Thesis (PH. D.) - University of Michigan, 1941. / Reprinted from the Transactions of the American mathematical society, vo. 52, no.l 3 ... November, 1942. Bibliography: p. 521.

Thesis (PH. D.) - University of Michigan, 1940. / Reprinted from Annals of mathematics, vol. 42, no. 1, January, 1941. Bibliography: p. 246.

Academisch proefschrift--Vrije Universiteit, Amsterdam. / "Stellingen": [2] p. laid in. Bibliography: p. 20-21.

Thesis--University of Kansas. / Includes bibliographical references.

Certain properties of the solution u of the equation Pu = v in a Banach space will be investigated. It will be assumed that v is a prescribed element of the space, P is a transformation defined on a closed subset in the space and consisting of the sum of a linear transformation and a contraction mapping, and that P and v depend on a real variable λ. which assumes values over the half-open positive interval 0 < λ ≤ λₒ. Then a theorem will be proved, establishing the existence and uniqueness of the solution u(λ) of P(λ)u(λ) = v(λ) . Under the hypothesis that P and v possess asymptotic expansions as λ→0, it will be shown that asymptotic solutions exist, that they are asymptotically unique, and that they possess asymptotic expansions which may be determined by a recursive process from those of P and v. The results obtained will be applied to particular types of Banach spaces, such as finite-dimensional Euclidean spaces, spaces of Lebesgue-square-summable functions and of continuous functions over a closed interval. / Science, Faculty of / Mathematics, Department of / Graduate

The objective of this thesis is to determine the linear transformations of a Grassmann product space which sends the set of nonzero Grassmann product vectors (also called pure vectors) into itself. Let U be an n-dimensional vector space over a field F and let r be an integer such that 0 ≤ r≤ n. The r-th Grassmann product space will be denoted by Ar(U). Subspaces of Ar(U) consisting entirely of pure vectors are called pure subspaces. With each non-zero pure vector of Ar(U) we associate an r-dimensional subspace of U. By studying the set of subspaces of U corresponding to a basis set of a pure subspace of Ar(U) we are able to deduce the form of this pure subspace. In this way we are able to classify the pure subspaces of Ar(U), arriving at only two essentially distinct types. We next study the maximal pure subspaces, i. e. the pure subspaces which are not contained in larger pure subspaces. They are of importance because the assumptions on the linear transformations under consideration imply that a maximal pure subspace is mapped into another maximal pure subspace. The form of the transformation is now almost completely determined by examining the incidence relations between pairs of maximal pure subspaces before and after the transformation is applied. Some algebraic manipulations are then needed in order to display the form of the transformation completely. With the suitable assumptions, our results state that the transformations under consideration are induced by linear transformations of the vector space U, except possibly when 2r = n. When 2r = n two types of transformations are possible. This arises from the fact that the two types of maximal pure subspaces have the same dimensions, (unlike the situation when 2r ≠ n). One type of transformation (those induced by linear maps of U)does not alter the type of pure subspaces, while the other interchanges the two types. / Science, Faculty of / Mathematics, Department of / Graduate

The chief aim of this work is to investigate cosmological consequences of a hypothesis put forward by Morrison and Gold in 1956. These authors postulate the existence of equal amounts of matter and antimatter in our universe. Abandoning the principle of equivalence, they attribute negative gravitational mass to anti-nucleons. The result is a drastic alteration in the field equation for the gravitational potential. In the first three chapters Newtonian Cosmology is developed from basic principles. The equations describing a universe consisting of matter are set up and solved. In chapter IV the hypothesis of Morrison and Gold is introduced, and the resulting model for the universe is compared with models obtained in chapter III. It is concluded that within the framework of the model considered, the hypothesis of Morrison and Gold is incompatible with the observational evidence, because it leads to an age of the universe of between 1.3 and 1.9 billion years, which is less than the age derived from other geological and astrophysical data. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

In this paper we obtain a generalization of the well known Riesz Representation Theorem to the case where the underlying space X is an infinite dimensional product of locally compact, regular and σ-compact topological spaces. In the process we prove that our measures on X correspond to projective limit measures of projective systems of regular Borel measures on the coordinate spaces. An example is given to show that σ-compactness of the coordinate spaces is necessary. / Science, Faculty of / Mathematics, Department of / Graduate

Let U, V be two vector spaces of dimensions n and m, respectively, over an algebraically closed field F; let U⊗V be their tensor product; and let Rk(U⊗V) be the set of all rank k tensors in U⊗V, that is Rk(U⊗V) = {[formula omitted] are each linearly independent in U and V respectively}. We first obtain conditions on two vectors X and Y that they be members of a subspace H contained in Rk(U⊗V). In chapter 2, we restrict our consideration to the rank 2 case, and derive a characterization of subspaces contained in R2(U⊗V). We show that any such subspace must be one of three types, and we find the maximum dimension of each type. We also find the dimension of the intersection of two subspaces of different types. Finally, we show that any maximal subspace has a dimension which depends only on its type. / Science, Faculty of / Mathematics, Department of / Graduate

Given a set X , let P(X) be the collection of all subsets of X . A nonempty sub-collection u, of P(X) is called a generalized topology for X and the ordered pair (X, u) is called a generalized topological space or an abstract space or simply a space. Elements of u are said to be u-open and their complements are said to be u-closed. We define u-closure, u-limit point, ….. and so on in the natural way. Most of the basic notions in point set topology are defined analogously. It is expected that many important results in point set topology will not be carried over and a number of interesting properties will be lost or weakened. Nevertheless, some of them will still hold true despite the absence of the finite intersection axiom and the arbitrary union axiom for the collection of subsets. The primary objective of this thesis is to investigate which theorems in point set topology still remain valid in our more general setting. A secondary objective is to provide some counterexamples showing certain basic results in point set topology turn out to be false in the setting. It should be noted that other basic notions which are not discussed here at all can be defined similarly. However, in order to attain desirable and interesting conclusions, additional conditions must be imposed. / Science, Faculty of / Mathematics, Department of / Graduate