Thesis (MSc)--Stellenbosch University, 2012 / ENGLISH ABSTRACT: This thesis presents the FATKID Finite Automata Toolkit. While a lot of toolkits currently exist which can manipulate and process nite state automata, this toolkit was designed to e ectively and e ciently generate, manipulate and process large numbers of nite automata by distributing the work ow across machines and running the computations in parallel. Other toolkits do not currently provide this functionality. We show that this framework is user-friendly and extremely extensible. Furthermore we show that the system e ectively distributes the work to reduce computation time. / AFRIKAANSE OPSOMMING: In hierdie tesis bespreek ons die FATKID Eindige Automaat Toestel. Al- hoewel daar reeds toestelle bestaan wat automate kan genereer, manupileer, en bewerkings daarmee kan uitvoer, is daar egter geen toestelle wat dit op die skaal kan doen wat ons vereis deur die proses te versprei na 'n aantal nodes nie. Ons vereis 'n stelsel wat mew baie groot aantalle automate werk. Die stelsel moet dan die gewensde prosesse in 'n verspreide omgewing, en in parallel uitvoer om verwerkingstyd te verminder. Ons sal wys dat ons stelsel nie net hierdie doel bereik nie, maar ook dat dit gebruikers-vriendelik is en maklik om uit te brei.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Adolescents are regarded as a high risk group in South Africa with the highest human immunodeficiency virus (HIV) incidence occurring in this group. Prevention among adolescents is therefore a key in decreasing the HIV burden. This thesis aims to assist in the design of trials by simulating the potential outcomes of a combination prevention trial in adolescents. We develop a stochastic individual-based model stratified by sex and age. We then use this model to determine the impact of various prevention packages on HIV incidence among adolescents participating in a hypothetical trial over a three year period. The trial that is simulated involves an intervention arm, in which adolescents are offered a choice of a prevention methods (including medical male circumcision (MMC), oral pre-exposure prophylaxis (PrEP) and antiretroviral-based vaginal microbicides (ARV-VM)), and a control arm. We predict that the impact of a full prevention package on HIV incidence would be a 46% per personyear( PPY) (95% CI 45–47%) risk reduction. The combination of MMC and PrEP has a substantial impact on HIV incidence in males, with a 51% PPY (95% CI 49–53%) relative risk of HIV infection. Offering women the choice of PrEP, a microbicide gel or a microbicide in the form of a vaginal ring would be less effective, with a 57% PPY (95% CI 56–58%) relative risk of HIV acquisition. This is not substantially different from the relative risk estimated when the vaginal ring alone is offered, as the ring is assumed to be the most accept able of the three prevention methods. We determine a sample size requirement of approximately 1013 in each arm of a trial would achieve 80% power to detect a statistically significant reduction in HIV risk. We find that the relative risk is sensitive to the assumed degree of correlation between condom use and the acceptability of the prevention method. We also find that the most efficient trial design may be to offer both MMC and PrEP to males but to offer only a microbicide ring to females. Further work is required to better understand the processes by which adolescent prevention method choices are made. / AFRIKAANSE OPSOMMING: Adolessente word beskou as ‘n hoe risiko groep in Suid Afrika, met die hoogste menslike immuniteitsgebrekvirus (MIV) insidensie in hierdie groep. Voorkoming van MIV onder adolessente is daarom noodsaaklik om die MIV las te verminder. Die doel van hierdie tesis is om te help met die ontwerp van studies deur die moontlike uitkomste van ‘n kombinasie-voorkoming studie in adolessente te simuleer. Ons het ‘n stogastiese individu-gebaseerde model, gestratifiseer met betrekking tot seks en ouderdom, ontwikkel. Ons het toe die model gebruik om die impak van ‘n verskeinheid van voorkomingspakette op MIV insidensie onder adolessente wat deelneem aan ‘n hipotetiese proef oor ‘n drie jaar periode, te bepaal. Die proef wat gesimuleer word behels a intervensie groep, waarin die jong volwassenes ‘n keuse van voorbehoedings metodes (insluitende mediese manlike besnydenis (MMB), pre-blootstelling profilakse (PrBP) en anti-retrovirale vaginale mikrobisiedes (ARV-VM)) aangebied word, en ‘n kontrole groep. Ons voorspel dat die impak van ‘n volle voorkomingspaket op MIV insidensie ‘n 46% per persoon-jaar (PPJ) (95% VI 47–47%) risiko vermindering sal wees. Die kombinasie van MMB en PrBP het ‘n substansiele impak op MIV insidensie onder mans, met ‘n relatiewe risiko van MIV infeksie van 51% PPJ (95% VI 49–53%). Om die keuse van PrBP, ‘n mikrobisiede gel of ‘n mikrobisiede in die vorm van ‘n vaginale ring aan vrouens te bied, is minder effektief, met ‘n relatiewe risiko van MIV infeksie van 57% PPJ (95% VI 56%–58%). Hierdie verskil nie