connecting the
source s with the terminal t. s, t-network reliability problems have been
proven to be #P-complete. In this thesis we present some old techniques
which have existed since the 1950’s, as well as four new algorithms for calculating
the network reliability. Because these algorithms are all coded in
Mathematica as a common platform, they can be compared in a fair way.
We first consider the exhaustive enumeration method. Then we explain
in detail the series-parallel reduction which is applied in the contraction deletion
algorithm. Let ps[i] and cs[i] be the number of cardinality i pathsets
and cutsets respectively. It was long known that knowing these parameters
yields the network reliability at once. Two algorithms of Wild (which more
generally concern arbitrary set filters) can be used to calculate the numbers
ps[i] and cs[i] more efficiently than previous approaches.
The comparison is based on CPU time where several random networks have
been tested. The results are presented in the form of graphs and tables and
we demonstrate some of the algorithms by examples.; AFRIKAANSE OPSOMMING: Netwerk betroubaarheid is die waarskynlikheid dat ’n operasionele pad die
bron s met die terminale t verbind. In praktiese gevalle was die meeste
probleme met netwerkbetroubaarheid #P-volledig. In hierdie tesis sal ons
’n aantal ou tegnieke wat sedert die vyftigerjare bestaan, sowel as nuwe
algoritmes vir die berekening van netwerkbetroubaarheid in Mathematica
kodeer. Op hierdie manier raak die algoritmes se werkverrigtings vergelykbaar.
Ons kyk eers na die uitputtende enumeratiewe metode. Vervolgens verduidelik
ons die serie-parallelle reduksie wat toegepas word in die stelling vir
sametrekking-skrapping. Uiteindelik is ons ook in staat om vier nuwe metodes
aan te bied.
Die vergelyking is gebaseer op die SVE-tyd wat deur netwerke benodig
word. Die resultate word aangebied in die vorm van grafieke en tabelle
en ons demonstreer enkele voorbeelde van die algoritmes.
Thesis (MSc)--Stellenbosch University, 2020.
2020-04-01T00:00:00ZTorsion bounds for Drinfeld modules with complex multiplication
http://hdl.handle.net/10019.1/108144
Torsion bounds for Drinfeld modules with complex multiplication
Rabenantoandro, Andry Nirina
ENGLISH ABSTRACT: The main objective of the present thesis is to prove an analogue for Drinfeld
modules of a theorem due to Clark and Pollack. The cardinality of the
group of K-rational torsion points of an elliptic curve EjK with complex multiplication
defined over a number field K of degree d is uniformly bounded
by Cd log log d for some absolute and effective constant C > 0, i.e. the constant
C > 0 depends neither on E nor on K. Let F be a global function field
over Fq and A the ring of elements of F regular away from a fixed prime
¥. Let r 1 be an integer. We prove that there exists a positive constant
CA,r > 0 depending only on A and r such that for any field extension L of
degree d over F and any Drinfeld A-module jjL of rank r with complex multiplication
defined over L and such that the endomorphism ring of j is the
maximal order in its CM field, the cardinality of the A-module of L-rational
torsion points of j is bounded by CA,rd log log d. The constant depends neither
on j nor on L. For a given A and r the constant CA,r is effective and
we get an explicit formula for it. The above result is not the full analogue
of Clark and Pollack’s theorem but rather a weaker version since it requires
the endomorphism ring of j to be the maximal order in its CM field. However,
when A = Fq[T], F = Fq(T) and r = 2 we obtain the full analogue of
Clark and Pollack’s result by proving the analogue of what they called the
Isogeny Torsion Theorem in [CP15].; AFRIKAANSE OPSOMMING: Die hoofdoel van hierdie tesis is om ’n analoog vir Drinfeld modules te
bewys van ’n stelling te danke aan Clark en Pollack wat die volgende beweer.
