This thesis is intended to be an introductory mathematical presentation of analog computation. An attempt was made to explain in concise mathematical language, how an electronic analog computer works, why it works, and the simplicity of its use. The components of the computer are considered as operational blocks, each block performing an indicated operation. Hence, the electrical knowledge presented is meager. The methods of solution and the corresponding computer solutions obtained for several types of mathematical problems are presented; such as, the determination of the characteristic vectors and characteristic values of a given matrix. In each case, a 15-amplifier Heath Kit analog computer model number ES-400 was used. Since this type of computer contains no devices for multiplying variable quantities, the only types of problems that could be considered were those that can be represented by a system of linear, ordinary differential equations with constant coefficients. However, similar techniques are applicable to the analogous non-linear systems and systems with variable coefficients, on a fully-equipped analog computer. / Master of Science

We give a method, by solving a nonlinear system of equations, for Gauss harmonic interpolation formulas which are useful for approximating, the solution of the Dirichlet problem. We also discuss approximations for integrals of the form I(f) = (1/2πi) ∫_L (f(z)/(z-α)) dz. Our approximations shall be of the form Q(f) = Σ_k=1ⁿ A_kf(τ_k). We characterize the nodes τ₁, τ₂, …, τ_n, to get the maximum precision for our formulas. Finally, we propose a general problem of approximating for linear functionals; our results need further development. / Ph. D.

If (X,Ƭ) is a topological space, then a quasi-uniformity U on X is compatible with Ƭ if the quasi-uniform topology, Ƭ_u = Ƭ. This paper is concerned with local properties of quasi-uniformities on a set X that are compatible with a given topology on X. Chapter II is devoted to the construction of Hausdorff completions of transitive quasi-uniform spaces that are members of the Pervin quasi-proximity class. Chapter III discusses locally complete, locally precompact, locally symmetric and locally transitive quasi-uniform spaces. Chapter IV is devoted to function spaces of quasi-uniform spaces. Chapter V and the Appendix are concerned with the topological homeomorphism groups of quasi-uniform spaces. / Ph. D.

The fundamental definitions of radiative transfer theory are given and the two-dimensional equation of transfer is derived, density of radiation is defined, and two-dimensional two-intensity transfer model is presented. An operational interpretation of the latter model is given interms of military truck transport supply and the functional dependencies of the terms in the transfer equations are evaluated. For this interpretation the density equations are given and the study state and time dependent solutions of the density equations are discussed in polar coordinates. This work was conducted for the U. S. Army Transportation Research Command, Fort Eustis, Virginia, 1961, Task 9R38-11-009-02. / Master of Science

The effects of polymer particle size, temperature, and time on the continued condensation of two polyamides: poly(hexamethylene adipamide) and poly(hexamethylene sebacamide), and a polyester, poly(ethylene terephthalate), were studied experimentally. The polyamides were held at elevated temperatures from 120 to 180 °C for periods of 5 to 20 hours in a nitrogen atmosphere. A similar procedure was followed with the polyester except that the range of temperature was 160 to 200 °C. The number-average molecular weights of the polymers before and after treatment were calculated from the polymer intrinsic viscosities. The thermal behavior of selected polymers was also examined by differential thermal analysis. Poly(hexamethylene adipamide) showed an increase in the number-average molecular weight from approximately 10,000 to 22,000 when treated at 180 °C for 20 hours. Under the same conditions, poly(hexamethylene sebacamide) showed a change from about 10,000 to 14,000. Poly- (ethylene terephthalate) treated at 200 °C for 20 hours exhibited an increase from approximately 18,000 to 34,000. Two kinetic equations were derived and were successfully applied to the experimental data. From these equations the specific reaction rates were obtained. The temperature dependency of the reaction rates was expressed in the form of the Arrhenius equation. The effect of particle size on the reaction was noted. Also noted were changes in polymer thermal behavior as the reaction temperature was raised. At low temperatures the transport of reaction by-products from the interior to the surface of solids controlled the reaction. At high temperatures chemical kinetics determined the reaction. / Doctor of Philosophy

