A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy (by production of original research) in Mathematics School of Mathematics University of the Witwatersrand Johannesburg December 2012 / Generating functions and asymptotic analysis have been used in four di er- ent situations to establish new results for extremely well studied structures. Later in this thesis a more detailed individual abstract for each of these studies is provided. The four situations are: A. Durfee square areas in integer partitions. B. A study of the relationship between integer compositions and their constituent partitions by specifying the asymptotic expectation of the number of such partitions in arbitrary composition. C. Similar to B above but focusing more on the generating functions rather than on the expectations derived therefrom. D. In the area of plane partitions with additional structure imposed upon them.

Thesis (ED. D.)--Temple University. / Includes bibliographical references (p. 132-134).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.

No description available.

This paper will investigate the geometry that results when Euclid's fifth postulate is replaced by the Lobachevskian assumption. The investigations will be limited to the plane.

The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Title from PDF of title page (University of Missouri--Columbia, viewed on January 14, 2010). Thesis advisor: Dr. Alex Iosevich. Includes bibliographical references.

Thesis--Universiteit te Amsterdam, 1927. / Includes bibliographical references.

Thesis (M. Sc.)--University of Canterbury, 2009. / Typescript (photocopy). "September 2009." Includes bibliographical references (p. 115-116) and index. Also available via the World Wide Web.

The property that a graph has an embedding in the projective plane is closed under taking minors. Thus by the well known Graph Minor theorem of Robertson and Seymour, there exists a finite list of minor-minimal graphs, call it L, such that a given graph G is projective planar if and only if G does not contain any graph isomorphic to a member of L as a minor. Glover, Huneke and Wang found 35 graphs in L, and Archdeacon proved that those are all the members of L, but Archdeacon's proof never appeared in any refereed journal. In this thesis we develop a modern approach and technique for finding the list L, independent of previous work. Our approach is based on conditioning on the connectivity of a member of L. Assume G is a member of L. If G is not 3-connected then the structure of G is well understood. In the case that G is 3-connected, the problem breaks down into two main cases, either G has an internal separation of order three or G is internally 4-connected. In this thesis we find the set of all 3-connected minor minimal non-projective planar graphs with an internal 3-separation. For proving our main result, we use a technique which can be considered as a variation and generalization of the method that Robertson, Seymour and Thomas used for non-planar extension of planar graphs. Using this technique, besides our main result, we also classify the set of minor minimal obstructions for a-, ac-, abc-planarity for rooted graphs. (A rooted graph (G,a,b,c) is a-planar if there exists a split of the vertex a to a' and a' in G such that the new graph G' obtained by the split has an embedding in a disk such that the vertices a', b, a', c are on the boundary of the disk in the order listed. We define b- and c-planarity analogously. We say that the rooted graph (G,a,b,c) is ab-planar if it is a-planar or b-planar, and we define abc-planarity analogously.)

The purpose of this study was to develop a course of study on Plane Geometry for the 8th grade, based upon the Instructional Systems Analysis technique. The systems model used was that developed by Bartel.The entire study involved identifying, selecting, and sequencing learning tasks. An elaborate review of resource materials on Plane Geometry yielded one hundred and fifty-five related tasks.The instrument developed to gather pertinent data had two parts to it. Part one contained seventeen items relating to the personal profile of the respondent. Part two consisted of the one hundred and fifty-five identified, verified, and correctly stated manipulative tasks. The tasks were grouped into two broad divisions. Each respondent rated each task on a 3-point scale according to the level of instructional importance he/she judged the task(s) should receive. A rating of 3 represented the highest level of importance; 2, average; and 1, least.The survey utilized the services of sixty-eight Nigerians currently on the Technical Teachers Training Program (TTTP) in the U.S. Information regarding the institutions and population of Nigerians on the program was obtained from the Training and Development Program (TDP) office in Washington, D.C. From the list of thirty-two possible institutions, twelve institutions met the criterion for participation.A Five-Interval Sampling Technique (FIST) was used to select subjects from the participating institutions. This gave the researcher a possible seventy subjects. Each subject's participation was purely voluntary. Thus the 73.6 percent return was considered satisfactory.The interpretation of data involved two analyses. Analysis one dealt with biographical information on each participant. The second analysis computed the summary of task ratings on the one hundred and fifty-five instrument items.Selection of tasks for inclusion in the course of study reported in this study was based on the judgments of the subjects. Higher mean values were used as the basis for selection. Overall, thirty top tasks were selected and sequenced for the purpose of the course of study under investigation.