Mathematics Commons^™

Microcomputer Algorithms For Prime Number Testing, Susan Dale Barton

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This paper gives a survey of different methods of prime number testing. Emphasis has been given to algorithms based upon Fermat's Theorem: if p is an odd prime number, then p divides a^p-a. All of the computer programs described in this paper have been written for use on microcomputers and so the feasibility of using microcomputers is also discussed. Finally, numbers of various forms have been considered for primality with special attention given to Mersenne and Fermat numbers. It is hoped that some of the information contained in this paper may provide worthwhile enrichment ideas for mathematics educators.

Cluster Analysis For Acid Rain Data In Norway, Ali Ghafourian

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This paper gives a description of three well known clustering methods, and discusses the advantages and disadvantages of each. Then, the results of these three clustering methods are compared through examining them on a specific set of data.

The Shape Seeker Algorithm, Ladawn Haws

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This paper gives a brief description of several well known fuzzy objective function clustering algorithms, and discusses the convergence properties of this type of algorithm. The Shape Seeker algorithm, an adaptive norm algorithm, is then described in detail, and convergence established. It is compared to the other algorithms by examining the clusterings it produces on several data sets.

Numerical Methods (Finite Element) For Time-Dependent Partial Differential Equations, Masaji Watanabe

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This paper surveys reasons why the Ritz method and the Galerkin method are not efficient and why these methods can not be applied directly, for time dependent problems. It also introduces methods that are used for those problems. For a linear boundary value problem defined by a positive definite symmetric (self-adjoint) operator, the existence and the convergence of the Ritz approximation are guaranteed. In non-symmetric case, Lax-Milgram lemma assures the existence and the convergence of the Galerkin approximation for H1/2(Ω)-elliptic operator. Since time-dependent problems are hyperbolic or parabolic, the existence and the convergence of approximations by those methods are not …

The Frobenius Theorem, Hiroshi Nagao

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Many theorems in differential geometry which deal with the existence of certain geometrical structures or properties depend upon various existence and uniqueness theorems for differential equations. Because of its wide range of applications one of the most important of these theorems is the Frobenius Theorem for systems of total differential equations. There are four different forms of the Frobenius Theorem. In applications of the theorem one form is often preferable to the others. In this report we -shall prove the Frobenius Theorem, establish the equivalence of these various forms, and discuss a few applications.

On The Existence Of Global Variational Principles, Ian M. Anderson, T. Duchamp

Mathematics and Statistics Faculty Publications

In studying physical phenomena one frequently encounters differential equations which arise from a variational principle, i.e. the equations are the Euler-Lagrangequations obtained from the fundamental (or action) integral of a problem in the calculus of variations. Because solutions to the Euler-Lagrange equations determine the possible extrema of the fundamental integral, the first step in the solution of a given problem in the calculus of variations is to obtain the appropriate Euler-Lagrange equations. This state of affairs suggests the so-called inverse problem, viz. does a given differential equation arise from a variational principle and, if so, what is the Lagrangian for …

Determining Initial Values For Stiff Systems With Incomplete Initial Data, Yech-Kuang Ning

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Stiff differential equations frequently arise in physical problems due to the existence of greatly differing time constants. One way of solving stiff differential equations is taking the step size very small, another way is using an A-stable or stiffly stable implicit method. The purpose of this thesis is to discuss the method of determining initial values for stiff systems. The algorithm was motivated by a problem from ionospheric physics in which some of the initial values are unknown. If we give those unknown initial values an arbitrary number and integrated with those values, we have an initial transient. However, if …

Shallow Water Problem, Shu-Chen Li

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The equations in shallow water over an obstacle are nonlinear hyperbolic differential equations. We use the method of lines and GEARB package to numerically solve this problem.

The Qr Algorithm, Hsiao-Yin Edith Chu

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

In this section, we will consider two methods for computing an eigenvector and in addition the associated eigenvalue of a matrix A.

On The Sampling Distribution Of Entropy Function, Pan Fu-Charo

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Since the introduction of Shannon's Entropy Function in a non-probabilistic setting [l], there has been an expectancy that it will provide a useful measure for the fuzzy cluster validity problem. We hope to demonstrate here (empirically) by Monte Carlo simulation that the entropy Hc(U) of fuzzy c-partitions (U) defined below has expectaction

E(H_c(U)) = ∑^c_k=2 1/k

and variance

Var(H_c(U)) = 1/n(∑^c_k=2 1/k₂ - (c-1)/(c+1)(π²/6) - 1).

Furthermore, for sufficiently large n, that H is approximately normal.

