The behavioral properties shown by people when they make selections between different choices will be studied. Based on empirical and logical data a mathematical axiomatic model is built. D. Luce is a major contributor in this area. This thesis is based on his works and those of his many co-authors. Three approaches will be considered that lead to the rank-dependent utility representation of binary gambles composed only of gains (losses) relative to a status quo. The proofs involve the theory of functional equations which is a very powerful tool giving the precise numerical representations.

The behavioral properties shown by people when they make selections between different choices will be studied. Based on empirical and logical data a mathematical axiomatic model is built. D. Luce is a major contributor in this area. This thesis is based on his works and those of his many co-authors. Three approaches will be considered that lead to the rank-dependent utility representation of binary gambles composed only of gains (losses) relative to a status quo. The proofs involve the theory of functional equations which is a very powerful tool giving the precise numerical representations.

Thesis (Ph. D.)--University of Missouri-Columbia, 2007. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on December 28, 2007) Includes bibliographical references.

No description available.

It is widely believed that Quantum Chromodynamics (QCD) is the theory that describes the strong interaction. In the infrared region of the theory, the perturbative expansion breaks down and so, other techniques must be used. One such technique is the study of the Schwinger-Dyson equations. In this thesis is presented such a study. It is shown that the ghost sector of QCD may be crucial to the understanding of the infrared behaviour. Conventionally, the Slavnov- Taylor identity is used to truncate the Schwinger-Dyson equations but it is found that for the ghost-gluon vertex, such an identity cannot be used in an appropriate manner. In order to extract information, a new technique is presented, based on the powerlaw behaviour of the two-point functions in the infrared. By demanding consistency in the full equations in Landau gauge and multiplicative renormalisability, it is found that in general, the gluon propagator dressing function cannot diverge and the ghost propagator function cannot vanish in the infrared. Further, it is shown that the powerlaw behaviour depends on a certain kinematical limit of only one function connected with the ghost-gluon vertex.

A research project submitted in partial fulfilment of the requirements of the degree of Masters in Science Education (Mathematics Education) University of the Witwatersrand Johannesburg South Africa May 2016 / This study investigated learners’ mathematical discourse on the hyperbola using commognitive theory, with particular focus on the use of words, narratives, routines and visual mediators. Data was collected by means of task-based interviews with nine Grade 10 and 11 learners from a township school in Johannesburg, South Africa. An analytical tool, named the Discourse Profile of the Hyperbola, was adapted from the Arithmetic Discourse Profile of Ben-Yahuda et al (2005) and was used to analyse learners’ mathematical discourse. The study focused on three representations of the hyperbola, namely, the formulae (equation); the graph and the table. Learners’ views and definition of the asymptote, in relation to the graph, emerged as a central theme in the analysis. The analysis also focused on the mismatch between what is said and what is done by learners, for example most learners sketched the graph of a hyperbola showing a vertical asymptote yet talked as if there is no vertical asymptote. Most routines were ritualized, for example learners failed to link iconic and symbolic mediators they had used in responding to tasks. However, there were traces of exploratory routines from a few learners, evidenced by links between equations, and identifying the hyperbola from unfamiliar tasks. While a few learners used literate words, colloquial word use was dominant. The discourse of learners was found to be visual. For example, some reasoned that an equation with a fraction represents a hyperbola while an equation not expressed in standard form does not represent a hyperbola. Some learner narratives are not endorsed by the community of mathematicians.

This research report confirms the value of the Framework of Growth Points in a learner’s mathematical development in the area of functions in equation form. The study also shows that learners advance through the various growth points in a progressive, sequential fashion, which mirrors the results of Ronda’s study, on which a part of this study was based. The study was carried out in a high school in Johannesburg. Learners in Grades 9, 10 and 11 were required to do an assessment which tested for their achievement in different growth points. This study also explores the discourse of learners while they talked about the tasks in the assessment. A smaller sample of learners was interviewed so that the researcher could explore the nature of their discourse. This research report shows that there are patterns in the discourse of learners which can be related to the growth points that they achieve.