substansieel van die beraamde relatiewe risiko in die geval waar slegs die vaginale ring gebied word nie, aangesien daar aanvaar word dat die ring die mees aanvaarde van die drie voorkomingsmetodes is. Ons het bepaal dat ‘n steekproef van ongeveer 1013 individue in elke arm van die proef nodig is om ‘n 80% kans te he om ‘n statisties betekenisvolle afname in MIV-risiko te bespeur. Ons vind dat die relatiewe risiko sensitief is tot die aanvaarde graad van die korrelasies tussen kondoom-gebruik en die aanvaarding van die voorkomings metodes. Ons het ook gevind dat dit mag wees dat die mees doeltreffende proef ontwerp is om beide MMB en PrBP vir mans en slegs ‘n mikrobisiede ring vir vrouens te bied. Verdere werk word benodig om die prosesse waarby jong volwassenes keuses maak oor voorkomingsmetodes te verstaan.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: In this dissertation we present a software tool, Artemis, that symbolically executes Java virtual machine bytecode to find bugs and automatically generate test cases to trigger the bugs found. Symbolic execution is a technique of static software analysis that entails analysing code over symbolic inputs—essentially, classes of inputs—where each class is formulated as constraints over some input domain. The analysis then proceeds in a path-sensitive way adding the constraints resulting from a symbolic choice at a program branch to a path condition, and branching non-deterministically over the path condition. When a possible error state is reached, the path condition can be solved, and if soluble, value assignments retrieved to be used to generate explicit test cases in a unit testing framework. This last step enhances confidence that bugs are real, because testing is forced through normal language semantics, which could prevent certain states from being reached. We illustrate and evaluate Artemis on a number of examples with known errors, as well as on a large, complex code base. A preliminary version of this work was successfully presented at the SAICSIT conference held on 1–3 October 2012, in Centurion, South Africa. / AFRIKAANSE OPSOMMING: In die dissertasie bied ons ’n stuk sagtewaregereedskap, Artemis, aan wat biskode van die Java virtuele masjien simbolies uitvoer om foute op te spoor en toetsgevalle outomaties voort te bring om die foute te ontketen. Simboliese uitvoering is ’n tegniek van statiese sagteware-analise wat behels dat kode oor simboliese toevoere—in wese, klasse van toevoer—geanaliseer word, waar elke klas geformuleer word as beperkinge oor ’n domein. Die analise volg dan ’n pad-sensitiewe benadering deur die domeinbeperkinge, wat volg uit ’n simboliese keuse by ’n programvertakking, tot ’n padvoorwaarde by te voeg en dan nie-deterministies vertakkings oor die padvoorwaarde te volg. Wanneer ’n moontlike fouttoestand bereik word, kan die padvoorwaarde opgelos word, en indien dit oplaasbaar is, kan waardetoekennings verkry word om eksplisiete toetsgevalle in ’n eenheidstoetsingsraamwerk te formuleer. Die laaste stap verhoog vertroue dat die foute gevind werklik is, want toetsing word deur die normale semantiek van die taal geforseer, wat sekere toestande onbereikbaar maak. Ons illustreer en evalueer Artemis met ’n aantal voorbeelde waar die foute bekend is, asook op ’n groot, komplekse versameling kode. ’n Voorlopige weergawe van die´ werk is suksesvol by die SAICSIT-konferensie, wat van 1 tot 3 Oktober 2012 in Centurion, Suid-Afrika, gehou is, aangebied.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: Cloud computing is seen as one of top 10 disruptive changes in IT for the next decade by leading research analysts. Consequently, enterprises are starting to investigate the effect it will have on the strategic direction of their businesses and technology stacks. Because of the disruptive nature of the paradigm shift introduced by it, as well as the strategic impact thereof, it is necessary that a structured approach with regard to risk, value and operational cost is followed with the decision on its relevance, as well as the selection of a platform if needed. The purpose of this thesis is to provide a reference model and its associating framework that can be used to evaluate private cloud management platforms, as well as the technologies associated with it. / AFRIKAANSE OPSOMMING: Wolk berekening word deur vooraanstaande navorsing ontleders as een van die top 10 ontwrigtende veranderings vir IT in die volgende dekade beskou. Gevolglik begin korporatiewe ondernemings met ondersoeke om te bepaal wat die invloed daarvan op hulle strategiese rigting en tegnologië gaan wees. Die ontwrigtende aard van die paradigma skuif, asook die strategiese impak daarvan, noodsaak ’n gestruktureerde ondersoek na die toepaslikheid en keuse van ’n platform, indien nodig, met betrekking tot risiko, waarde en operasionele koste. Die doel van hierdie tesis is om ’n verwysings model, en ’n raamwerk wat dit implementeer, saam te stel wat dan gebruik kan word om privaat wolk berekening platforms te evalueer.