In this report we propose a method which generates the required fuzzy matrix …

Topological Properties Of C(X, Y), Chris Alan Schwendiman

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The purpose of this paper is to examine the important topological properties of the function spaces BC(X,Y) and C(X,Y). Emphasis is given to the relationship between the metrizability of X and the separability of these function spaces. The paper is divided into three major parts: the preliminary definitions and theorems; the relationship between the topological properties of X and BC (X ,Y), for compact X; and the extension of the results of Part II for X not compact and for the case when we have C(X, Y).

On The Structure Of Divergence-Free Tensors, Ian M. Anderson

Mathematics and Statistics Faculty Publications

Contravariant rank two tensors which are divergence‐free on one index and which are constructed from the metric tensor, an auxiliary collection of arbitrary tensor fields, and the first and second partial derivatives of these quantities are classified. The results generalize existing mathematical arguments in support of the Einstein field equations

Stochastic Integration, Diane Knipfer

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Differential equations have been used to model physical systems, but in many processes this has not been sufficient due to the presence of random occurrences in the system. One method of dealing with this problem is to model the system as a stochastic or random process.

A stochastic process, x, is a function mapping the product of a probability space, Ω, and a subset of the real numbers, TcR, into the real numbers, x: Ω*T→R. In many physical situations, T can be thought of as representing time and Ω as all possible outcomes of the process. For a fixed t …

Positive Controllability Of Systems With Nearly-Non-Negative Matrices, Theodore Sonne Perry

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This paper analyzes the controllability of constant coefficient linear differential equations and presents two proofs of a major theorem on controllability. Properties of nearly-non-negative matrices are discussed and in particular a theorem on the behavior of the exponential matrix of nearly-non-negative matrices is proven. These results are then used to prove that the reachable set for systems with nearly-non-negative matrices is limited to the positive hyperoctant.

Interpretation And Application Of Elements Of Differential Geometry And Lie Theory, James R. Brannan

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Basic concepts of differential geometry and Lie theory are introduced. Lie transformation groups are applied to linear systems of differential equations and the problem of describing rigid body orientation. Linear Hamiltonian systems are then treated as a Lie system of differential equations. This theory is applied to a particular Hamiltonian system arising from a problem in control theory, the linear state regulator problem.

Shooting Method For Two-Point Boundary Value Problems, John D. Baumann

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The purpose of this paper is to develop the shooting method as a technique for approximating the solution to the two-point boundary value problem on the interval [a,b] with the even order differential equation {i.e. n is even)

u⁽ⁿ⁾(t) + f(t, u(t), u⁽ⁱ⁾(t, ),..., u^(n-1)(t)) = 0

and boundary conditions

u(a) = A

u(b) = B

and with at most n-2 other boundary conditions specified at either a or b. The basic proceedure will be illustrated by the following example.

Consider the two-point boundary value problem (0.1) (0.2) (0.3) with the additional boundary conditions …

A Characterization Of The Einstein Tensor In Terms Of Spinors, Ian M. Anderson, D. Lovelock

Mathematics and Statistics Faculty Publications

All tensors of contravariant rank two which are divergence‐free on one index, concomitants of a spinor field σ_iAX′ together with its first two partial derivatives, and scalars under spin transformations are constructed. The Einstein and metric tensors are the only candidates.

An Evaluation Of Truncated Sequential Test, Ryh-Thinn Chang

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The development of sequential analysis has led to the proposal of tests that are more economical in that the Average Sample Number (A. S. N.) of the sequential test is smaller than the sample size of the fixed sample test. Although these tests usually have a smaller A. S. N. than the equivelent fixed sample procedure, there still remains the possibility that an extremely large sample size will be necessary to make a decision. To remedy this, truncated sequential tests have been developed.

A method of truncation for testing a composite hypotheses is studied. This method is formed by mixing …

Multivalue Methods, Clayton V. Henrie

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Methods of solving ordinary differential equations with initial conditions are of a great importance to engineers and scientists. Many of these equations can be solved by well-known analytical techniques, but a greater number of physically significant differential equations cannot be so solved. Thus, the solutions of these equations must be approximated numerically. It is the purpose of this paper to investigate the techniques used in solving differential equations with initial conditions by "multivalue methods."

An Evaluation Of Bartlett's Chi-Square Approximation For The Determinant Of A Matrix Of Sample Zero-Order Correlation Coefficients, Stephen M. Hattori

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The single equation least-squares regression model has been extensively studied by economists and statisticians alike in order to determine the problems which arise when particular assumptions are violated. Much literature is available in terms of the properties and limitations of the model. However, on the multicollinearity problem, there has been little research, and consequently, limited literature is available when the problem is encountered. Farrar & Glauber (1967) present a collection of techniques to use in order to detect or diagnose the occurrence of multicollinearity within a regression analysis. They attempt to define multicollinearity in terms of departures from a hypothesized …