We begin this paper with a survey of characterizations of pseudocompact spaces and relate pseudocompactness to other forms of compactness such as light compactness, countable compactness, weak compactness, etc. Some theorems on properties of subspaces of pseudocompact spaces are presented. In particular, conditions are given for the intersection of two pseudocompact spaces to be pseudocompact. First countable pseudocompact spaces are investigated and turn out to be maximally pseudocompact and minimally first countable in the class of completely regular spaces. We define a pseudocompactification of a space X to be a pseudocompact space in which Xis embedded as a dense subspace. In particular, for a completely regular space X, we consider the pseudocompactification αX = (βX - ζX) U X. We investigate this space and in general all pseudocompact subspaces of βX which contain X. There are many pseudocompact spaces between X and βX, but we may characterize αx as follows: 1) αx is the smallest subspace of βX containing X such that every free hyperreal z-ultrafilter on X is fixed in αx. 2) αx is the largest subspace of βX containing X such that every point in αX - X is contained in a zero set which doesn't intersect X. The space αx also has the nice property that any subset of X which is closed and relatively pseudocompact in X is closed in αx. The relatively pseudocompact subspaces of a space are important and are investigated in Chapter 4. We further relate relative pseudocompactness to the hyperreal z-ultrafilter on X and obtain the following characterizations of a relatively pseudocompact zero set: 1) A zero set Z is relatively pseudocompact if and only if Z is contained in no hyperreal z-ultrafilter. 2) A zero set Z is relatively pseudocompact if and only if every countable cover of Z by cozero sets of X has a finite subcover. In the next chapter we consider locally pseudocompact spaces and obtain results analogous to those for locally compact spaces. Then we relate pseudocompactness and the property of being C* - or C-embedded in a space X. Included in this is a study of certain weak normality properties and their relationship to pseudocompact spaces. We develop two types of one-point pseudocompactifications and investigate the properties of each. It turns out that a space X is never C* -embedded in its one-point pseudocompactification. Also one space has the property that closed pseudocompact subsets are closed in the one-point pseudocompactification while the other may not have this property but will be completely regular. We present survey material on products of pseudocompact spaces and unify these results. As an outgrowth of this study we investigate certain functions which are related to pseudocompactness. / Doctor of Philosophy

The classical theorem of N. Wiener and P. Lévy states that if f(x) has an absolutely convergent Fourier series and W(z) is an analytic function whose domain contains the range of f(x), then W[f(x)] also has an absolutely convergent Fourier series. The main result of this paper is an analog of the Wiener-Lévy Theorem in which we consider analytic transformations acting upon kernels of integral operators of the form (Tφ)(x) = ∫_-π^π K(x,y)φ(y)dy and the so-called s-numbers of K and W[K] take the place of the classical Fourier coefficients of f and W[f]. Theorem: Let W(z) be analytic in a region R and let K(x,y) map the square [-π,π] x [-π,π] continuously into R. If {φ_n}_n=1^∞ and {ψ_n}_n=1^∞ are a full set of continuous singular functions for K(x,y) and ∑_n=1^∞ [s_n (K)]^P | | φ_n | |_∞^P| | ψ<sub>n</sub | |_∞^P < ∞ for some 0 < p ≤ 1, then ∑_n=1^∞ [s_n (W[K])]^P < ∞ or, expressed alternatively, W[K] belongs to the Schatten class C_p. The classical Wiener-Lévy Theorem is obtained as a corollary in the special situation when p = 1 and K(x,y) ≡ f(x-y) is a difference kernel. For the case 1 < p < 2 we generalize a Fourier series result of L. Alpar to the following theorem in integral operator theory. Theorem: Let W(z) be analytic (but not necessarily single-valued) in a region R and let K(x,y) satisfy the following conditions: a) K(x,y) maps the square [-π,π] x [-π,π] continuously into R. b) K(x,y) is 2π-periodic in x (or y). c) K(x,y) satisfies an integrated Lipschitz condition of order α relatively uniformly in x (or y) where p⁻¹ < α ≤ 1, 1 < p < 2. Then, if W(z) returns to its initial determination after z travels completely around any curve C_y (or C_x) of the form z = K(x,y), -π ≤ x (or y) ≤ π, then W[K(x,y)] ε C_p. To round out the paper, we show that analytic transformations preserve smoothness conditions of Lipschitz and bounded variation type, and consequently we are able to give a number of sufficiency conditions for analytic functions of kernels to be in various kernel classes. Finally, we investigate the converse of the Wiener-Lévy Theorem and how analogues of it relate to integral operators. The paper concludes with suggestions of several interesting questions warranting further study. / Ph. D.