Die kardinaliteit van die groep K-rasionale torsiepunte van ’n elliptiese
kromme EjK met komplekse vermenigvuldiging gedefinieÃ ´nr oor ’n
getalveld K van graad d is eenvormig begrens deur Cd log log d vir ’n absolute
en effektiewe konstante C > 0, dit wil sê die konstante C > 0 hang nie
van E of van K af nie. Laat F ’n globale funksieveld oor Fq wees en A die
ring van elemente van F reëlmatig weg vanaf ’n vaste priem ¥. Laat r 1
’n heelgetal wees. Ons bewys dat daar ’n positiewe konstante CA,r > 0 is
afhangende slegs van A en r sodanig dat vir enige velduitbreiding L van
graad d oor F en enige Drinfeld A-module jjL van rang r met ingewikkelde
vermenigvuldiging gedefinieer o or L e n s odanig dat die endomorphism
ring van j is die maksimale orde in sy CM-veld, die kardinaliteit
van die A-module van L-rasionale torsiepunte van j begrens word deur
CA,rd log log d. Die konstante hang nie van j of van L af nie. Vir ’n gegewe
A en r die konstante CA,r is effektief en ons kry ’n eksplisiete formule
daarvoor. Die bogenoemde resultaat is nie die volledige analoog van Clark
en Pollack se stelling nie, maar eerder ’n swakker weergawe, aangesien dit vereis dat die endomorfisme van j die maksimale orde in sy CM-veld. Wanneer
A = Fq[T], F = Fq(T) en r = 2, verkry ons die volledige analoog van
Clark en Pollack se resultaat deur die analoog te bewys van wat hulle die
Isogeny Torsion Stelling in [CP15] genoem het.
Thesis (PhD)--Stellenbosch University, 2020.
2020-04-01T00:00:00ZInfluence of rotation on transversely isotropic piezoelectric rod coated with a thin film
http://hdl.handle.net/10019.1/107438
Influence of rotation on transversely isotropic piezoelectric rod coated with a thin film
Selvamani, Rajendran; Makinde, Oluwole Daniel
In this paper, the influence of rotation on axisymmetric waves of a piezoelectric rod coated with a thin film is studied using constitutive form linear theory elasticity and piezo-electric equations. Potential functions are introduced to uncouple the equations of motion in radial and axial directions. The surface area of the rod is coated by a perfectly conducting material. The frequency equations are obtained for longitudinal and flexural modes of vibration and are studied numerically for PZT-4 ceramics. The computed non-dimensional frequency, phase velocity, relative frequency shift, electromechanical coupling and electric displacement are presented in the form of dispersion curves. This type of study is important in the construction of
rotating sensors and gyroscope.
CITATION: Selvamani, R. & Makinde, O. D. 2018. Influence of rotation on transversely isotropic piezoelectric rod coated with a thin film. Engineering Transactions, 66(3):211–227, doi:10.24423/EngTrans.859.20180726.; The original publication is available at http://et.ippt.gov.pl
2018-01-01T00:00:00ZOn the distribution of subtree orders of a tree
http://hdl.handle.net/10019.1/107420
On the distribution of subtree orders of a tree
Ralaivaosaona, Dimbinaina; Wagner, Stephan
We investigate the distribution of the number of vertices of a randomly chosen subtree of a tree. Specifically, it is proven that this distribution is close to a Gaussian distribution in an explicitly quantifiable way if the tree has sufficiently many leaves and no long branchless paths. We also show that the conditions are satisfied asymptotically almost surely for random trees. If the conditions are violated, however, we exhibit by means of explicit counterexamples that many other (non-Gaussian) distributions can occur in the limit. These examples also show that our conditions are essentially best possible.
