Thesis (Ph. D.)--University of Wisconsin--Madison, 1969. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Let U be an n-dimensional vector space over an algebraically closed field. Let [formula omitted] denote the [formula omitted] space spanned by all Grassmann products [formula omitted]. Subsets of vectors of [formula omitted] denoted by [formula omitted] and [formula omitted] are defined as follows [formula omitted]. A vector which is in [formula omitted] or is zero is called pure or decomposable. Each vector in [formula omitted] is said to have rank one. Similarly each vector in [formula omitted] has rank two. A subspace of H of [formula omitted] is called a rank two subspace If [formula omitted] is contained in [formula omitted]. In this thesis we are concerned with investigating rank two subspaces. The main results are as follows: If dim [formula omitted] such that every nonzero vector [formula omitted] is independent in U. The rank two subspaces of dimension less than four are also characterized. / Science, Faculty of / Mathematics, Department of / Graduate

Graduate

Thesis (M.A.)--Carleton University, 2001. / Includes bibliographical references (p. 90-98). Also available in electronic format on the Internet.

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D.Ing. / Aerial Vehicle (UAV)-based observation system, by using the principles of strapdown inertial measurement and navigation systems. Effort is concentrated around the mathematical implementation thereof and analysis and proof of the concept in a computer simulation environment. Although the principles of the strapdown system approach to camera-to-target vector estimation are universal to any type of airborne platform that can carry the observation payload, the application thereof is specifically tailored for a UAV system. More specifically, the operational scenario and UAV parameters of a typical close-range UAV system that is used for artillery observation, is used in the derivation of the models and equations. The secondary objective of this research is to derive a realizable mathematical implementation for this strapdown system based camera-to-target vector estimation methodology, by performing a systematic tradeoff between the use of Euler angles and quaternions for describing the camera-to-target vector, and by incorporating the principles of Kalman filtering. This dissertation fully describes the approach that was followed in the derivation of the strapdown system equations for the camera-to-target vector estimation. The mathematical models and principles used are universal for any airborne targeting application with a real-time video down-link. The results as presented in this dissertation, prove that the methodology provides satisfactory results in both a pure digital computer simulation environment, as well as in a digital computer simulation that is hybridized with experimentally determined sensor outputs. It has led to a realizable and workable implementation that could form the basis of practical implementation thereof in operational targeting systems. It further proves that the slant range between a camera and a stationary target on the ground, can be estimated effectively without the use of a laser rangefinder.

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If r > 1 is an integer then U(r) denotes the vector space of r-fold symmetric tensors and Pr[U] is the set of rank 1 tensors in U(r). Let U be a finite-dimensional vector space over an algebraically closed field of characteristic not a prime p if r = p[formula omitted] for some positive integer k. We first determine the maximal subspaces of rank 1 symmetric tensors. Suppose h is a linear mapping of U(r) such that h(Pr[U]) ⊆ Pr[U] and ker h ⋂ Pr[U] = 0. We have shown that every such h is induced by a non-singular linear mapping of U, provided dim U > r+1 . This work partially answers a question raised by Marcus and Newman (Ann. of Math., 75, (1962) p.62.). / Science, Faculty of / Mathematics, Department of / Graduate

We begin by constructing a vector space over a field F , which we call a (G,σ) space of the set W = V₁xV₂... xVn , a cartesian product, where Vi is a finite-dimensional vector space over an arbitrary field F , G is a subgroup of the full symmetric group Sn and σ is a linear character of G . This space generalizes the spaces called the symmetry class of tensors defined by Marcus and Newman [1]. We can obtain the classical spaces, namely the Tensor space, the Grassman space and the symmetric space, by particularizing the group G and the linear character σ in our (G,σ) space. If (v₁,v₂,..., vn ) ∈ W , we shall denote the "decomposable" element in our space by v₁Δv₂…Δvn and call it the (G,σ) product or the Pure product if there is no confusion regarding G and σ, of the vectors v₁,v₂,..., vn . This corresponds to the tensor product, the skew symmetric product and the symmetric product in the classical spaces. The purpose of this thesis is to determine a necessary and sufficient condition for the vanishing of the (G,σ) product of the vectors v₁,v₂,..., vn in the general case. The results for the classical spaces are well-known and are deduced from our main theorem. We use the "universal mapping property" of the (G,σ) space to prove the necessity of our condition. These conditions are stated in terms of determinant-like functions of the matrices associated with the set of vectors v₁,v₂,...,vn. / Science, Faculty of / Mathematics, Department of / Graduate

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