Thesis (PhD)--University of Stellenbosch, 2001. / ENGLISH ABSTRACT: In this thesis we examine some aspects of automatic face recognition, with specific reference to the eigenface technique. We provide a thorough theoretical analysis of this technique which allows us to explain many of the results reported in the literature. It also suggests that clustering can improve the performance of the system and we provide experimental evidence of this. From the analysis, we also derive an efficient algorithm for updating the eigenfaces. We demonstrate the ability of an eigenface-based system to represent faces efficiently (using at most forty values in our experiments) and also demonstrate our updating algorithm. Since we are concerned with aspects of face recognition, one of the important practical problems is locating the face in a image, subject to distortions such as rotation. We review two well-known methods for locating faces based on the eigenface technique.e These algorithms are computationally expensive, so we illustrate how the Synthetic Discriminant Function can be used to reduce the cost. For our purposes, we propose the concept of a linearly interpolating SDF and we show how this can be used not only to locate the face, but also to estimate the extent of the distortion. We derive conditions which will ensure a SDF is linearly interpolating. We show how many of the more popular SDF-type filters are related to the classic SDF and thus extend our analysis to a wide range of SDF-type filters. Our analysis suggests that by carefully choosing the training set to satisfy our condition, we can significantly reduce the size of the training set required. This is demonstrated by using the equidistributing principle to design a suitable training set for the SDF. All this is illustrated with several examples. Our results with the SDF allow us to construct a two-stage algorithm for locating faces. We use the SDF-type filters to obtain initial estimates of the location and extent of the distortion. This information is then used by one of the more accurate eigenface-based techniques to obtain the final location from a reduced search space. This significantly reduces the computational cost of the process. / AFRIKAANSE OPSOMMING: In hierdie tesis ondersoek ons sommige aspekte van automatiese gesigs- herkenning met spesifieke verwysing na die sogenaamde eigengesig ("eigen- face") tegniek. ‘n Deeglike teoretiese analise van hierdie tegniek stel ons in staat om heelparty van die resultate wat in die literatuur verskyn te verduidelik. Dit bied ook die moontlikheid dat die gedrag van die stelsel sal verbeter as die gesigte in verskillende klasse gegroepeer word. Uit die analise, herlei ons ook ‘n doeltreffende algoritme om die eigegesigte op te dateer. Ons demonstreer die vermoë van die stelsel om gesigte op ‘n doeltreffende manier te beskryf (ons gebruik hoogstens veertig eigegesigte) asook ons opdateringsalgoritme met praktiese voorbeelde. Verder ondersoek ons die belangrike probleem om gesigte in ‘n beeld te vind, veral as rotasie- en skaalveranderinge plaasvind. Ons bespreek twee welbekende algoritmes om gesigte te vind wat op eigengesigte gebaseer is. Hierdie algoritme is baie duur in terme van numerise berekeninge en ons ontwikkel n koste-effektiewe metode wat op die sogenaamde "Synthetic Discriminant Functions" (SDF) gebaseer is. Vir hierdie doel word die begrip van lineêr interpolerende SDF’s ingevoer. Dit stel ons in staat om nie net die gesig te vind nie, maar ook ‘n skatting van sy versteuring te bereken. Voorts kon ons voorwaardes aflei wat verseker dat ‘n SDF lineêr interpolerend is. Aangesien ons aantoon dat baie van die gewilde SDF-tipe filters aan die klassieke SDF verwant is, geld ons resultate vir ‘n hele verskeidenheid SDF- tipe filters. Ons analise toon ook dat ‘n versigtige keuse van die afrigdata mens in staat stel om die grootte van die afrigstel aansienlik te verminder. Dit word duidelik met behulp van die sogenaamde gelykverspreidings beginsel ("equidistributing principle") gedemonstreer. Al hierdie aspekte van die SDF’s word met voorbeelde geïllustreer. Ons resultate met die SDF laat ons toe om ‘n tweestap algoritme vir die vind van ‘n gesig in ‘n beeld te ontwikkel. Ons gebruik eers die SDF-tipe filters om skattings vir die posisie en versteuring van die gesig te kry en dan verfyn ons hierdie skattings deur een van die teknieke wat op eigengesigte gebaseer is te gebruik. Dit lei tot ‘n aansienlike vermindering in die berekeningstyd.

