Browsing by Author "Winter, Paul August."

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An analysis of learners' engagement in mathematical task.
(1988) Winter, Paul August.; Craig, A. P.
The present project is part of a larger research programme focussed on the analysis of change; one aspect being educational transformation and in particular an emphasis on the explication of the contentless processes (eg. logical operations, reasoning styles, analysis and synthesis) which underlie both learning and teaching at university level. The present project is aimed at an analysis of the teaching-learning dialectic in mathematics courses. This analysis has two major focal points, that is, making explicit the often tacit and mostly inadequate and/or inappropriate rules for engaging in mathematical tasks which the under-prepared learner brings to the teaching-learning situation, and secondly the teaching strategies which may enable these learners to overcome their past (erroneous) knowledge and skills towards the development of effecient, autonomous mathematical problem-solving strategies. In order to remedy inadequate and inappropriate past learning and/or teaching, the present project presents a set of mediational strategies and regulative cues which function both for the benefit of the teacher and the learner in a problematic teaching-learning situation and on the meta and epistemic cognitive levels of information processing. Furthermore, these mediational strategies and regulative cues fall on a kind of interface between contentless processes and the particular content of the teaching-learning dialectic of mathematics in particular, as well as between the ideal components of any instructional process and the particular needs and demands of under-prepared learners engaged in mathematical tasks.
The eigen-chromatic ratio of classes of graphs : molecular stability, asymptotes and area.
(2017) Mayala, Roger Mbonga.; Winter, Paul August.; Namayanja, Proscovia.
This dissertation involves combining the two concepts of energy and the chromatic number of classes of graphs into a new ratio, the eigen-chromatic ratio of a graph G. Associated with this ratio is the importance of its asymptotic convergence in applications, as well as the idea of area involving the Rieman integral of this ratio, when it is a function of the order n of the graph G belonging to a class of graphs. The energy of a graph G, is the sum of the absolute values of the eigenvalues associated with the adjacency matrix of G, and its importance has found its way into many areas of research in graph theory. The chromatic number of a graph G, is the least number of colours required to colour the vertices of the graph, so that no two adjacent vertices receive the same colour. The importance of ratios in graph theory is evident by the vast amount of research articles: Expanders, The central ratio of a graph, Eigen-pair ratio of classes of graphs , Independence and Hall ratios, Tree-cover ratio of graphs, Eigen-energy formation ratio, The eigen-complete difference ratio, The chromatic-cover ratio and "Graph theory and calculus: ratios of classes of graphs". We combine the two concepts of energy and chromatic number (which involves the order n of the graph G) in a ratio, called the eigen-chromatic ratio of a graph. The chromatic number associated with the molecular graph (the atoms are vertices and edges are bonds between the atoms) would involve the partitioning of the atoms into the smallest number of sets of like atoms so that like atoms are not bonded. This ratio would allow for the investigation of the effect of the energy on the atomic partition, when a large number of atoms are involved. The complete graph is associated with the value 1 2 when the eigen-chromatic ratio is investigated when a large number of atoms are involved; this has allowed for the investigation of molecular stability associated with the idea of hypo/hyper energetic graphs. Attaching the average degree to the Riemann integral of this ratio (as a function of n) would result in an area analogue for investigation. Once the ratio is defned the objective is to find the eigen-chromatic ratio of various well known classes of graphs such as the complete graph, bipartite graphs, star graphs with rays of length two, wheels, paths, cycles, dual star graphs, lollipop graphs and caterpillar graphs. Once the ratio of each class of graph are determined the asymptote and area of this ratio are determined and conclusions and conjectures inferred.
Energy of graphs : the Eigen-complete difference ratio with its asymptotic domination and area aspects.
(2015) Ojako, Samson Ogaga-Oghene.; Winter, Paul August.
Abstract available in PDF file.
Matrices of graphs and designs with emphasis on their integral eigen-pair balance characteristic.
(2014) Jessop, Carol Lynne.; Winter, Paul August.
The interplay between graphs and designs is well researched. In this dissertation we connect designs and graphs entirely through their associated matrices – the incidence matrix for designs and the adjacency matrix for graphs. The properties of graphs are immediately adopted by their associated designs, and the linear algebra of the common matrix, will apply to both designs and graphs sharing this matrix. We apply various techniques of finding the eigenvalues of the matrices associated with graphs/designs, to determine the eigenvalues of well-known classes of graphs, such as complete graphs, complete bipartite graphs, cycles, paths, wheels, stars and hypercubes. Graphs which are well connected, or edge-balanced, in terms of a centrally defined set of vertices, appear to give rise to a conjugate pair of eigenvalues. The association of integers, conjugate pairs and edge-balance with the eigenvalues of graphs provide the motivation for the new concepts of eigen-sum and eigen-product balanced properties of classes of graphs and designs. We combine these ideas by considering eigen bibalanced classes of graphs, where robustness and the reciprocity of the eigen-pair a,b allowed for the ratio of the eigen-pair sum to the eigen-pair product ab, and the asymptotic behaviour of this ratio (in terms of large values of the size of the graph/designs). The product of the average degree of a graph with the Riemann integral of the eigen bi-balanced ratio of the class of graphs is introduced as the area of a class of graphs/designs associated with the eigenpair. We observe that unique area of the class of complete graphs appears to be the largest. Also, the interval of asymptotic convergence of the eigen bi-balanced ratio, of uniquely eigen-bibalanced classes of graphs, appears to be [1,0]. We construct a new class of graphs, called q-cliqued graphs, involving q maximal cliques of size q, connected, and hence edge-balanced, to a central vertex. We apply the eigenvector method to find a general conjugate eigen-pair associated with the q-cliqued graphs and then determine the eigen-pair characteristics above for this class of graphs. The eigen-bi-balanced ratio associated with a conjugate pair of eigenvalues of the class of q-cliqued graphs, is the same as the eigen-bi-balanced ratio of the class of the complements of these graphs. The q-cliqued graphs are also designs, and we use the case q 10 as an application of a hypothetical entomological experiment involving 10 treatments and 10 blocks. We use the design's graphical characteristics to determine a possible scheduling situation which involves the chromatic number of its associated graph. PREFACE The experimental work described in this dissertation was carried out in the School of Mathematics, University of Natal, Durban, from February 2011 to November 2013, under the supervision of Dr Paul August Winter. These studies represent original work by the author and have not otherwise been submitted in any form for any degree or diploma to any tertiary institution. Where use has been made of the work of others it is duly acknowledged in the text
Spanning trees : eigenvalues, special numbers, and the tree-cover ratios, asymptotes and areas of graphs.
(2014) Adewusi, Fadekemi Janet.; Winter, Paul August.
This dissertation deals with spanning trees associated with graphs. The number of spanning trees of a graph can be found by considering the eigenvalues of the Laplacian matrix associated with that graph. Special classes of graphs are also considered, such as fan and wheel graphs, where their spanning tree numbers are connected to special numbers, like the Lucas and Fibonacci numbers. We use the eigenvalues of the complete graph and its associated circulant matrix to create a unit-trigonometric equation which generates a sequence and diagram similar to that of the famous Farey sequence. A new ratio is introduced: the tree-cover ratio involving spanning trees and vertex coverings and is motivated by the fact that such a ratio, associated with complete graphs, has the asymptotic convergence identical to that of the secretary problem. We use this ratio to introduce the idea of tree-cover asymptotes and areas and determine such values for known classes of graphs. This ratio, in communication networks, allows for the investigation of the outward social connectivity from a vertex covering to the rest of the network when a large number of vertices are involved.