The motivating idea behind the thesis is the study of the relationship of the degrees of the vertices of a graph and the structure of a graph. To each graph (indirected, without multiple edges) one can associate a graphical sequence by arranging the degrees of the vertices in their natural order. Conversely, an arbitrary sequence of numbers is graphical if it is a graphical sequence for some graph. In the first chapter general properties of graphical sequences are studied. We give conditions under which a sequence can be lengthened or shortened and have the property of being graphical be preserved. The concept of a 'transfer' is introduced to show how all realizations of a graphical sequence can be obtained from a given realization. Also in chapter one we show how graphical sequences can be used to characterize concepts like 'connected', 'block' and 'arbitrarily traceable'. If a graphical sequence has one 'realization' up to isomorphism then the sequence and the graph are called simple. Since simple graphs are determined up to isomorphism by the degrees of the vertices it is hoped that simple graphs will reveal in some measure the effect of the degree sequence on the structure of a graph. Thus, in chapter two, we attempt to characterize simple graphs--the central problem of the thesis. Simple trees, simple disconnected graphs, and simple graphs with cut points and no pendant vertices are characterized. (This means that characterizing simple blocks will solve the problem). Probably the most useful result is that a connected, simple graph must be of radius ≤ two and diameter ≤ three The third chapter is devoted to the problem of counting the number of non-isomorphic realizations of a given graphical sequence. Generating functions are used and several interesting special cases are given. These latter are in turn used to establish certain bounds on the number of realizations for sequences of a given length. / Ph. D.

A new approach to Kneser's Theorem, which achieves a simplification of the analysis through the introduction of maximal sets, the basic sequence of maximal e-transformations, and the limit set, B*, is presented. For two sets of non-negative integers, A and B, with C∈A⋂B, the maximal sets, Aᴹ and Bᴹ, are the largest supersets of A and B, respectively, such that Aᴹ + Bᴹ = A + B. By shifting from A and B to Aᴹ and Bᴹ to initiate the analysis, the maximal properties of Aᴹ and Bᴹ are exploited to simplify the analysis. A maximal e-transformation is a Kneser e-transformation in which the image sets are maximized in order to preserve the properties of maximal sets. The basic sequence of maximal e-transformation is a specific sequence of maximal a-transformations which is exclusively used throughout the analysis. B* is the set of all non-negative elements of sM which are not deleted by any transformation in the basic sequence of maximal e-transformations. Whether or not B* = {O} divides the analysis into two cases. One significant result is that B* = {O} implies δ (A + B) = δ (A, B) where δ(A + B) is asymptotic density of A + B and δ (A, B) is the two-fold asymptotic density of A and B. The second major result describes the structure of A + B when δ(A + B) < δ(A, B). With B* ≠ {0} it is shown, using only elementary properties of greatest common divisor and residue classes, that there exists C⊆ A+ B, 0εC, such that δ(C) ≥ δ(A, B) -1/g where g is the greatest common divisor of B* and C is asymptotically equal to C^(g), the union of all residue classes, mod g, which have a representative in C. The existence of C provides the crucial step in obtaining an equivalent form of Kneser’s Theorem: If A and B are two subsets of non-negative integer, 0εA⋂B, and δ(A + B) < δ(A, B), then there exists a positive integer g such that A + B is asymptotically equal to (A + B)^(g) and δ(A + B) = δ ((A + B)^(g)) ≥ δ (A^(g) , B^(g)) - 1/g ≥ δ(A, B) -1/g. / 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.