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Computational molecular evolution is a field that attempts to characterize how genetic sequences evolve over phylogenetic trees – the branching processes that describe the patterns of genetic inheritance in living organisms. It has a long history of developing progressively more sophisticated stochastic models of evolution. Through a probabilist’s lens, this can be seen as a search for more appropriate ways to parameterize discrete state continuous time Markov chains to better encode biological reality, matching the historical processes that created empirical data sets, and creating useful tools that allow biologists to test specific hypotheses about the evolution of the organisms or the genes that interest them. This dissertation is an attempt to fill some of the gaps that persist in the literature, solving what we see as existing open problems. The overarching theme of this work is how to better model variation in the action of natural selection at multiple levels: across genes, between sites, and over time. Through four published journal articles and a fifth in preparation, we present amino acid and codon models that improve upon existing approaches, providing better descriptions of the process of natural selection and better tools to detect adaptive evolution. / AFRIKAANSE OPSOMMING: Komputasionele molekulêre evolusie is ’n navorsingsarea wat poog om die evolusie van genetiese sekwensies oor filogenetiese bome – die vertakkende prosesse wat die patrone van genetiese oorerwing in lewende organismes beskryf – te karakteriseer. Dit het ’n lang geskiedenis waartydens al hoe meer gesofistikeerde waarskynlikheidsmodelle van evolusie ontwikkel is. Deur die lens van waarskynlikheidsleer kan hierdie proses gesien word as ’n soektog na meer gepasde metodes om diskrete-toestand kontinuë-tyd Markov kettings te parametriseer ten einde biologiese realiteit beter te enkodeer – op so ’n manier dat die historiese prosesse wat tot die vorming van biologiese sekwensies gelei het nageboots word, en dat nuttige metodes geskep word wat bioloë toelaat om spesifieke hipotesisse met betrekking tot die evolusie van belanghebbende organismes of gene te toets. Hierdie proefskrif is ’n poging om sommige van die gapings wat in die literatuur bestaan in te vul en bestaande oop probleme op te los. Die oorkoepelende tema is verbeterde modellering van variasie in die werking van natuurlike seleksie op verskeie vlakke: variasie van geen tot geen, variasie tussen posisies in gene en variasie oor tyd. Deur middel van vier gepubliseerde joernaalartikels en ’n vyfde artikel in voorbereiding, bied ons aminosuur- en kodon-modelle aan wat verbeter op bestaande benaderings – hierdie modelle verskaf beter beskrywings van die proses van natuurlike seleksie sowel as beter metodes om gevalle van aanpassing in evolusie te vind.

