Mathematics Commons^™

Some Problems In Combinatorial Number Theory., Frank Ernest Higgins

Mathematics & Statistics ETDs

In this paper we consider three unrelated problems in combinational number theory. The first of these pertains to minimal residual polynomials, the second to certain group factorizations and the third to the addition of positive integers in their Zeckendorf representations.

In section II We show that for any given positive integer m there exists a unique integer monic polynomial P of least degree and whose coefficients satisfy certain inequalities with the property that m divides P(x) for every integer x. This polynomial is called the minimal residual polynomial modulo m. We then show that this polynomial assumes various forms which …

A Maximum Principle For Time-Lag Control Problems With Bounded States, Gary R. Bunce

Mathematics & Statistics ETDs

A maximum principle is obtained for control problems involving system equations with a constant time lag in the control and state variables. The arcs under consideration are subject to constraints of the form

The results are obtained using the method of M. R. Hestenes.

Operators As Elements In A Partially Ordered Linear Algebra., Edward Wayne Davenport

Mathematics & Statistics ETDs

R. DeMarr {unpublished) has begun a study of linear operators as elements in a Dedekind O-complete partially ordered linear algebra {dsc-pola). The order structure necessary to produce elements in a dsc-pola which behave as specific operators on a specific space of functions is studied.

A New Test For Normality, Richard Leroy Roller

All Master's Theses

This paper presents a new test for normality which is based on a complete characterization of the normal distribution. Motivation for the test is given in terms of a proof of this characterization. The test is derived and evaluated by computer-simulated sampling from alternative distributions. The empirical powers of the test generated from such samplings are tabled and compared to nine commonly used tests. Evaluation of the proposed test is discussed and further avenues of investigation are suggested.

Orthogonality In Normed Spaces, Martin R. Mccarthy

All Master's Theses

This paper presents three definitions of orthogonality in normed spaces. Each definition is shown equivalent to the inner product being zero when restricted to an inner product space. The definitions arise from such properties in two space as the diagonals of a rectangle being equal and the Pythagorean Theorem. The third definition shows that the idea of an inner product can be generalized under certain conditions.

Some Applications Of Decompositions Of The Identity To Ordered Vector Spaces., John Annulis

Mathematics & Statistics ETDs

We study decompositions of the identity (for definition see Vulikh [13]) in Dedekind a-complete vector lattices. In Chapter 1, we define and develop a notion of continuous elements with respect to a decomposition of the identity. We shall also develop a spectral theory for these elements. In Chapter 2, two Stieltjes-type type integrals are defined with respect to decompositions of the identity. The relationship of these integrals to the continuous elements is then explored. In Chapter 3, we prove two results in ordered vector spaces. The first result is that every infinite dimensional Dedekind complete vector lattice has a base …

A Maximum Principle For An Optimal Control Problem With Integral Constraints., Vernon Bakke

Mathematics & Statistics ETDs

In this problem necessary conditions are obtained for an optimal control problem ·whose state variables are given in terms of integral equations. The conditions are obtained separately for Volterra equations and Fredholm equations. The main result for each case is the maximum principle and multiplier rule. For the Volterra equations, transversality conditions are obtained.

A New Confidence Interval For The Mean Of A Normal Distribution, David Lee Wallace

All Master's Theses

A typical problem in statistical inference is the following: An experimenter is confronted with a density function f(x; ϴ) which describes the underlying population of measurements. The form of f may or may not be known, and ϴ is a parameter (possibly vector-valued) which describes the population. The statistician's job is to estimate or to test hypotheses about the unknown parameter ϴ. In this paper, we shall consider interval estimation of the mean of the normal density function.

Convergence Rates For The Central Limit Theorem For Random Sums, Christopher E. Olson

Mathematics & Statistics ETDs

Let (X_i} be a sequence of independent, identically-distributed random variables with EX²_i < ꝏ and E(X_i - EX_i)² = 1.