We present the techniques for the calculation of one- and two-loop integrals contributing to the virtual corrections to 2→2 scattering of massless particles. First, tensor integrals are related to scalar integrals with extra powers of propagators and higher dimension using the Schwinger representation. Integration By Parts and Lorentz Invariance recurrence relations reduce the number of independent scalar integrals to a set of master integrals for which their expansion in є = 2 — D/2 is calculated using a combination of Feynman parameters, the Negative Dimension Integration Method, the Differential Equations Method, and Mellin-Barnes integral representations. The two-loop matrix-elements for light-quark scattering are calculated in Conventional Dimensional Regularisation by direct evaluation of the Feynman diagrams. The ultraviolet divergences are removed by renormalising with the MS scheme. Finally, the infrared singular behavior is shown to be in agreement with the one anticipated by the application of Catani's formalism for the infrared divergences of generic QCD two-loop amplitudes.

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-10-16T11:31:20Z No. of bitstreams: 1 arquivototal.pdf: 956564 bytes, checksum: 874abf5457408fd33da298fe5e5cd62b (MD5) / Rejected by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br), reason: corrigir referência on 2015-10-16T12:08:50Z (GMT) / Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-10-16T12:11:23Z No. of bitstreams: 1 arquivototal.pdf: 956564 bytes, checksum: 874abf5457408fd33da298fe5e5cd62b (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-10-16T12:12:12Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 956564 bytes, checksum: 874abf5457408fd33da298fe5e5cd62b (MD5) / Made available in DSpace on 2015-10-16T12:12:12Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 956564 bytes, checksum: 874abf5457408fd33da298fe5e5cd62b (MD5) Previous issue date: 2014-04-04 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This paper presents a study on functional equations, considering its relevance to the teaching of mathematics, with the ultimate goal of presenting a proposal to contribute to the improvement of teaching this topic. A summary of the history of functional equations is presented. Then, Chapter 1 consists of the study of the Cauchy Functional Equations, Chapter 2 deals with equations Jensen, Pexider, d 'Alembert. In chapter 3 we show some applications of functional equations. / Este trabalho apresenta um estudo sobre equações funcionais, considerando sua relevância para o ensino da Matemática, tendo como objetivo nal apresentar uma proposta que contribua para a melhoria do ensino deste tópico. É apresentado um resumo sobre a história das equações funcionais. Em seguida, o capítulo 1 é constituído pelo estudo das equações Funcionais de Cauchy, o capítulo 2 trata das equações de Jensen, Pexider e d'Alembert. No capitulo 3 mostramos algumas aplicações das equações funcionais.

Submitted by Jean Medeiros (jeanletras@uepb.edu.br) on 2017-08-11T12:24:07Z No. of bitstreams: 1 PDF - Cleyson Cassimiro de Souza.pdf: 10161350 bytes, checksum: 0c61b0d485f05a94d08de746a02323cf (MD5) / Approved for entry into archive by Secta BC (secta.csu.bc@uepb.edu.br) on 2017-08-29T15:45:00Z (GMT) No. of bitstreams: 1 PDF - Cleyson Cassimiro de Souza.pdf: 10161350 bytes, checksum: 0c61b0d485f05a94d08de746a02323cf (MD5) / Made available in DSpace on 2017-08-29T15:45:00Z (GMT). No. of bitstreams: 1 PDF - Cleyson Cassimiro de Souza.pdf: 10161350 bytes, checksum: 0c61b0d485f05a94d08de746a02323cf (MD5) Previous issue date: 2017-06-16 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we will present some special cases of functional equations of one variable together with specific algorithms that will help us to find solutions for these types of equations. In order to fill some of the lack of Portuguese written materials on the subject, we first explore basic concepts of functions and limits and then actually enter into the functional equations of one variable. Finally, we will present some exercises in which the theory seen during the work can be applied. / Neste trabalho, apresentaremos alguns casos especiais de equações funcionais de uma variável juntamente com algoritmos específicos que nos ajudarão a encontrar soluções para estes tipos de equações. Com o objetivo de suprir um pouco da carência de materiais escritos em português sobre o tema, exploramos primeiramente conceitos básicos de funções e limites para em seguida ingressar de fato nas equações funcionais de uma variável. Por fim, apresentaremos alguns exercícios nos quais a teoria vista durante o trabalho pode ser aplicada.