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On The Fourier Developments Of A Certain Class Of Theta Quotients, M. A. Basoco

Department of Mathematics: Faculty Publications

In this paper we shall be concerned with the functions φ^k_a(z) defined by the relation
(1) φ^k_a(z) ≡{d/(dz)logℓ_α (z,q)}^k = {[ℓά(z,q)] ÷ [ℓα(z,q)]^k, α + 0, 1, 2, 3,
where ℓ_α(z, q) is a Jacobi theta function and k is a positive integer. In the first place, we shall derive the Fourier developments which represent these functions in a certain strip of the complex plane; it …

A Research In Sixth Grade Children's Solution Of Verbal Arithmetic Problems, Herbert Schroder

Master's Theses or Doctor of Nursing Practice

The purpose of this research in sixth grade children's solution of verbal problems in arithmetic is to investigate the mental processes that lie back of pupils' answers in arithmetic. That is, how do pupils solve verbal problems in arithmetic? The investigation involves several related problems such as; to what extent is the pupils' method of solution influenced by irrelevant data, cues, details, and numbers used in a problem?

The Basic Analogue Of Kummer’S Theorem, J. A. Daum

Department of Mathematics: Faculty Publications

About one hundred years ago, E. E. Kummer proved the formula
(1) ₂F₁ [_{1 + a – b} ^{a, b; -1}] = [Γ(1 + a - b)Γ(1 + a/2) ÷ Γ(1+a)Γ(1+a/2 - b)
which has since been known as Kummer's theorem. This appears to be the simplest relation involving a hypergeometric function with argument ( - 1).
All the relations in the theory of hypergeometric series _rF₈ which have analogues in the theory of basic serie3 are those in which the argument is ( + 1 ) …

Studies Arising From A Problem In The Calculus Of Variations, John Gonzalez

Mathematics & Statistics ETDs

In mathematics, generalization is progress; so much so that oftentimes one loses sight of the fact that generalization is the result of arduous work in the consideration of the particular. In no other branch of mathematics is this better exemplified than in the Calculus of Variations. The beginning of a systematic development of the theory of the Calculus of Variations really started with the two Bernoulli brothers (1654-1748) in their discussion of the brachistochrone problem in 1696. The method devised by them were sufficiently powerful in the attack of a large number of problems. Euler (1707-85) further elaborated the geometrical …

Families Of Circles, Mary Olive Norman

Bachelors’ Theses

No abstract provided.

A Problem In Reflection Of Light, M. Imogene Palen

Bachelors’ Theses

A circular mirror has a. ray of light from a source at one point on the circumference which is reflected twice. The first ray will by extension, if necessary, intersect the second reflected ray. Increasing the angle of reflection by revolving the source of light so that the first ray falls on all points of the circumference gives a locus, to the successive points made by the intersection of the first ray and the second reflected ray. The problem of this thesis is to picture the locus and find its equation.

On Certain Basic Series, John Daum

Department of Mathematics: Faculty Publications

The identity
(1) Σ_n=1^∞ (qⁿ)/[(1-qⁿ)²] {(1÷(1-q)) + (1÷(1-q²)) + … + (1 ÷ (1 -qⁿ))} = Σ_n=1^∞ [(n²qⁿ)
was deduced from arithmetical considerations by E. T. Bell. About five years ago, W. N. Bailey proved the relation
(2) Σ_n=0^∞ [(1-q)(1-q²)…(1-qⁿ] ÷ [(1-z)(1-qz)…(1-qⁿz)] x [(zⁿ + 1) ÷ (1 - q^{n …}

The Unit Plan Versus The Textbook Plan Of Teaching Arithmetic To Seventh And Eighth Grade Pupils, Arthur P. Harvey

Master's Theses or Doctor of Nursing Practice

The present day trends in education seem to indicate the desirability of its becoming more and more a socialized education. This trend will no doubt have a more significant bearing on some subjects or studies in the school curriculum than on other subjects. The Kansas public elementary schools have recently adopted a Unit Program in School Studies, which includes history, civics, geography, science, and hygiene in an attempt to contribute better to the socialized idea. This Unit Program, however, does not make direct provision for English, spelling, arithmetic, and penmanship .Naturally, perhaps, comes the question to many teachers of why …

Upon The Asymptotic Representation Of Certain Entire Functions In Distant Portions Of The Plane, Abraham Franck

Mathematics & Statistics ETDs

The purpose of this paper is to study two particular entire functions which satisfy the conditions set up in a theorem due to Newsom. It is to be hoped that this report may be preliminary to the invention of a method which will lend itself toward the solution of certain general problems.

