The first two chapters provide the necessary prerequisites. In the third and fourth chapters we demonstrate than an affine Hjelmslev plane (or A. H. plane) is coordinatized by a biternary ring; and that given a biternary ring, one can construct an affine Hjelmslev plane. In the fifth and sixth chapters we introduce the notions of an ordering of an A. H. plane and an ordering of a biternary ring. In the seventh chapter we show that an ordering of an A. H. plane H induces an ordering on the coordinate biternary ring. In the eighth chapter we show that a given ordering of a biternary ring M induces an ordering on the A.H. plane constructed over M. In the remaining chapters we examine the associated ordinary affine plane of an A. H. plane, the case where an A. H. plane is Desarguesian, and give an example of an ordered non-Desarguesian A. H. plane. / Thesis / Master of Science (MSc)

<p> Extensive bedding plane exposures in the Ludlowville shales along Cazenovia Creek near Spring Brook, New York display the spatial distribution of the skeletal remains from a marine faunal assemblage. Fossils typically occur in aggregates that are subcircular in plan view and plano-convex in cross-section with the convex side down. The clusters measure 1 meter in diameter and 2 centimeters thick at the center. This dispersion pattern has led to a general consideration of the different mechanisms responsible for creating fossil aggregations. Possible mechanisms, a spectrum from biological to geological, have been categorized into reproductive, ecological, postmortem redistributional, and preservational modes of formation. </p> <p> Quantitative sampling of the most abundant species, Ambocoelia umbonata, in four successive 5 millimeter layers within three clusters was carried out to determine which process is responsible for cluster formation. Between level variation in shell parameters demonstrates that fragmentation, distortion and valve ratios are independent of trends in position, density, and disarticulation. The trends are not controlled by geological agents, but rather result from ecological conditions. Furthermore, the size distributions of Arnbocoelia are bimodal and have to be explained on a biological basis. This has led to an interpretation of cluster development involving initiation by occasional spat survival on a somewhat "lethal" substrate, subsequent succession and regulation by ecological requirements, and final termination due to failure of spat recruitment probably because of fecal and/or decay toxin buildup. </p> / Thesis / Master of Science (MSc)

In this thesis, first we propose a new approach to solve Semi Infinite Linear Optimization Problems (SILP). The new algorithm uses the idea of adding violated cut or cuts at each iteration. Our proposed algorithm distinguishes itself from Luo, Roos, and Terlaky's logarithmic barrier decomposition method, in three aspects: First, the violated cuts are added at their original locations. Second, we extend the analysis to the case where multiple violated cuts are added simultaneously, instead of adding only one constraint at a time. Finally, at each iteration we update the barrier parameter and the feasible set in the same step. In terms of complexity, we also show that a good approximation of an optimal solution will be guaranteed after finite number of iterations. Our focus in this thesis is mainly on the implementation of our algorithm to approximate an optimal solution of the SILP. Our numerical experiences show that unlike other SILP solvers which are suffering from the lack of accuracy, our algorithm can reach high accuracy in a competitive time. We discuss the linear algebra involved in efficient implementation and describe the software that was developed. Our test problem set includes large scale second order conic optimization problems. / Thesis / Master of Applied Science (MASc)

Several different approaches to determining second-order moments in plane frames were studied during this research. The focus of the research was to compare the moments predicted by four different commercially available computer analysis programs and the current design specification, the AISC LRFD moment magnification method. For this research, the second-order moments for ten commonly designed frames were compared. An overview of various second-order analysis procedures is presented first. The solution procedure utilized by each computer program and the AISC moment magnification method are explained. Also, the frames considered in the research are described. Next the frames are analyzed and the results between each of the computer programs and the current design specifications are compared. Finally, conclusions are drawn concerning the consistency of the second-order moments predicted by each of the solution procedures and recommendations for their use are discussed. In general, each of the four computer analysis programs evaluated and the AISC moment magnification method can consistently and adequately predict the second-order moments in plane frames. / Master of Science

One of the marvelous phenomena of coordinate geometry is the equivalence of Desargues' Theorem to the presence of an underlying division ring in a projective plane. Supplementing this correspondence is the general theory of intersection theorems, which, restricted to desarguian projective planes P, corresponds precisely to the theory of integral rational identities, restricted to division rings D. The first chapter of this paper introduces projective planes, develops the concept of an intersection theorem, and expounds upon the Theorem of Desargues; the discussion culminates with a proof of the desarguian phenomenon in the second chapter. The third chapter characterizes the automorphisms of P and introduces the theory of polynomial identities; the fourth chapter expands this discussion to rational identities and cements the ``dictionary''. The last section describes a measure of complexity for these intersection theorems, and the paper concludes with a curious spawn of the correspondence. / Master of Science