Maintaining Topological Properties On The Brink Of Destruction, Kay J. Hamm

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Throughout the paper we consider the setting where f is a continuous function (a mapping) whose domain X and range Y are both Hausdorff spaces. Our object is to determine conditions on the map f which insure that when X has a certain topological property Q, then Y will also have property Q. For example, if X is metrizable, then it does not necessarily follow that Y is a metric space; but if f is a perfect map, then metrizability is preserved. Chapter III is devoted to the study of this metrizability problem. In particular, we present Frink's [ 2] …

Functional Analysis Techniques In Numerical Analysis, Kenneth D. Schoenfeld

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

In this paper we will consider the problem of selecting the best, or optimal, numerical method of solution to a given mathematical problem. The admissible numerical methods will be a clearly defined set for each problem. Obviously, in order to find the best method in this set, we must have a clear mathematical formulation of just what "best" means; this will be the intent of Theorem 0.1. Intuitively, the best method will be understood to be the one which minimizes the maximum possible error where this error will be measured in terms of the norm of a given Hilbert space. …

Infinite Product Spaces Under The Tychonoff And Goofynoff Topologies, James A. Capps

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The only topology considered for the infinite product of topological spaces in most current topology texts and research papers is the Tychonoff topology. Yet there is another topology which seems to be a much more topologically natural generalization of the usual "box" topology of finite products. We call this natural generalization the Goofynoff topology and exploit its properties. The use of the word "Goofynoff" (pronounced Goof'-n-off) is not universal and does not refer to any person of that name. In the few references to this topology that can be found, it is usually called simply the Box Topology. None of …

Classifications Of Plane Continua, Steven Ray Matthews

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

In the course of studying continua in the plane it has been asked if a given continuum has uncountably many disjoint duplications in the plane, and if so, what are the consequences of the existence of such a collection. The object of this paper is to study these problems and to develop some machinery useful in their resolution. In Section I, we review the definition of convergence and homeomorphic convergence of point sets in a metric space S. We then consider the space, π of all continuous functions from a compact metric space P to a separable metric space Q …

A Monte Carlo Evaluation Of A Nonparametric Technique For Estimating The Hazard Function, Sheng Jia Lin

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

This research is primarily concerned with the estimation of the Hazard functions, the Hazard function is the failure rate at time t, and is defined as -R '(t)/R(t), so it plays an important role in Reliability.

In order to compare and evaluate the estimation methods, it is convenient to select one distribution in this research. Since the Weibull distribution is a useful distribution in Reliability, the Weibull distribution is used in this paper.

Torus Knots, David S. Bradley

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

We have compiled here many interesting results concerning a particular collection of knots called torus knots. Torus knots are merely simple closed curves imbedded in an unknotted torus T² in E³. We show that the fundamental groups of T², π(T²), is the direct product of the additive group of integers with itself. The ordered pair (p, q) in Z x Z determines an equivalence class of loops on the torus, and we show in Section II that the class [(p, q)] contains a loop whose image is a simple closed curve if and …

Integral Representation Theorems, Leiko Hatta

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Since F. Riesz showed in 1909 that the dual of C[0,1] is BV[0,1] (the functions of bounded variation on [0, 1] with || g ||_BV = V(g)) via the Stieltjes integral, obtaining representations for linear operators in various settings has been a problem of interest. This paper shows the historical manner of representations, the road map type theorems and representations obtained via the v-integral.

The Poincaré Conjecture, Joseph D. Peck

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The central theme for this paper is provided by the following three statements:

(1) Every compact connected 1-manifold is S1.

(2) Every compact connected simply connected 2-manifold is S2.

(3) Every compact connected simply connected 3-manifold is S3.

We provide proofs of statements (1) and (2). The veracity of the third statement, the Poincaré Conjecture, has not been determined. It is known that should a counter-example exist it can be found by removing from S3 a finite collect ion of solid tori and sewing them back differently. We show that it is not possible to find a counterexample by removing …

Bayesian Estimate Of System Reliability, Naresh Shah

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

A Bayesian estimate of reliability for each component in the system of n-components, each exponentially distributed, is developed which utilizes the basic notion of loss in estimation theory. Here we assume that each component is independently dis­tributed. In reliability estimation, the loss associated with over­estimation is usually greater than the loss associated with under­estimation; and hence loss function can be a very useful tool. The prior distribution and loss function of reliability considered in this paper are flexible to be compatible with other situations in which reliability estimates are required. When the loss function is symmetric and no prior information …

A Survey In Mean Value Theorems, David A. Neuser

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A variety of new mean value theorems are presented along with interesting proofs and generalizations of the standard theorems.

Three proofs are given for the ordinary Mean Value Theorem for derivatives, the third of which is interesting in that it is independent of of Rolle's Theorem. The Second Mean Value Theorem for derivatives is generalized, with the use of determinants, to three functions and also generalized in terms of n-th order derivatives.

Observing that under certain conditions the tangent line to the curve of a differentiable function passes through the initial point, we find a new type of mean value …