The von Neumann finiteness problem for k[G] is still open. Kaplansky proved it in characteristic zero. He used the nonvanishing of the trace: tr(e) = 0 implies e = 0 for any idempotent e ∊ k[G]. Assume now that char k = p > 0. Now tr can vanish on nonzero idempotents. Instead, we study the lifted trace ltr. For e = e² ∊ k[G], define ltr(e) by ê(1) where ê = Σ{x ∊ G}ê(x)x lifts e. Here ê is an infinite series with |ê(x)|_p→0, where each ê(x) lives in the Witt vector ring of k. We prove that ltr(e) depends on e only, it is a p-adic integer and ltr(e) = ltr(f) if f is equivalent to e. Also ltr(e) ∊ Q and ltr(e) = 0 implies e = 0 if G is polycyclic-by-finite. We conjecture that -log_p|ltr(e)|_p < |supp(e)|. We prove this for e central and for e = e² ∊ k[G] with |G| ≤ 30. In the last section, we give the example of an idempotent e such ath supp(f) is infinite for all f ~ e. Finally we estimate |<supp(e)>| for central idempotents e. / Ph. D.

Let X be a locally compact σ compact Hausdorff space. Let µ be a complete regular Borel measure defined on the Borel sets of X. It is shown that there is a base for the topology of X consisting of open sets whose boundaries are of µ measure zero. Let (S, p) be a metric space. It is shown that a function on X whose range is a subset of S can be uniformly approximated by µ almost everywhere continuous simple functions if, and only if, the function itself is µ almost everywhere continuous and its range is a totally bounded subset of S. S is then specialized to be a Banach algebra and several consequences are obtained culminating in the study of the ideal structure of the ring of ail µ almost everywhere continuous functions on X whose ranges are totally bounded subsets of a Banach algebra which is either the reals, complexes or quaternions. / Ph. D.

A topological space X is invertible at p ∈ X if for every· neighborhood U of p in X, there is a homeomorphism h on X onto X such that h(X - U) ⊆ U. X is continuously invertible at p ∈ X if for every neighborhood U of p in X there is an isotopy {h_t , 0 ≤ t ≤ 1, on X onto X such that h₁(X - U) ⊆ U. It is proved that, if X is a locally compact space which is invertible at a point p which has an open cone neighborhood, and if the inverting homeomorphisms may be taken to be the identity at p, then X is continuously invertible at p. A locally compact Hausdorff space X, invertible at two or more points which have open cone neighborhoods in X, is characterized as a suspension. A locally compact Hausdorff space X which is invertible at exactly one point p, which has an open cone neighborhood U such that U - p has two components, while X - p is connected, is characterized as a suspension with suspension points identified. Let Cⁿ be an n-conplex with invert point p. Let U be an open cone neighborhood of p in Cⁿ, and let L be the link of U in Cⁿ. Then it is shown that H_p(Cⁿ) is isomorphic to a subgroup of H_p-1(L). Invertibility properties of the i-skeleton of an n-complex are discussed, for i < n. Also, a method is described by which an n-complex which is invertible at certain points may be expressed as the union of subcomplexes, ca.ch of which is invertible at the same points. One-complexes with invert points are characterized as either a suspension over a finite set of points or a union of simple closed curves [n above ⋃ and i = 1 below that symbol], such that Sᵢ ⋂ Sⱼ = p, i ≠ j. It is proved that, if C² may be expressed as the monotone union of closed 2-cells. Also if the link of an open cone neighborhood of an invert point in a 2 - complex C² is planar, C² may be embedded in E³. / Doctor of Philosophy

A bitopological space (X,τ,μ) is a set X with two topologies. The study of bitopological spaces was initiated by J. C. Kelly. In this thesis, we study pairwise-separation axioms as defined by J. C. Kelly, C. W. Patty, and F. P. Lane. In addition, definitions for semi-compactness, semi-paracompactness, and bicontinuous functions are proposed and are related to the definitions of pairwise-separated spaces. Finally, quasi-pseudo-metric spaces are defined, and a number of quasi-pseudo-metrization theorems are summarized. / Master of Science

Many general properties of real continuous functions defined on the closed interval [0,1] on the real line have been studied in the past. The present work started with a review of some known analytical results of this class C. An extended class A of positively continuous functions was then defined and its properties compared with that of C. While many of the characteristics of C were inherited from A, some properties of C are not shared by A. The first part of the second section (chapter) is devoted to a study of some elementary algebraic properties of A and C. Results obtained for the two classes showed differences. The rest of the section (chapter) deals with the algebraic structures of some subclasses of F, the set of all real-valued functions of [0,1] on the real line. The concept of an ideal in F was introduced for the class of real functions from [0,1] to the real line. In the last section (chapter), the concept of areas of function, C_f, defined as the closure of the graph of a function, is used to study the properties of elements of I. Integrals of continuous functions in I are completely determined by their C_f’s. Some topological implications of a few analytical subclasses of I were also revealed. This section concluded with an important theorem that fully characterizes the G_f of a real function in I by a closed set in the closed square U = {(x,y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}. / Master of Science