Phylogenetic models generally assume a homogeneous, time reversible, stationary process. These assumptions are often violated by the real, far more complex, evolutionary process. This thesis is centered on non-homogeneous, lineage-specific, properties of molecular sequences. It consist several related but independent studies. LineageSpecificSeqgen, an extension to the Seq-Gen program, which allows generation of sequences with changes in the proportion of variable sites, is introduced. This program is then used in a simulation study showing that changes in the proportion of variable sites can hinder tree estimation accuracy, and that tree reconstruction under the best-fit model chosen using a relative test can result in a wrong tree. In this case, the less commonly used absolute model-fit was a better predictor of tree estimation accuracy. This study found that increased taxon sampling of lineages that have undergone a change in the proportion of variable sites was critical for accurate tree reconstruction and that, in contrast to some earlier findings, the accuracy of maximum parsimony is adversely affected by such changes. This thesis also addresses the well-known long-branch attraction artifact. A nonparametric bootstrap test to identify changes in the substitution process is introduced, validated, and applied to the case of Microsporidia, a highly reduced intracellular parasite. Microsporidia was first thought to be an early branching eukaryote, but is now believed to be sister to, or included within, fungi. Its apparent basal eukaryote position is considered a result of long-branch attraction due to an elevated evolutionary rate in the microsporidian lineage. This study shows that long-branch estimates and basal positioning of Microsporidia both correlate with increased proportions of radical substitutions in the microsporidian lineage. In simulated data, such increased proportions of radical substitutions leads to erroneous long-branch estimates. These results suggest that the long microsporidian branch is likely to be a result of an increased proportion of radical substitutions on that branch, rather than increased evolutionary rate per se. The focus of the last study is the intriguing case of Mesostigma, a fresh water green alga for which contradicting phylogenetic relationships were inferred. While some studies placed Mesostigma within the Streptophyta lineage (which includes land plants), others placed it as the deepest green algae divergence. This basal positioning is regarded as a result of long-branch attraction due to poor taxon sampling. Reinvestigation of a 13- taxon mitochondrial amino acid dataset and a sub-dataset of 8 taxa reveals that site sampling, and in particular the treatment of missing data, is just as important a factor for accurate tree reconstruction as taxon sampling. This study identifies a difficulty in recreating the long-branch attraction observed for the 8-taxon dataset in simulated data. The cause is likely to be the smaller number of amino acid characters per site in simulated data compared to real data, highlighting the fact that there are properties of the evolutionary process that are yet to be accurately modeled.

Phylogenetic models generally assume a homogeneous, time reversible, stationary process. These assumptions are often violated by the real, far more complex, evolutionary process. This thesis is centered on non-homogeneous, lineage-specific, properties of molecular sequences. It consist several related but independent studies. LineageSpecificSeqgen, an extension to the Seq-Gen program, which allows generation of sequences with changes in the proportion of variable sites, is introduced. This program is then used in a simulation study showing that changes in the proportion of variable sites can hinder tree estimation accuracy, and that tree reconstruction under the best-fit model chosen using a relative test can result in a wrong tree. In this case, the less commonly used absolute model-fit was a better predictor of tree estimation accuracy. This study found that increased taxon sampling of lineages that have undergone a change in the proportion of variable sites was critical for accurate tree reconstruction and that, in contrast to some earlier findings, the accuracy of maximum parsimony is adversely affected by such changes. This thesis also addresses the well-known long-branch attraction artifact. A nonparametric bootstrap test to identify changes in the substitution process is introduced, validated, and applied to the case of Microsporidia, a highly reduced intracellular parasite. Microsporidia was first thought to be an early branching eukaryote, but is now believed to be sister to, or included within, fungi. Its apparent basal eukaryote position is considered a result of long-branch attraction due to an elevated evolutionary rate in the microsporidian lineage. This study shows that long-branch estimates and basal positioning of Microsporidia both correlate with increased proportions of radical substitutions in the microsporidian lineage. In simulated data, such increased proportions of radical substitutions leads to erroneous long-branch estimates. These results suggest that the long microsporidian branch is likely to be a result of an increased proportion of radical substitutions on that branch, rather than increased evolutionary rate per se. The focus of the last study is the intriguing case of Mesostigma, a fresh water green alga for which contradicting phylogenetic relationships were inferred. While some studies placed Mesostigma within the Streptophyta lineage (which includes land plants), others placed it as the deepest green algae divergence. This basal positioning is regarded as a result of long-branch attraction due to poor taxon sampling. Reinvestigation of a 13- taxon mitochondrial amino acid dataset and a sub-dataset of 8 taxa reveals that site sampling, and in particular the treatment of missing data, is just as important a factor for accurate tree reconstruction as taxon sampling. This study identifies a difficulty in recreating the long-branch attraction observed for the 8-taxon dataset in simulated data. The cause is likely to be the smaller number of amino acid characters per site in simulated data compared to real data, highlighting the fact that there are properties of the evolutionary process that are yet to be accurately modeled.