CITATION: Ralaivaosaona, D. & Wagner, S. 2018. On the distribution of subtree orders of a tree. Ars Mathematica Contemporanea, 14(1):129-156, doi:10.26493/1855-3974.996.675.; The original publication is available at https://amc-journal.eu
2018-01-01T00:00:00ZDrivers of species turnover vary with species commonness for native and alien plants with different residence times
http://hdl.handle.net/10019.1/107400
Drivers of species turnover vary with species commonness for native and alien plants with different residence times
Latombe, Guillaume; Richardson, David M.; Pysek, Petr; Kucera, Tomas; Hui, Cang
Communities comprising alien species with different residence times are natural experiments allowing the assessment of drivers of community assembly over time. Stochastic processes (such as dispersal and fluctuating environments) should be the dominant factors structuring communities of exotic species with short residence times. In contrast, communities should become more similar, or systematically diverge, if they contain exotics with increasing resident times, due to the increasing importance of deterministic processes (such as environmental filtering). We use zeta diversity (the number of species shared by multiple assemblages) to explore the relationship between the turnover of native species and two categories of alien species with different residence times (archaeophytes [introduced between 4000 BC and 1500 AD] and neophytes [introduced after 1500 AD]) in a network of nature reserves in central Europe. By considering multiple assemblages simultaneously, zeta diversity allows us to determine the contribution of rare and widespread species to turnover. Specifically, we explore the relative effects of assembly processes representing isolation by distance, environmental filtering, and environmental stochasticity (fluctuating environments) on zeta diversity using Multi‐Site Generalized Dissimilarity Modelling (MS‐GDM). Four clusters of results emerged. First, stochastic processes for structuring plant assemblages decreased in importance with increasing residence time. Environmental stochasticity only affected species composition for neophytes, offering possibilities to predict the spread debt of recent invasions. Second, native species turnover was well explained by environmental filtering and isolation by distance, although these factors did not explain the turnover of archaeophytes and neophytes. Third, native and alien species compositions were only correlated for rare species, whereas turnover in widespread alien species was surprisingly unrelated to the composition of widespread native species. Site‐specific approaches would therefore be more appropriate for the monitoring and management of rare alien species, whereas species‐specific approaches would suit widespread species. Finally, the size difference of nature reserves influences not only native species richness, but also their richness‐independent turnover. A network of reserves must therefore be designed and managed using a variety of approaches to enhance native diversity, while controlling alien species with different residence times and degrees of commonness.
CITATION: Latombe, G., et al. 2018. Drivers of species turnover vary with species commonness for native and alien plants with different residence times. Ecology, 99(12):2763-2775, doi:10.1002/ecy.2528.; The original publication is available at https://esajournals.onlinelibrary.wiley.com
2018-01-01T00:00:00ZUpscaling biodiversity : estimating the species–area relationship from small samples
http://hdl.handle.net/10019.1/107397
Upscaling biodiversity : estimating the species–area relationship from small samples
Kunin, William E.; Harte, John; He, Fangliang; Hui, Cang; Jobe, R. Todd; Ostling, Annette; Polce, Chiara; Sizling, Arnost; Smith, Adam B.; Smith, Krister; Smart, Simon M.; Storch, David; Tjorve, Even; Ugland, Karl‐Inne; Ulrich, Werner; Varma, Varun
The challenge of biodiversity upscaling, estimating the species richness of a large area from scattered local surveys within it, has attracted increasing interest in recent years, producing a wide range of competing approaches. Such methods, if successful, could have important applications to multi‐scale biodiversity estimation and monitoring. Here we test 19 techniques using a high quality plant data set: the GB Countryside Survey 1999, detailed surveys of a stratified random sample of British landscapes. In addition to the full data set, a set of geographical and statistical subsets was created, allowing each method to be tested on multiple data sets with different characteristics. The predictions of the models were tested against the “true” species–area relationship for British plants, derived from contemporaneously surveyed national atlas data. This represents a far more ambitious test than is usually employed, requiring 5–10 orders of magnitude in upscaling. The methods differed greatly in their performance; while there are 2,326 focal plant taxa recorded in the focal region, up‐scaled species richness estimates ranged from 62 to 11,593. Several models provided reasonably reliable results across the 16 test data sets: the Shen and He and the Ulrich and Ollik models provided the most robust estimates of total species richness, with the former generally providing estimates within 10% of the true value. The methods tested proved less accurate at estimating the shape of the species–area relationship (SAR) as a whole; the best single method was Hui's Occupancy Rank Curve approach, which erred on average by <20%. A hybrid method combining a total species richness estimate (from the Shen and He model) with a downscaling approach (the Šizling model) proved more accurate in predicting the SAR (mean relative error 15.5%) than any of the pure upscaling approaches tested. There remains substantial room for improvement in upscaling methods, but our results suggest that several existing methods have a high potential for practical application to estimating species richness at coarse spatial scales. The methods should greatly facilitate biodiversity estimation in poorly studied taxa and regions, and the monitoring of biodiversity change at multiple spatial scales.