The Evolution Of The Concept Of Axiom, William Fred Jones

Mathematics & Statistics ETDs

The field of knowledge known as mathematics is composed of the totality of mathematical systems. A mathematical system consists of a body of propositions known as the axioms. The concern of this paper is with the axioms; in fact, with the change which has taken place during the centuries in the ideas of mathematicians about the axioms.

New Formulae For The Determination Of The Yield Of A Bond, Marvin Roberts

Mathematics & Statistics ETDs

A written contract to pay a certain amount of money on a specified redemption date, and to pay equal periodical dividends, is called a bond from a mathematical standpoint. The principal mentioned in the contract is its face value, or par value. The amount redeemed, or the redemption value, is denoted by C, the dividends by D, and the principal by F. A bond is redeemed at a par if C and F are the same, and at a premium if C is greater than F. The divided rate, or bond rate, is the interest rate named in the bond. …

A Study Of Fermat's Last Theorem., Jerome L. Krumpelman

Electronic Theses and Dissertations

This work is intended primarily for the student of mathematics who may not necessarily wish to delve into the complete theory of the problem. but who may wish to get a picture of the work necessary to study and understand it. For a complete history of the problem until 1919, refer to Chapter XXVI of Dickson's History of the Theory of Numbers, II, and for the work primarily connected with Irregular Cyclotomic Fields and Fermat's Last Theorem through part of 1927, refer to Algebraic Numbers, II, National Research Council Bulletin, # 62. For a Short presentation of the problem refer …

The Existence Theorem Of Ordinary Differential Equations, Harris J. Dark

Master's Theses

There are a great many devices for solving differential equations of certain special forms. But there is a large number of classes of differential equations that are not included in these special forms and cannot be integrated by quadratures or other purely elementary methods. When mathematicians were forced to abandon their cherished hope of finding a method for expressing the solution of every differential equation in terms of a finite number of known functions or their integrals they turned their attention to the question of whether a differential equation in general had a solution at all, and, if so, of …

A Remedial Program In The Four Fundamental Operations In Decimal Fractions, Hallie Ruth Weber

Master's Theses or Doctor of Nursing Practice

Can the results of the teaching of the four fundamental operations in arithmetic be improved? This is an important question that confronts most teachers of arithmetic. How can the teacher eliminate the difficulties which interfere with pupil progress, and reduce the number of failures in arithmetic? A group diagnostic test will show not only the level of ability of pupils in the various operations, but will point out weaknesses so that the teacher may organize her work in such a way as to give time to those who are below the standard and omit deadening drill for those who are …

Variation Of Moments With Distributions Of Masses And Areas, Mayo Glenwood Shults

Master's Theses or Doctor of Nursing Practice

The purpose of this investigation has been to determine the variations of the first, second, third and fourth moments as functions of distributions of masses and areas. These moments were found about a line or plane perpendicular to the horizontal axes of the figures, and at various distances along the axes. For a complete understanding of the problem, an explanation of various terms will be made in relation to mechanics statistics.

Gauss' Hypergeometric Equation, William R. Smith

Master's Theses

As early as the seventeenth century the English mathematician, john Wallis (1616-1703), used the term "hypergeometric" to describe a series which he was studying. This series, ∑(a)(a+b)(a+2b)…(a+n-1b), is quite different from the usual geometric series, hence the term, "hyper" (=above) plus "geometric," was used to signify that the series was of greater complexity than the geometric series. Wallis did not consider his series a power series or a function of x.

In 1769 this series received a remarkable development at the hands of Loonhard Euler who, following the example of Wallis, applied the word "hypergeometric" to it. He observed that …

Upon The Asymptotic Representation Of Entire Functions Where The General Coefficient Is The Product Of Two Gamma Functions, James R. Ellis

Mathematics & Statistics ETDs

The essential purpose of this paper is to obtain further information in regard to the asymptotic representation of a power series.

The Segregation Of Real Roots Of Lower Orders, Ross W. Bland

Master's Theses or Doctor of Nursing Practice

The purpose of this thesis is the presentation of the better algebraic methods of the segregation of real roots of equations of lower orders and a brief historical sketch of the advance made in the development of algebra. Mention is also made of a few of the men who have made notable contributions to this field of mathematics and the time, as nearly as it is possible to determine it, when new ideas were discovered.

