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.

M.S.

This paper presents new concepts in the use of infinite and infinitesimal numbers in real analysis. theory is based upon the hyperreal number system developed by Abraham Robinson in the 1960's in his invention of "nonstandard analysis". paper begins with a short exposition of the construction of the hyperreal nU1l1ber system and the fundamental results of nonstandard analysis which are used throughout the paper. The new theory which is built upon this foundation organizes the set hyperreal numbers through structures which on an infinite base logarithm. Several new relations are introduced whose properties enable the simplification of calculations involving infinite and infinitesimal The paper explores two areas of application of these results to standard problems in elementary calculus. The first is to the evaluation of limits which assume indeterminate forms. The second is to the determination of convergence of infinite series. Both applications provide methods which greatly reduce the amount of con1putation necessary in many situations. / Master of Science

In this paper we investigate several aspects of computing control system radii for finite element approximations of control systems governed by partial differential equations. Finite element approximations of the heat equation (parabolic), the wave equation (hyperbolic) and the equations of thermoelasticity (mixed) are used as test cases. Balanced realizations, reduced order models and other transformed models are also studied. / Master of Science

This thesis presents a comparison of correlation methods in risk analysis. A theoretical solution is given to the correlation problem along with a discussion of each method. Each method is compared to a developed test case and two other cost projects. Restrictions on correlation coefficients are also given followed by the advantages and disadvantages of each method. / Master of Science

We prove that no disjoint union of any number of copies of even dimensional complex projective space can bound a smooth oriented compact manifold with boundary. We prove this by defining and computing certain algebraic invariants for smooth oriented manifolds. A non-diffeomorphic relationship is established between boundary manifolds and complex projective space by contrasting invariants computed for these spaces. / Master of Science