CITATION: Kunin, W. E., et al. 2018. Upscaling biodiversity : estimating the species–area relationship from small samples. Ecological Monographs, 88(2):170-187, doi:10.1002/ecm.1284.; The original publication is available at https://esajournals.onlinelibrary.wiley.com
2018-01-01T00:00:00ZWeighing evidence
http://hdl.handle.net/10019.1/107348
Weighing evidence
Muller, M. A.
Often different pieces of evidence of different degrees of credibility are submitted in legal proceedings. Then a court “weighs” the evidence and arrives at a conclusion. But how should evidence be weighed? The use of probability theory comes to mind. Examples in this article illustrate this approach.
CITATION: Muller, M.A. 2019. Weighing evidence. Advocate, 32(3):60-61.; The original publication is available at https://www.sabar.co.za/law-journals/
2019-12-01T00:00:00ZExplicit bound on Siegel zeros of imaginary quadratic fields
http://hdl.handle.net/10019.1/107160
Explicit bound on Siegel zeros of imaginary quadratic fields
Razakarinoro, Faratiana Brice
ENGLISH ABSTRACT: Please refer to full text for abstract.; AFRIKAANSE OPSOMMING: Raadpleeg asseblief volteks vir opsomming.
Thesis (MSc)--Stellenbosch University, 2019.
2019-12-01T00:00:00ZWeighted centrality, and a further approach to categorical commutativity
http://hdl.handle.net/10019.1/107175
Weighted centrality, and a further approach to categorical commutativity
Shaumbwa, Vaino Tuhafeni
ENGLISH ABSTRACT: We investigate weighted commutators, that is, weighted subobject commutator and weighted
normal commutator, as well as commutators in the sense of Huq, Higgins, Ursini and Smith, which are all special cases of weighted commutators. One of the main aims is to establish further properties of weighted commutators, and explore new relationships among commutators. In a normal Mal'cev category C with finite colimits, we show that the Huq commutator of a pair of local representations (i.e. equivalence relations considered as subobjects in a category of points over fixed object) is the local representation of the Smith commutator of the equivalence relations corresponding to the original local representations. We also show that the weighted normal commutator can be obtained as the image of the kernel functor applied to the Huq commutator
of another type of morphisms in a category of points over a fixed object. In addition, the weighted normal commutator is characterized as the largest monotone ternary operation C defined on subobjects in a finitely cocomplete normal Barr-exact Mal'cev category, such that: (a) C(X; Y; W ) X ^ Y ; (b) C(f(X); f(Y ); f(W )) = f(C(X; Y; W )); for subobjects (X; x); (Y; y); and (W; w) of an object A; and every morphism f whose domain is A. The weighted subobject
commutator is characterized in a similar way, and furthermore, known characterizations of Higgins, Huq, and Ursini commutators are recovered as special cases.