On Some Special Classes Of Partially Ordered Linear Algebras., Taen-Yu Dai

Mathematics & Statistics ETDs

R. DeMarr (unpublished) has begun a study of Banach algebras as subalgebras of partially ordered linear algebras which are Dedekind a -complete. In [3] he has shown that the real Banach algebra of norm-bounded linear operators (mapping a real Banach space into itself) can be made into a partially ordered linear algebra which is Dedekind a -complete. This leads us to study a more gen­eralized function algebra by using the order structure. In this paper from analytical point of view we will study some special classes of partially ordered linear algebras which are Dedekind a -complete. In chapter 1 we …

Stacking Methods And Mixing Properties Of Measure-Preserving Transformations., Sarah Lee Meyer Christiansen

Mathematics & Statistics ETDs

In this paper we study ergodic and mixing properties of transformations. Stacking methods are employed to construct the transformations used in this paper. Number theoretical methods are developed and used to study mixing properties. Two new mixing properties B-mixing and S-mixing are defined, and are shown to be between partial mixing and weak mixing. Finally a transformation is constructed which is weak mixing but not S-mixing, and hence not partially mixing. Several applications to additive number theory are given.

Random Evolutions On Diffusion Processes, Donald Quiring

Mathematics & Statistics ETDs

Let {V(t,ω), t ≥ O, ω ε Ω} be a diffusion process on the real line with infinitesimal operator 1/2σ²(⋅)D² + m(⋅)D. Markov processes {V_n, n = 1,2,....} on the real line are constructed in such a way that the paths of V_n are step functions with jump size n^-1/2 and

P_O [lim sup |V_n(s)-V(s)| = 0] =1

n∞ 0≤s≤t,

where P_O assigns probability one to paths starting at the origin at t = 0.

Let {T_V(t), t≥0, vε R} be a family of linear contraction operators …

Toeplitz Operators On Locally Compact Abelian Groups, Henry A. Krieger, C.A. Schaffner

All HMC Faculty Publications and Research

The problem of global optimization of M incoherent phase signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for $N = 2$, and the optimal signal sets are determined for $M = 2,3,4,6$ and 12.

The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ${\bf s}_j $, $j = 1,2, \cdots ,M$, on the unit sphere $S_1 $ in $C^N $. If $W_{jk} $ is the half space determined by ${\bf s}_j …

I. Existence Of Eigenvalues For Integral Equations; Ii. A Collocation Method For Boundary Value Problems, Robert Dodd Russell

Mathematics & Statistics ETDs

I. The existence of eigenvalues is shown for certain classes of integral equations with continuous kernels. A number of interesting and useful results are thereby treated in a unified and relatively elementary way. The simplicity of these new proofs make the results accessible to introductory courses on the theory of integral equations.

II. Collocation with piecewise polynomial functions is developed as a method for solving two-point bour:rlary value problems. Convergence is shown for a general class of linear problems and a rather broad class of nonlinear problems. Some computational examples are presented to illustrate the wide applicability and efficiency of …

Goursat Problems For The Abstract Equation Urs = Lu, Walter J. Roth

Mathematics & Statistics ETDs

Transitive Subgraphs Of Undirected And Directed Graphs., Charles C. Harner

Mathematics & Statistics ETDs

A graph G is transitive if it is transitive when considered as a relation; i.e., if the edges ab and be are in G, so is ac. In this paper, several natural questions concerning transitivity and related concepts are investigated.

Because the terminology and notation of graph theory is not standard, a brief section of definitions and notation is included in Section II. Section III is a study of the transitive cut number of a graph, the minimum number of edges which must be removed from the graph so that the remaining subgraph is transitive. Undirected graphs on n points …

The Ideal And New Interval Topologies On Lattice Ordered Groups., Frieda Koster Holley

Mathematics & Statistics ETDs

In his 1967 edition of Lattice Theory, G. Birkhoff posed the question: "When is a directed group a topological group and a topological lattice in its new interval topology?'. This problem and the similar problem involving the ideal topology are investigated.

