In this thesis lessons learned from the use of computer algebra systems and machine assisted theorem provers are developed in order to give an insight into both the problems and their solutions. Many algorithms in computational algebra and automated deduction (for example Grobner basis computations and Knuth-Bendix completion) tend to produce redundant facts and can contain more than one proof of any particular fact. This thesis introduces proof diagrams in order to compare and contrast the proofs of facts which such procedures generate. Proof diagrams make it possible to analyse the effect of heuristics which can be used to guide implementations of such algorithms. An extended version of an inference system for Knuth-Bendix completion is introduced. It is possible to see that this extension characterises the applicability of critical pair criteria, which are heuristics used in completion. We investigate a number of executions of a completion procedure by analysing the associated proof diagrams. This leads to a better understanding of the heuristics used to control these examples. Derived rales of inference are also investigated in this thesis. This is done in the formalism of proof diagrams. Rewrite rules for proof diagrams are defined: this is motivated by the notion of a transformation tactic in the Nuprl proof development system. A method to automatically extract 'useful' derived inference rales is also discussed. 'Off the shelf' theorem provers, such as the Larch Prover and Otter, are compared to specialised programs from computational group theory. This analysis makes it possible to see where methods from automated deduction can improve on the tools which group theorists currently use. Problems which can be attacked with theorem provers but not with currently used specialised programs are also indicated. Tietze transformations, from group theory, are discussed. This makes it possible to link ideas used in Knuth-Bendix completion programs and group presentation simplification programs. Tietze transformations provide heuristics for more efficient and effective implementations of these programs.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciencias Fisicas e Matematicas / Made available in DSpace on 2012-10-16T21:34:21Z (GMT). No. of bitstreams: 0Bitstream added on 2016-01-08T14:04:14Z : No. of bitstreams: 1 91789.pdf: 1241872 bytes, checksum: c6eb5914edbe4ecbb40ed1bbcba1fec1 (MD5) / ARANTE, Jurema Maria Costa. Sobre a sym-universalidade de palavras primitivas. Florianópolis, 1981. 51p. Dissertação (Mestrado em Matemática) - Curso de Pós-Graduação em Matemática, Universidade Federal de Santa Catarina.

This study was conducted in one of the education districts in the Eastern Cape Province of South Africa. The purpose was to analyse the effectiveness of the concrete/semi-concrete/abstract (CSA) approach and drill-practice instructional strategies on Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions. The following two objectives were set. First, to identify the learners’ challenges in studying addition and subtraction of algebraic fractions in grade 10; and second to analyse the effectiveness of the CSA approach and drill-practice instructional strategies on Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions. Both threshold concepts and troublesome knowledge, Polya’s problem-solving techniques, CSA Approach theory and Drill-practice theory were all pertinent as a theoretical framework for the study. Positivism research paradigm was adopted for the study and it afforded the researcher opportunity to employ quantitative research approach. Based on the research question of this study, an experimental design was chosen as a suitable descriptive design. Purposive sampling method was used to select three schools which involved 135 grade 10 mathematics learners. Stratified random sampling method was thereafter employed to select 45 learners from each school for the study. The learners were grouped in each school as top, average and weak based on their performance in Algebra in term one. Pre-questionnaire and post-questionnaire were used to obtain data regarding challenges learners experience in simplifying addition and subtraction of algebraic fractions. Ethical clearance from the relevant school and university authorities were obtained. On the first two days, the researcher briefed the school authorities and learners and explained to them the purpose and details of the study. Day three was used to administer the pre-questionnaire test, thereafter, the next ten days were used to teach addition and subtraction of both numeric and algebraic fractions with same and different numerators and denominators. The next two days were used for revision and the last day was used to administer the postquestionnaire test out 25 marks. The respondent rate was 98.5%. The data collected were analysed by using SPSS version 16.10. Both descriptive and inferential statistics were used to analyse the data. The pre-questionnaire scores revealed that majority of the learners’ perceived fractions as two separate entities and as a result add or subtract numerator to numerator and denominator to denominator. It was also discovered that learners had a challenge in finding LCM of algebraic fractions. A t-Test for independent means was used to test the following hypotheses at 𝛼 = 0.05: 𝐇𝟎: The CSA approach and drill-practice intervention has no significant effect on Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions; 𝐇𝟏: The CSA approach and drill-practice will significantly enhance Grade 10 learners’ ability to simplify addition and subtraction of algebraic fractions. The t-Test revealed a p-value of 0.139 which was statistically significant at 𝛼 = 0.05. Therefore, the researcher rejected the null hypothesis and concluded that the CSA approach and drill-practice have significantly enhanced the Grade 10 learners’ ability to simplify algebraic fractions.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Matemática Pura e Aplicada, Florianópolis, 2013 / Made available in DSpace on 2013-07-16T21:11:41Z (GMT). No. of bitstreams: 1 317399.pdf: 794727 bytes, checksum: 2e84c7d9ae9b5bab25109d3a9697a381 (MD5) / Álgebras de Nichols são ferramentas importantes para a classificação de álgebras de Hopf pontuadas (veja [3]). Uma álgebra de Nichols é, em suma, uma álgebra de Hopf trançada e graduada. Ao considerarmos a categoria dos módulos de Yetter-Drinfeld sobre uma álgebra de Hopf com antípoda bijetora, cria-se o ambiente para definir álgebras de Hopf trançadas nessa categoria (o que pode ser feito em uma categoria trançada qualquer). Esse trabalho desenvolve esse problema, isto é, dada uma álgebra de Hopf H com antípoda bijetora sobre um corpo k, nossos principais objetivos são estudar álgebras de Hopf trançadas na categoria dos módulos de Yetter-Drinfeld sobre H e mostrar a existência e a unicidade da álgebra de Nichols de um módulo de Yetter-Drinfeld sobre H.<br> / Abstract : Nichols algebras play an important role to classify pointed Hopf algebras. If we consider the category of modules of Yetter-Drinfeld over a Hopf algebra H with bijective antipode, we get a braided category and so it is possible to define a braided Hopf algebra there. In this work, we consider this kind of problem, i. e., given a Hopf algebra H with bijective antipode (over a field k), we consider the category of modules of Yetter-Drinfeld over it. We study braided Hopf algebras in this category and also we prove the existence and uniqueness of the Nichols algebra of a Yetter-Drinfeld module.