Another aim is to extend the notion of commuting morphisms to a more general context, and in particular, to a subtractive category with finite joins of subobjects, where we show that
commuting morphisms are related to the notion of internal partial subtraction structures. Furthermore, we show that several results about central morphisms, commutative objects, and abelian objects, which usually require a category to be at least (strongly) unital, also hold in the
context of (regular) subtractive category with finite joins of subobjects.; AFRIKAANSE OPSOMMING: Ons ondersoek geweegde kommutator, dit wil sê geweegde subobjekkommutator en geweegde normale kommutator, sowel as kommutators in die sin van Huq, Higgins, Ursini en Smith, wat almal spesiale gevalle van geweegde kommutator is. Een van die hoofdoelwitte is om verdere eienskappe van geweegde kommutator te vestig, en om nuwe verhoudings tussen kommutator te ondersoek. In 'n normale Mal'cev-kategorie C met eindige kolimiete, wys ons dat die Huqkommutator
van 'n paar plaaslike voorstellings (dit wil sê ekwivalensieverhoudinge wat as subobjekte in 'n kategorie punte oor 'n vaste objek beskou word) die plaaslike voorstelling van die Smith is kommutator van die ekwivalensieverhoudinge wat ooreenstem met die oorspronklike plaaslike voorstellings. Ons wys ook dat die geweegde normale kommutator verkry kan word as die beeld
van die "kernel" funktor wat op die Huq-kommutator van 'n ander soort morfismes toegepas word in 'n kategorie punte oor 'n vaste objek. Daarbenewens word die geweegde normale kommutator gekenmerk as die grootste monotone ternêre werking C gedefinieër op sub-objekte in 'n
eindelik klaargemaakte normale Barr-exact Mal'cev-kategorie, sodat: (a) C(X; Y; W ) X ^ Y ; (b) C(f(X); f(Y ); f(W )) = f(C(X; Y; W )); vir subobjekte (X; x); (Y; y); en (W; w) van 'n objek A; en elke morfisme f waarvan die domein A is. Die geweegde subobjekkommutator word op 'n soortgelyke manier gekenmerk, en voorts word bekende karakterisering van Higgins, Huq en Ursini-kommutators as spesiale gevalle herwin. 'n Ander doel is om die idee van die pendel van morfismes uit te brei na 'n meer algemene konteks, en veral tot 'n subtraktiewe kategorie met 'n eindige samevoeging van sub-onderwerpe, waar ons wys dat die pendel-morfisme verband hou met die idee van interne gedeeltelike aftrekstrukture. Verder toon ons dat verskeie resultate oor sentrale morfismes, kommutatiewe objekte en abeliese objekte, wat gewoonlik vereis dat 'n kategorie ten minste (sterk) uniaal is, ook in die konteks van 'n (gereelde) aftrekkategorie met 'n beperkte samevoeging van subobjekte geld.
Thesis (PhD)--Stellenbosch University, 2019.
2019-12-01T00:00:00ZConcrete foundations of the theory of Noetherian forms
http://hdl.handle.net/10019.1/107103
Concrete foundations of the theory of Noetherian forms
Van Niekerk, Francois Koch
ENGLISH ABSTRACT: This thesis concerns certain investigations in abstract algebra that bring together
the ideas of the category of algebraic structures and the lattice of substructures. A central notion in such investigation is that of a noetherian form. Originally, noetherian forms were introduced to provide a self-dual axiomatic context for establishing homomorphism theorems for (non-abelian) group-like structures. It is known that the form of “subobjects” over any variety is a noetherian form exactly when the variety is semi-abelian. An unexpected result in this thesis is that there is a noetherian form over any variety. In particular, this shows that the context of a noetherian form is much wider than originally thought. One of the aims of the thesis is to explore methods of constructing new noetherian forms out of existing forms; the mentioned result is obtained as an application of one of these constructions. Another aim is to show how the self-dual analogue of products in noetherian forms, called “biproducts” (first introduced in the author’s MSc thesis), are related to products. Finally, in this
thesis we study the notion of an n-complemented lattice. This notion arose from studying subgroup lattices of finite abelian groups.; AFRIKAANSE OPSOMMING: Hierdie tesis handel oor sekere ondersoeke in abstrakte algebra wat die idees van die kategorie van algebraïse strukture en die tralie van substrukture by mekaar bring.