Phylogenetic models generally assume a homogeneous, time reversible, stationary process. These assumptions are often violated by the real, far more complex, evolutionary process. This thesis is centered on non-homogeneous, lineage-specific, properties of molecular sequences. It consist several related but independent studies. LineageSpecificSeqgen, an extension to the Seq-Gen program, which allows generation of sequences with changes in the proportion of variable sites, is introduced. This program is then used in a simulation study showing that changes in the proportion of variable sites can hinder tree estimation accuracy, and that tree reconstruction under the best-fit model chosen using a relative test can result in a wrong tree. In this case, the less commonly used absolute model-fit was a better predictor of tree estimation accuracy. This study found that increased taxon sampling of lineages that have undergone a change in the proportion of variable sites was critical for accurate tree reconstruction and that, in contrast to some earlier findings, the accuracy of maximum parsimony is adversely affected by such changes. This thesis also addresses the well-known long-branch attraction artifact. A nonparametric bootstrap test to identify changes in the substitution process is introduced, validated, and applied to the case of Microsporidia, a highly reduced intracellular parasite. Microsporidia was first thought to be an early branching eukaryote, but is now believed to be sister to, or included within, fungi. Its apparent basal eukaryote position is considered a result of long-branch attraction due to an elevated evolutionary rate in the microsporidian lineage. This study shows that long-branch estimates and basal positioning of Microsporidia both correlate with increased proportions of radical substitutions in the microsporidian lineage. In simulated data, such increased proportions of radical substitutions leads to erroneous long-branch estimates. These results suggest that the long microsporidian branch is likely to be a result of an increased proportion of radical substitutions on that branch, rather than increased evolutionary rate per se. The focus of the last study is the intriguing case of Mesostigma, a fresh water green alga for which contradicting phylogenetic relationships were inferred. While some studies placed Mesostigma within the Streptophyta lineage (which includes land plants), others placed it as the deepest green algae divergence. This basal positioning is regarded as a result of long-branch attraction due to poor taxon sampling. Reinvestigation of a 13- taxon mitochondrial amino acid dataset and a sub-dataset of 8 taxa reveals that site sampling, and in particular the treatment of missing data, is just as important a factor for accurate tree reconstruction as taxon sampling. This study identifies a difficulty in recreating the long-branch attraction observed for the 8-taxon dataset in simulated data. The cause is likely to be the smaller number of amino acid characters per site in simulated data compared to real data, highlighting the fact that there are properties of the evolutionary process that are yet to be accurately modeled.

Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007. / We study refinable functions where the dilation factor is not always assumed to be 2. In our investigation, the role of convolutions and refinable step functions is emphasized as a framework for understanding various previously published results. Of particular importance is a class of polynomial factors, which was first introduced for dilation factor 2 by Berg and Plonka and which we generalise to general integer dilation factors. We obtain results on the existence of refinable functions corresponding to certain reduced masks which generalise similar results for dilation factor 2, where our proofs do not rely on Fourier methods as those in the existing literature do. We also consider subdivision for general integer dilation factors. In this regard, we extend previous results of De Villiers on refinable function existence and subdivision convergence in the case of positive masks from dilation factor 2 to general integer dilation factors. We also obtain results on the preservation of subdivision convergence, as well as on the convergence rate of the subdivision algorithm, when generalised Berg-Plonka polynomial factors are added to the mask symbol. We obtain sufficient conditions for the occurrence of polynomial sections in refinable functions and construct families of related refinable functions. We also obtain results on the regularity of a refinable function in terms of the mask symbol factorisation. In this regard, we obtain much more general sufficient conditions than those previously published, while for dilation factor 2, we obtain a characterisation of refinable functions with a given number of continuous derivatives. We also study the phenomenon of subsequence convergence in subdivision, which explains some of the behaviour that we observed in non-convergent subdivision processes during numerical experimentation. Here we are able to establish different sets of sufficient conditions for this to occur, with some results similar to standard subdivision convergence, e.g. that the limit function is refinable. These results provide generalisations of the corresponding results for subdivision, since subsequence convergence is a generalisation of subdivision convergence. The nature of this phenomenon is such that the standard subdivision algorithm can be extended in a trivial manner to allow it to work in instances where it previously failed. Lastly, we show how, for masks of length 3, explicit formulas for refinable functions can be used to calculate the exact values of the refinable function at rational points. Various examples with accompanying figures are given throughout the text to illustrate our results.