Maps have been extensively studied and are important in many research fields. A map is a 2-cell embedding of a graph on an orientable surface. Motivated by a new way to read the information provided by the skeleton of a map, we introduce new objects called plane permutations. Plane permutations not only provide new insight into enumeration of maps and related graph embedding problems, but they also provide a powerful framework to study less related genome rearrangement problems. As results, we refine and extend several existing results on enumeration of maps by counting plane permutations filtered by different criteria. In the spirit of the topological, graph theoretical study of graph embeddings, we study the behavior of graph embeddings under local changes. We obtain a local version of the interpolation theorem, local genus distribution as well as an easy-to-check necessary condition for a given embedding to be of minimum genus. Applying the plane permutation paradigm to genome rearrangement problems, we present a unified simple framework to study transposition distances and block-interchange distances of permutations as well as reversal distances of signed permutations. The essential idea is associating a plane permutation to a given permutation or signed permutation to sort, and then applying the developed plane permutation theory. / Ph. D. / This work is mainly concerned with studying two problems. The first problem starts with a graph <i>G</i> consisting of vertices and lines (called edges) linking some pairs of vertices. Intuitively, if the graph <i>G</i> can not be drawn on the sphere without crossing edges, it may be possibly drawn on a torus (i.e., the surface of a doughnut) without crossing edges; if it is still impossible, it may be possible to draw the graph <i>G</i> on the surface obtained by “gluing” several tori together. Once a graph <i>G</i> is drawn on a surface without crossing edges, there is a cyclic order of those edges incident to each vertex of the graph. Suppose you are not satisfied with how the edges around a vertex are cyclically arranged, and you want to arrange them differently. A question that arises naturally would be: is the adjusted drawing still cross-free on the original surface, or do we need to glue more (or fewer) tori in order for it to be crossfree? The second problem stems from genome rearrangements. In bioinformatics, people try to understand evolution (of species) by comparing the genome sequences (e.g., DNA sequences) of different species. Certain operations on genome sequences are believed to be potential ways of how species evolve. The operations studied in this work are transpositions, block-interchanges and reversals. For example, a transposition is such an operation that swaps two consecutive segments on the given genome sequence. As a candidate indicator of how far away one species is from another from an evolutionary perspective, we can compute how many transpositions are required to transform the genome sequence of one species to that of the other. In this work, we propose a plane permutation framework, which works effectively on solving the above mentioned two problems. In addition, plane permutations themselves are interesting objects to study and are studied as well.

Textile process and textile structures that are suitable for composites are carefully studied and chosen to have weft knitted fabrics. The aim of this research is to knit the carbon and Dyneema® fibres in circular weft knitting to fit contour shapes. Carbon/Dyneema® can also be knitted in warp knitting machines to get properties in multi axial direction. But the fabric was flat and can be used only for 2D shape products which are having less drapabiity. According to previous research, weft knitting is the best suitable for complex preforms. Before knitting these fibres properties were studied in order to avoid the damage to the carbon fibres. The carbon fibres have high bending rigidity, low resistance to friction and are very brittle. A small damage to the carbon fibre in knitting subsequently affect directly on the composite properties. The strongest manmade fibre manufactured till date is Dyneema® and these fibres could be used in composites due to its performance, properties and light weight. But, the Dyneema® fibres are expensive when compared to common polyester, so polyester fibres are used to compare the properties and cost performance ratio. The critical bending of the carbon fibres causes friction between the fibres and also between fibre and machine. This was considered carefully during the knitting of carbon fibres and the idea chosen is mentioned in this thesis. Between the two layers of Dyneema®/polyester, carbon fibres are laid circularly in unidirectional and in un-crimped condition. This makes the carbon yarn to possess good mechanical properties. The 2 layers of Dyneema®/polyester fibres exchange the loops at certain points to increase the inter-laminar strength and decrease the carbon fibre distortion. This structure helps to withstand external load. It is also lighter than the carbon composite with additional properties. This makes much more space in the future for the Dyneema® fibres in the 3D carbon composite manufacturing. The internal carbon fibres are fully covered by the Dyneema® fibres to withstand the external impact load and not to damage the carbon fibres. So the loop length, stitch density, fibre volume fractions are considered before knitting.

Recent developments in post-stall maneuverability and thrust vectoring have opened up new possibilities in the field of air combat maneuvering. High angle of attack maneuvers like the Cobra, Herbst Reversal, and Chakra demonstrate that today's cutting edge fighters are capable of exploiting the post-stall flight regime for very dynamic and unconventional maneuvers. With the development and testing of Unmanned Combat Aerial Vehicles, even greater maneuvering ability is expected. However, little work has been done to make use of this increased ability by optimizing a wide range of combat maneuvers. The goal of this thesis was to begin that process by finding several time-optimal air combat maneuvers that could be employed by current and future high performance fighter aircraft.

We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple.

Neste trabalho, estendemos o método desenvolvido em (SALARINOGHABI, 2016),(SALARINOGHABI; TARI, 2017) para curvas no plano Minkowski. Tal método propõe um modo de estudar deformações de curvas planas levando em consideração a geometria das mesmas juntamente com suas singularidades. Abordamos detalhadamente todos os fenômenos locais que ocorrem genericamente em famílias de curvas a 2-parâmetros. Em cada caso, obtemos a geometria da curva deformada, ou seja, informações a respeito de inflexões, vértices e pontos lightlike. Obtemos também o comportamento da evoluta/cáustica de uma curva em pontos especiais e as bifurcações que podem aparecer ao deformá-la. Além disso, a fim de obter as deformações genéricas em uma inflexão lightlike de ordem 2, também classificamos submersões de R3 em R por meio de difeomorfismos na fonte que preservam a swallowtail e, utilizando tal classificação, estudamos a geometria plana da swallowtail, a qual provém de seu contato com planos, o qual por sua vez é medido pelas singularidades da função altura sobre a swallowtail. / In this work, we extend the method developed in (SALARINOGHABI, 2016),(SALARINOGHABI; TARI, 2017) to curves in the Minkowski plane. The method proposes a way to study deformations of plane curves taking into consideration their geometry as well as their singularities. We deal in detail with all local phenomena that occur generically in 2-parameters families of curves. In each case, we obtain the geometry of the deformed curve, that is, information about inflections, vertices and lightlike points. We also obtain the behavior of the evolute/caustic of a curve at special points and the bifurcations that can occur when the curve is deformed. Moreover, in order to obtain the generic deformations at a lightlike inflection point of order 2, we also classify submersions from R3 to R by diffeomorphisms in the source that preserve the swallowtail and, using such classification, we study the flat geometry of the swallowtail, which comes from its contact with planes, which in turn is measured by the singularities of the height function on the swallowtail.