‘n Sentrale idee van so ‘n ondersoek is dié van ‘n noetherse vorm. Noetherse vorms was oorspronklik bekendgestel om ‘n selfduale konteks te bied vir die skepping van homomorfisme stellings vir (nie-abelse) groepagtige strukture. Dit is bekend dat die vorm van “sub-objekte” oor ‘n variëteit ‘n noetherse vorm is presies wanneer die variëteit semi-abels is. ‘n Onverwagte resultaat in hierdie tesis is dat daar ‘n
noetherse vorm oor enige variëteit bestaan. In besonders wys dit dat die konteks van noetherse vorms baie wyer strek as oorspronklik gedink. Een van die doelwitte van die tesis is om metodes van konstruksies van nuwe noetherse vorms uit bestaande vorms te verken; die genoemde resultaat is verkry deur ’n toepassing van een van hierdie konstruksies. ‘n Ander doelwit is om die verwantskap tussen die selfduale
analoog van produkte in noetherse vorms, genoem “biprodukte” (soos bekendgestel in die skrywer se MSc tesis) en kategoriese produkte aan te toon. Laastens, in hierdie tesis bestudeer ons die idee van ‘n n-komplemente tralie. Hierdie idee het ontstaan deur om subgroep tralies van eindige abelse groepe te bestudeer.
Thesis (PhD)--Stellenbosch University, 2019.
2019-12-01T00:00:00ZContributions to the theory of Beidleman near-vector spaces
http://hdl.handle.net/10019.1/107165
Contributions to the theory of Beidleman near-vector spaces
Djagba, Prudence
ENGLISH SUMMARY: (Please refer to the abstract on the full text for symbols that did not translate well into this abstract). The study of nearfields was started in 1905 by L.E. Dickson. This thesis is a first step toward a detailed study of J.C. Beidleman near-vector spaces, as first introduced by Beidleman in 1966. Recalling well-known
results, we conduct a detailed study of finite nearfields by showing how to construct a finite Dickson nearfield and presenting the center of a finite Dickson nearfield that arises from the Dickson pair (q, n). Furthermore, as main results of this thesis, we present the following. We characterise
the finite dimensional Beidleman near-vector spaces. We develop an algorithm called EGE (Expanded Gaussian Elimination) which determines the smallest R-subgroup containing a given finite set of vectors v1, . . . , vk 2 Rm
where R is a proper nearfield and k,m are positive integers, defined as gen(v1, . . . , vk). We also classify all the subspaces of Rm by designing an algorithm called the Adjustment of the EGE algorithm. We study the concept of
seed number of an R-subgroup T (i.e., the minimal cardinality of all the possible finite sets of vectors that generate T) and R-dimension of gen(v1, . . . , vk) (i.e., the number of vectors obtained after the implementation of the EGE algorithm
on the finite set of vectors v1, . . . , vk). We evaluate the seed number of Rm for some positive integer m satisfying m jRj +1. Furthermore from the EGE algorithm we also study, for a given pair (a, b) in R2, the generalized distributive set defined as D(a, b) =
l 2 R : (a + b) l = a l + b l
,
where ” ” is the multiplication of the nearfield. We find that in contrast to the situation of D(R) = fl 2 R : (a+ b) l = a l+ b l for all a, b 2 Rg from the work of Zemmer in 1964, the generalized distributive set D(a, b) is
not always a subnearfield of R where R is a finite Dickson nearfield arising from the Dickson pair (q, n). We find a sufficient condition on a and b such that D(a, b) is a subfield of the finite field of order qn and develop an algorithm that tests whether D(a, b) is a subfield of Fqn or not. We then investigate D(a, b) where a, b and a + b are all in distinct g
qi1 q1 H (where g is a generator of F q n and H is the subgroup generated by gn) and we obtain a construction of a subfield of Fqn by making use of D(a, b).
Thesis (PhD)--Stellenbosch University, 2019.
