Mathematics Commons^™

Fredholm Solution Of Linear Integral Equations, Dewey F. Furness

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

An integral equation of the form

ζ(x) - λ∫ K(x, s) ζ(s) ds = f(x) a ≤ x ≤ b

is called a Fredholm equation. By the method of successive approximations a solution can be obtained if the parameter λ is sufficiently small. If the kernel, K (x, s), is degenerate then a solution can be obtained by reducing it to a system of linear algebraic equations. In the general case the kernel is represented as an infinite Fourier Series. With this representation the solution is obtained by combining the two methods mentioned. The solution is the solution of two …

Axiomatic Set Theory, Ronald P. Smit

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

An analysis of the well known paradoxes found in intuitive set theory has led to the reconstruction of set theory by axiomatic means. This exposition is devoted to Zermelo-Fraenkel set theory with some changes made by Suppes.

The first order predicate calculus is presupposed. In addition to the usual quantifiers admitted, a unique existential quantifier is used. The primitive notions of the set theory are the empty set and the two place membership predicate.

Decision Problems, Lowell Anderson

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

From an intuitive point of view the notion of effective procedure consists of a set of rules or instructions that enables one, in a finite number of steps and in a purely mechanical way, to answer yes or no to any one of a given class of questions. This procedure requires no intelligence to carry out the instructions and, in fact, it is con­ceivable that some mechanical contrivance may be constructed to carry out these instructions. Should such an effective procedure exist, that answers either yes or no, then the group of problems in question is said to be effectively …

Laplace, Finite Fourier Sine And Cosine, Transformations, Jan Eugene Wynn

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

The Laplace and finite Fourier sine transforms can be used to solve certain boundary value problems. Also the Laplace transform is a useful took in solving some integral and integrodifferential equations. This report is composed of transform solutions of seventeen such applied problems, while the author's first report is focused towards the theoretical aspect of these transforms.

Several different types of problems are solved in this report. Among these are Bessel's classical differential equation of index n, two electrical circuit problems, a beam problem, a vibrating string problem, a heat flow problem, and a temperature gradient problem.

One of the …

Error Structure Of Randomized Design Under Background Correlation With A Missing Value, Tseng-Chi Chang

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The analysis of variance technique is probably the most popular statistical technique used for testing hypotheses and estimating parameters. Eisenhart (12) presents two classes of problems solvable by the analysis of variance and the assumption underlying each class. Cochran (9) lists the assumptions and also discusses the consequences when these assumptions are not met. It is evident that if all the assumptions are not satisfied, the confidence placed in any result obtained in this manner is adversly affected to varying degrees according to the extent of the violation.

Simulation Of Mathematical Models In Genetic Analysis, Dinesh Govindal Patel

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

In recent years a new field of statistics has become of importance in many branches of experimental science. This is the Monte Carlo Method, so called because it is based on simulation of stochastic processes. By stochastic process, it is meant some possible physical process in the real world that has some random or stochastic element in its structure. This is the subject which may appropriately be called the dynamic part of statistics or the statistics of "change," in contrast with the static statistical problems which have so far been the more systematically studied. Many obvious examples of such processes …

A Development Of The Number System, Janet R. Olsen

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This paper is based on Landau's book "Foundations of Analysis" which constitutes a development of the number system founded on the Peano axioms for natural numbers.

Topological Groups, Nicolas Anthony Thireos

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A topological group is an abstract group which is also a topological space and in which the group operation are continuous. In group theory the algebraic binary operation of passage to a limit is studied in a similar manner. The two fundamental mathematical concepts of binary operation and passage to a limit are united and interrelated in the concept of topological group.

The concept of topological groups arose from the study of continuous transformations. However, topological groups can be studied quite independently from continuous transformations and the latter can be presented as applications of topological groups. The first person to …

Investigation Of The Properties Of The Iterations Of A Homeomorphism On A Metric Space, Murray B. Peterson, Jr.

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Considerable study has been made concerning the properties of the iterations of a homeomorphism on a metric space. Much of this material is scattered throughout the literature and understood solely by a specialist. The main object of this paper is to put into readable form proofs of theorems found in G.T. Whyburn's "Analytic Topology" pertaining to this topic in topology. Properties of the decomposition space of point-orbits induced by the iterations of a homeomorphism will compose a major part of the study. Some theorems will be established through series of lemmas required to fill in much of the detail lacking …

An Investigation Of The Range Of A Boolean Function, Norman H. Eggert, Jr.