In this thesis we begin by presenting an introduction on random matrices, their different classes and applications in quantum mechanics to study the characteristics of the eigenvectors of a particular random matrix model. The focus of this work is on one of the oldest and most well-known symmetry classes of random matrices - the Gaussian unitary ensemble. We look at how the different possible deformations of the Gaussian unitary ensemble could have an impact on the nature of the eigenvectors, and back up our results by numerical simulations to confirm validity. We will begin exploring the structure of the eigenvectors by employing the supersymmetry technique, a method for studying eigenvectors of complex quantum systems. In particular, we can analyse the moments of the eigenvectors, a quantity used in the classification of eigenvectors, in different random matrix models. Eigenvectors can either be extended, localised or critical and the scaling of the moments of the eigenvectors with matrix size N is used to determine the exact type. This enables one to study the transition of the eigenvectors from extended to localised and the intermediate stages. We consider different classes of random matrices, such as random matrices with an external source and structured random matrices. In particular, we studied the Rosenzweig-Porter model by generalising our previous results from a deterministic potential to a random one and study the impact of such an alteration to the model.

This thesis investigates class field theory for one dimensional fields and higher dimensional fields. For one dimensional fields we cover the cases of local fields and global fields of positive characteristic. For higher dimensional fields we study the case of higher local fields of positive characteristic. The main content of the thesis is divided into two parts. The first part solves several problems directly related to Neukirch's axiomatic class field theory method. We first prove the famous Hilbert 90 Theorem in the case of tamely ramified extensions of local fields in an explicit way. This approach can be of use in understanding the role of the ring structure as opposed to the role of multiplication only in local class field theory. Next, we prove that for every local field, its `class field theory' is unique. Lastly, we establish the Neukirch axiom for global fields of positive characteristic, which leads to a new approach to class field theory for such fields, an approach that has not appeared in the previous literature. There are two main successful directions in higher local class field theory, one by Kato and another by Fesenko. While Kato used a technical cohomological method, Fesenko generalised the Neukirch method and gave the first proof of the existence theorem. In the second part of the thesis we deal with the third method in class field theory that works in positive characteristic only, the Kawada-Satake method. We generalise the classical Kawada-Satake method to higher local fields of positive characteristic. We correct substantial mistakes in a paper of Parshin on such class field theory. We develop the first complete presentation of the theory based on the generalised Kawada-Satake method using advanced properties of topological Milnor K-groups. These advanced properties include Fesenko's theorem about relations of topological and algebraic properties of Milnor K-groups.

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M.Sc. (Mathematics) / Please refer to full text to view abstract

M.Sc. (Mathematics) / Die eienskappe van 'n ring het 'n bepalende invloed op die eienskappe van die module oor daardie ring. So kan belangrike klasse van ringe gekarakteriseer word met behulp van module oar sodanige ringe. Origens blyk dat eienskappe van R-module wat in die algemeen nie saamval nie. weI saamval as ons R beperk tot sekere klasse van ringe. In die literatuur word die verwantskappe tussen 'n moduul en sy ring van skalare wyd verspreid aangetref. Die oogmerke van hierdie skripsie is om die belangrikste resultate wat die verwantskap tussen ringe uit 'n gegewe klas met hulle module aandui in een bron saam te bring vir sekere belangrike klasse van ringe. 'n Literatuurlys word vir verdere insae verskaf. Geen aanspraak word op volledigheid gemaak nie. Definisies word deurgaans gegee ter wille van volledigheid en ter wille van terminologie. Waar stellings en lemmas egter direk uit definisies volg. word geen bewys gegee nie. Bewyse word weI gegee in gevalle waar 'n besondere strategie. helderheid of bondigheid dit regverdig.

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