A New Characterization Of Besicovitch Almost Periodic Functions., Abdallah N. Dabboucy

Mathematics & Statistics ETDs

Besicovitch and Bohr have given an internal description or Besicovitch almost periodic functions which involves the notion of “satisfactory uniformity” and is somewhat complicated. Raouf Doss has given an alternative simpler internal description, which, however, involves an infinity of independent conditions. Theorem. Let f E Lp(a,b) for all real numbers a,b. Then ff (BP-AP}

if end only if (1) the functions fn obtained by truncating f atn=l,2, … !·ll-converge to f; (2) lim)C+{) x llf-fl! BP =O; (3a) for every f: > 0 the set (x: l!f x -fl' P < eJ is relatively dense; and (3b) for every E: > 0

for all sufficiently large T, where E = {x: l\fx-fllB < e). Here fx(y) = f(x+y), XE denotes the characteristic function of E and 1 :Sp< oo. The theorem extends to LC connected groups, compactly generated Abelian groups and a somewhat stronger version of it extends to discrete countable Abelian groups.

Right Baer Semigroups., William Christopher Hardy

Mathematics & Statistics ETDs

In his paper, "A Semigroup Approach to Lattices, 11 (Canad . Jour. Math., Vol. 18 (1966)), M. F. Janowitz showed that every lattice with 0 and 1 is isomorphic to the lattice formed by the left annihilators of a suitably chosen Baer semigroup when they are ordered under set inclusion. A Baer semigroup, s~ was then said to coordinatize a lattice, L, if Lis isomorphic to the lattice of left annihilators of S. This dissertation presents a similar semigroup approach to lattices by showing that a poset, P, with O and 1 forms a meet semilattice if and only if …

Analysis Of Heat Transfer In A Two-Layer Circular Cylinder:Constant Flux On Outer Surface And Zero Flux On Inner Surface, Mary Elena Franklin

Mathematics & Statistics ETDs

The one-dimensional time-dependent equation of heat conduction is solved analytically for an infinite two-layer circular cylinder whose core may be either hollow or solid. On the outer surface of the cylinder, which has no heat loss due to convection, a constant heat flux from an external source of heating is applied uniformly. The layers are in perfect thermal contact, and there is no heat loss at their interface. At the smaller radius of the inner layer is a perfect insulator so that the heat flux on the inner surface of the two-layer cylinder is zero. The temperature is uniform initially …

Adaptive Nonlinear Prediction And Convergence Rates., Dirk Andrew Dahlgren

Mathematics & Statistics ETDs

Let x1,…,xm be radom variables with an arbitrary (initial) distribution functions F. For i>=m, define

Xi+1=f(theta,x1,…,x1-m+1)+oZi

Where theta is an unknown parameter in Euclidean r space and {Zi}inf m is a sequence of independent, identically distributed random variables with E[Zi]=0 and E[Z 2 i]=1. This type of process will be termed a generalized autoregressive process in the sequel.

A Combinational Analysis Of Finite Minimal Uniform Covers, Gustavus J. Simmons

Mathematics & Statistics ETDs

No abstract provided.

Pseudo-Lattices With Application To Topology And Analysis., Ih-Ching Hsu

Mathematics & Statistics ETDs

In recent times, lattice theory has pervaded many branches of mathematics. Usually, however, the structures of primary interest are not lattices themselves. The lattice structure is arrived at by passing to a quotient set.

Most Stable Tolerance Limits For Log-Convex Distributions, John Richard Ellefson

Mathematics & Statistics ETDs

Necessary and sufficient conditions are given for reversing Jensen's inequality for continuous convex functions, and the result is then stated for the discrete case. The main theorem of Hanson and Koopmans' (1964) pioneering tolerance limit paper is extended to the case of any finite number of order statistics from a probability distribution F for which either -log(1 - F) or - log F is convex. In the first case upper tolerance limits and in the second case lower tolerance limits are exhibited for any content p and any confidence ɣ, 0 < p, ɣ < 1 whenever the sample size is not less than two. When F is such that both functions are convex, two-sided, p-content tolerance intervals at confidence level rare constructed for any sample size greater than one. The distributions for which both functions are convex include the Pólya distribution functions of order two, which includes the normal and exponential families. The distribution of the coverage of the tolerance limits is calculated using an extension of Renyi’s (1953) work on exponential random variables. Goodman and Madansky's (1962) most stable criterion for tolerance intervals is characterized in terms of its second moment about a point, and some of its other properties are established. The results are applied to develop a method for finding the most stable one-sided tolerance limits. The general method is applicable to the problem of finding the most stable two-sided tolerance intervals, but the required explicit calculation of the probability distribution function is lacking. Instead the two-sided tolerance interval is found in terms of two one-sided tolerance limits.