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The purpose of this section is to define a boolean algebra and to determine some of the important properties of it.

A boolean algebra is a set B with two binary operations, join and meet, denoted by + and juxtaposition respectively, and a unary operation, complementation, denoted by ', which satisfy the following axioms:

(1) for all a,b ∑ B (that is, for all a,b elements of B) a + b = b + a and a b = b a, (the commutative laws),

(2) for all a,b,c ∑ B, a + b c =(a + b) (a + b) …

An Investigation Of The Properties Of Join Geometry, Louis John Giegerich Jr.

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This paper presents a proof that the classical geometry as stated by Karol Borsuk [1] follows from the join geometry of Walter Prenowitz [2].

The approach taken is to assume the axioms of Prenowitz. Using these as the foundation, the theory of join geometry is then developed to include such ideas as 'convex set', 'linear set', the important concept of 'dimension', and finally the relation of 'betweenness'. The development is in the form of definitions with the important extensions given in the form of theorems.

With a firm foundation of theorems in the join geometry, the axioms of classical geometry …

Anti-Associative Systems, Dick R. Rogers

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

A set of elements with a binary operation is called a system, or, more explicitly, a mathematical system. [2] The following discussion will involve systems with only one operation. This operation will be denoted by "⋅" and will sometimes be referred to as a product.

A Determination Of The Earth's Gravity Field In Spheroidal Coordinates, M. Spencer Hamilton Jr.

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The earth's gravity field G * at a point P in the region surrounding the earth's surface is defined as the force acting on a unit mass concentrated at P. This is a force resulting from two components: (1) G₁ due to the gravitational attraction of the earth's mass, and (2) G₂ due to the earth's rotation.

As a result of Newton's law of gravitation, G₁ can be written in integral form as follows:

G₁ = k ( (_V ( rdm/r³

where r = PQ, r = |r|, Q is a point which ranges …

Pfaffian Differential Expressions And Equations, K. Raman Unni

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

It is needless to point out the necessity and the importance of the study of Pfaffian differential expressions and equations. While it is interesting to consider from the pure mathematical point of view, their applications in many branches of applied mathematics are well known. To mention a few, one may observe that they arise in connection with line integrals (example, determination of work). They provide a more rational formulation of the foundations of thermodynamics as developed by the Greek mathematician Caratheodory. They also arise in the problem of determining the orthogonal trajectories. In many branches of engineering and other physical …

A Generalization Of The Rayleigh Distribution, Ruth H. Lemon

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

This papers is divided into numbered sections. The equations are numbered anew in each section, and equation numbers are always enclosed in parentheses. Merely the equation number is given when referring to an equation in the same section as the references; otherwise the section number is prefixed. Thus equation (4) refers to the fourth equation of the same section as the reference, and equation (2.2) refers to the second equation of the second section.

Studies On The Sampling Methodology Of Peas For Yield And Quality, Pratapsinha Chintamani Pendse

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

Pea¹ growers have much at stake in getting high yields of peas of prime quality. The income accruing from a pea crop grown for processors is determined by the yield as well as quality. Therefore the farmers' efforts are directed toward growing such a crop.

Research workers are interested in knowing the yield of peas with known tenderometer values which will indicate the quality of peas. Present methods of field harvesting are costly and time consuming which tend to limit the number of varieties that can be satisfactorily evaluated for trial.

A comparison of sampling techniques with present harvesting …

The Solution Of Boundary Value Problems By Use Of The Laplace Transformation As Compared With Classical Methods, Dan W. Stoddard

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

The purpose of this paper is to present a study of different methods of solving certain boundary value problems. In particular it will be concerned with solutions by classical methods and by operational methods. Of the various operational methods that may be considered, the Laplace transformation appears to be the best⁽¹⁾ and will be used in this paper.

In the 1951 Encyclopedia Americana Annual is this report on the activities in applied mathematics for the previous year:

Progress was made on the general problem of finding the eigenvalues of matrices and systems of differential equations. Considerable effort was also …