2019-12-01T00:00:00ZOn a generalisation of k-Dyck paths
http://hdl.handle.net/10019.1/107091
On a generalisation of k-Dyck paths
Selkirk, Sarah Jane
ENGLISH ABSTRACT: (Refer to full text abstract for symbols that did not transfer correctly). We consider a family of non-negative lattice paths consisting of the step set f(1; 1); (1;k)g called k-Dyck paths, which are enumerated by the generalised Catalan numbers 1 (k+1)n+1 (k+1)n+1
n . By removing the non-negativity condition but restricting the path to stay above the line y = t we obtain a family of lattice paths called kt-Dyck paths which are enumerated by `generalised generalised Catalan numbers
t + 1 (k + 1)n + t + 1 (k + 1)n + t + 1 n : We provide proofs of the enumeration of these paths by means of a bijection, the kernel method, the cycle lemma, and the symbolic method. Analysis of parameters associated with the paths is also performed using symbolic equations
{ particularly the number of peaks, the number of valleys, and the number of returns. These kt-Dyck paths nd application in enumerating a family of walks in the quarter plane (Z 0 Z 0) with step set f(1; 1); (1;k +1); (k; 0)g. Such walks can be decomposed into ordered pairs of kt-Dyck paths and thus their enumeration can be proved via a simple bijection. Through this bijection some parameters in kt-Dyck paths are preserved. Finally, we discuss two different families of lattice paths, S-Motzkin and T-
Motzkin paths, which are related to kt-Dyck paths when k = 2 along with t = 0 and t = 1. We provide bijections between these paths and other combinatorial objects, and perform analysis of parameters in these paths.; AFRIKAANSE OPSOMMING: (Verwys na volteks opsomming vir simbole wat nie korrek oorgeskryf het nie). Ons beskou 'n familie van nie-negatiewe roosterpaaie wat bestaan uit die stap-
versameling f(1; 1); (1;k)g genoem k-Dyck paaie, wat deur die veralgemeende
Catalan getalle, 1
(k+1)n+1
(k+1)n+1
n
, getel word. Deur die nie-negatiwiteitsvereiste
te verwyder, maar die pad tot bokant die lyn y = t te beperk kry ons 'n fami-
lie roosterpaaie genaamd kt-Dyck paaie wat getel word deur `veralgemeende'
veralgemeende Catalan getalle
t + 1
(k + 1)n + t + 1
(k + 1)n + t + 1
n
:
Ons lewer 'n bewys van die aftelling van hierdie paaie deur middel van 'n
bijeksie, die kernmetode, die sikluslemma en die simboliese metode. Analise
van parameters wat met die paaie geassosieer word, word ook uitgevoer met
behulp van simboliese vergelykings { veral die aantal pieke, die aantal valleie
en die aantal terugkomste.
Hierdie kt-Dyck paaie vind 'n toepassing in 'n familie van wandelinge in
die kwartvlak (Z 0 Z 0) met stapversameling f(1; 1); (1;k + 1); (k; 0)g.
Sulke wandelinge kan ontleed word in geordende pare kt-Dyck paaie en dus
kan hul aftelling deur middel van 'n eenvoudige bijeksie bewys word. Deur
hierdie bijeksie word 'n paar parameters in kt-Dyck paaie bewaar.
Laastens bespreek ons twee verskillende families van roosterpaaie, S-Motzkin
en T-Motzkin paaie, wat verband hou met kt-Dyck paaie wanneer k = 2 saam
met t = 0 en t = 1. Ons bied bijeksies tussen hierdie paaie en ander kom-
binatoriese voorwerpe, en doen 'n analise van parameters op hierdie paaie.
Thesis (MSc)--Stellenbosch University, 2019.
2019-12-01T00:00:00ZOn spanning sets and generators of near-vector spaces
http://hdl.handle.net/10019.1/106781
On spanning sets and generators of near-vector spaces
Howell, Karin-Therese; Sanon, Sogo Pierre
In this paper we study the quasi-kernel of certain constructions of near-vector spaces and the span of a
vector. We characterize those vectors whose span is one-dimensional and those that generate the whole space.
CITATION: Howell, K.-T. & Sanon, S. P. 2018. On spanning sets and generators of near-vector spaces. Turkish Journal of Mathematics, 42(6):3232 – 3241, doi:10.3906/mat-1807-155.; The original publication is available at http://journals.tubitak.gov.tr
2018-01-01T00:00:00Z