Semi-Groups And A Class Of Singular Perturbation Problems, Andrew Y. Schoene

Mathematics & Statistics ETDs

No abstract provided.

Enumeration Of Certain Binary Arrays, Douglas Jackson

Mathematics & Statistics ETDs

T.V. Narayana [9] and later L. Carlitz [3] have solved the following problem. Given positive integers n, r, L₁,…,L_n-1, a₁,…,a_n such that a_i ≥ r, i = 1,...,n and a_i + L_i ≥ a_i+l, i = 1,…,n-1, how many n x r matrices [a_ij] of positive integers are there which satisfy

1.) a_i1 = a_i i=1,...,n

2.) a_ij > a_i,j+1 i = 1,...,n j = 1,..., r-1

3.) a_ij + L_i ≥ a_i+1,j i=1,...,n-1 j=1,...,r.

Narayana gave a formula for …

The Convergence Of Numerical Solutions Of Hydrodynamic Shock Problems., Darrell Lee Hicks

Mathematics & Statistics ETDs

In 1944 von Neumann Conjectures that only averages of numerical solutions, produced by a discrete scheme he had proposed to solve hydrodynamic shock problems, would converge as the meshes were refined. The hydrodynamic shock problem is defined here as: Find a solution, allowing discontinuous solutions, to the three conservation laws, the increasing-entropy law, and the ideal-gas law when giben physically acceptable initial and boundary values. For discrete schemes similar to von Neumann’s, it is verified that the averages do converge and it appears that von Neumann was also correct in surmising that only the averages converge. A smoothing device named …

Formally Real Fields, Gail L. Carns

Mathematics & Statistics ETDs

This paper is concerned with the analysis of the set of orders on a formally real field using category and sheaf theory as tools. We topologize the set of orders in a way similar to that used in the prime spectrum of algebraic geometry and define a presheaf of groups using the notion of inverse limits. We then find necessary and sufficient conditions that these presheaves be sheaves. Next we notice that if F ⊆ Fⁱ respectively, fields, θ and θⁱ are the set of orders on F and Fⁱ respectively, and Γ and Γⁱ are …

Functional Relationships For Fredholm Integral Equations Arising From Pseudo-Transport Problems., Richard Crenshaw Allen Jr.

Mathematics & Statistics ETDs

In this dissertation we consider a class of integral equations of the general form

Ψ(z)=g(z) + ∫_y^x Y(z’)K(|z-z’|) Ψ(z’)dz’

Where K has an integral representation of the type

K(u)= ∫₀^∞ k(s’)e^-a(s’)uds’

With this integral equation, we associate the pseudo transport problem

(sgn s) ∂/∂z N(z,s) + a(s)N(z,s) = k(s)Y(z) ∫_-∞^∞ N(z,s’)ds’,

Y< =z< =x, |s| < ∞

with certain boundary conditions. (Equation (2) is a generalization or the equation for particle transport in a slab geometry). Equivalence between the problems (1) and (2) is established. Thus any results concerning the problem (2) produce corresponding results concerning the …

Algebraic Properties Of Endomorphisms Of Abelian Groups And Rings, Johnnie George Slagle

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The main objective of the thesis was to extend the definition of an M-Group to what is called an M-Ring. From this extension a system called an expanded ring follows naturally. To facilitate the development of the expanded ring, chapter I is devoted to developing properties on systems that are not quite rings where many interesting examples are constructed. In chapter II the definition of an M-Ring is given and some of its properties are derived. In chapter III some of the properties of expanded rings are proved, and examples of expanded rings are given to show their existence.