The convergence properties of the permanent exponential interpolation series f(Z) = 1^Zf(0) + (2^Z - 1^Z)Δf(0) + (3^Z - 2.2^Z + 1^Z/2!)Δ(Δ - 1)f(0) + … have been investigated. Using the following notation U_n(Z) = ∑ⁿ_k=0 (-1)^k(ⁿ_k)(n - i + 1)^Z, Δ⁽ⁿ⁾ f(0) = Δ(Δ-1)…(Δ - n + 1)f(0), the series can be written more compactly as f(Z) = ∑^∞₀ U_n(Z)/n!Δ⁽ⁿ⁾ f(0). It is shown that Δ⁽ⁿ⁾ f(0) can be represented as Δ⁽ⁿ⁾ f(0) = M_n(f) = 1/2πi ∫_Γ (e^ω - 1)⁽ⁿ⁾ F(ω)dω, where F(ω) is the Borel transform of f(Z) and Γ encloses the convex hull of the singularities of F(ω). It is further shown that the series ∑^∞₀ U_n(Z)/n! (e^ω - 1)⁽ⁿ⁾ forms a uniformly convergent Gregory-Newton series, convergent to e^Zω in any bounded region in the strip |I(ω)| < π/2. The Polya representation of an entire function of exponential type is then formed, and the method of kernel expansion (R. P. Boas, and R. C. Buck, Polynomial Expansions of Analytic Functions, Springer-Verlag, Berlin, 1964) yields the desired result. This result is summed up in the following: Theorem Any entire function of exponential type such that the convex hull of the set of singularities of its Borel transform lies in the strip |I(ω)| < π/2. admits the convergent exponential interpolation series expansion f(Z) = ∑^∞_n=0 U_n(Z)/n!Δ⁽ⁿ⁾ f(0) for all Z. / M.S.

In this thesis we investigate the properties of M-spaces and M*-spaces, which are generalized metric spaces. Chapter II is devoted to preliminary results, and in Chapter III we prove the characterization for M-spaces theorem of K. Morita [12]. This theorem states that a space X is an M-space if and only if there exists a quasi-perfect map from X onto a metrizable space T. Chapter IV is concerned with the relationships between M-spaces and M*-spaces. We first prove an M-space is an expandable, M*'-space and then show that every normal, expandable, M*-space is an M-space. Using Katetov's Theorem, we show that in a collectionwise normal space, X is an M-space if and only if it is an M*-space. We conclude by generalizing this to the following. In a normal space X, X is an M-space if and only if it is an M*-space. Chapter V is concerned with the study of M-spaces and M*-spaces under quasi-perfect maps. We also prove the Closed Subspace Theorem for M-spaces and M*-spaces and establish the Locally Finite Sum Theorem for M-spaces and M*-spaces. In Chapter VI, we give an example of a T₂, locally compact M-space X, which is not normal and therefore not metrizable. We also give an example of a T₂, locally compact M*-space Y, which is not an M-space, but is however the image of X under a quasi-perfect mapping. / Master of Science

The problem of detecting deep space objects with certain probabilities was investigated. A mathematical model was then developed from given problem specifications that deals with the trade-off of various parameters involved in the detection problem. A software package that allows the user to input data interactively was written to implement the model. The completed program as well as an analysis of the tested results are included. / M.S.

Topological groupoids with"approximate" inverses are studied. In the compact case, these"approximate" inverses turn out to be true inverses. Examples of groupoids wL:h"approximate" inverses are given in the section dealing with function spaces. Using the classical construction of Haar as a guide, we succeed in obtaining a (non-trivial) regular, right-invariant measure over a locally compact left group satisfying the conditions: a) open sets are preserved by left translation b) each group component is open. In the section dealing with integrals, we consider a compact metric topological semigroup that is right simple 3nd possesses a right contractive metric (ρ(xz,yz) ≤ ρ(x,y)). It is shown that such a structure always carries a non-trivial right-invariant integral. Throughout the entire development, associativity is invoked only once. The investigation concludes with a section dealing with sufficient conditions under which binary-topological systems become topological groups. A mob-group is defined to be a T_o-space which is also an algebraic group. A theorem in the last section states that a mob-group is a topological group iff given any open set W about the identity, W ∩ W⁻¹ has non-void interior. / Ph. D.

This thesis is an investigation of the concept of paracompactness. It presents the history of paracompactness, analyzes this concept from several diverse points of view and tries to establish the relationship between these different views. The starting point is the work of Tukey [40]. The important problem of the metrization of topological spaces is presented as an application of the concept of paracompactness. / Master of Science