Linearity In Factor Analysis, Opal Emmons

Master's Theses or Doctor of Nursing Practice

The uniqueness of mathematics in that it is a science complete in itself makes it possible to apply mathematical tools to sciences which, in themselves vary widely in content. There is, however, a particular difficulty which must be surmounted. A given mathematical procedure may be valid in the science of mathematics but invalid in another science, such as psychology. The reason for this is that sometimes there is a discrepancy between the implicit assumptions in the field of mathematics and those implicit in the field of application. The purpose of this investigation is twofold: (1) To point out that linearity …

Asymptotic Series, Bernard Martin

Master's Theses or Doctor of Nursing Practice

The purpose of this thesis is to follow the development of the asymptotic series from the beginning made by Euler, in his solution of problems, to the modern applications of asymptotic series, also to present a definition and an analysis of asymptotic series, explain some different theories of their use, and show the ways in which they find common usage. The use of asymptotic series is a comparative new development in the field of applied mathematics. The divergent series were not used as a method of calculation until Euler showed that they gave close approximations in the solution of problems. …

Some Applications Of The Principles Of Isomorphism And Variation To The Teaching Of Geometry, Ralph Mock

Mathematics & Statistics ETDs

In summary, it should be stressed that the numerous examples of this paper are important only in so much as they illustrate the use of the devices and concepts emphasized in the study. It is our premise that plane geometry, as usually taught, does not possess the value which it might have. Certain changes in content as well as in presentation would materially improve the course. This study, then, presents suggestions for improving the ordinary course in two directions. First, by presenting logical propositions from the students' experiences which are partially isomorphic to the theorems of plane geometry, it is …

Solution To Theory Of Equations For Teachers And Graduate Students, Arthur Lawrence Birkholz

Bachelors’ Theses

This thesis is intended primarily for teachers and graduate students who often find it necessary to review all or part of the basic theorems in the study of equations. Dickson's excellent book on the subject, "First Course in the Theory of Equations", is a recognized authority, so to facilitate a study of this book the writer has worked the problems suggested by Dickson, because these problems, more than any explanation could, illustrate and simplify the theorems on equations.

The user of this work-must bear in mind that there may be several ways of working many of the problems, and hence …

The Efficiency Of Approximation Formulae For Determining The Rate Of Interest In Amortization Schedules, Wade Ellis

Mathematics & Statistics ETDs

The problem of this study is to determine the relative efficiency of the more important formulae for approximating the rate of interest involved in an amortization plan when the amount of the debt, the amount of the periodic payment, and the number of periods are known. By efficiency this meant the degree of accuracy obtainable, and the time required for solution.

The Construction Of Triangles Having Given Three Elements, Mary Eugene Wolter

Bachelors’ Theses

It is with great pleasure and satisfaction that I dedicate these following pages to those who are especially interested in mathematics, to those who are more geometrically than algebraically inclined, and in particular to my students in mathematics. It is through my students of the past that I have been assisted in bringing forth this work, by applying such knowledge as I have acquired in my teaching experience. To the pupils who in the future will come under my guidance and direction I hope to impart the benefits and fruits that should be the outgrowth of the labor and study …

The Summation Of The Series X+X/4+X/9+...+X/N-+.., William J. Nee

Bachelors’ Theses

The purpose of this thesis is to determine the sum of the series x+4²/4+x³/9+x⁴/16+....xⁿ/n². In order to determine whether or not the series is summable, we must first find out if it is a convergent series and so the first part of this paper will deal with some of the tests of convergence. After this matter is settles, we shall select a value of x, in this case x=1 and determine the ^∞∑/1 1/n2

Orthogonal Polynomials With Orthogonal Derivatives, M. S. Webster

Department of Mathematics: Faculty Publications

We are concerned with the following assertion:
THEOREM. If {φ_n(x)} and { φ_n’(x)} are orthogonal systems of polynomials, then {φ_n(x)} may be reduced to the classical polynomials of Jacobi, Laguerre, or Hermite by means of a linear transformation on x.

Graphical Representations Of Singular Solutions Of Differential Equations, Kathryn Lois Pitman

Bachelors’ Theses

The first men to detect singular solutions of differential equations were Leibniz, Brook Taylor and Clairaut. The direct method of attack used by each man in attaining these solutions is not known, but a short history of each man and what he has contributed to mathematics can be given. Gottfried Wilhelm von Leibniz (1646-1716) was born in Leipsic, Germany, and was the son of a professor of law in a nearby university. He was well-educated in law himself, expecting to follow in the profession of his father. But not being under the necessity of earning his living, he enrolled at …

A Brief Consideration Of The Different Methods Of Integration, Donald Golden

Master's Theses or Doctor of Nursing Practice

Any student of mathematics is familiar with the importance of the process of integration. Integration is as fundamental to analysis as the basic principles of the number theory are to arithmetical calculation. Some of the common applications of integration are, finding the distance a falling body has travelled during a particular interval of time; to determine the equation of a curve, given different conditions (such as slope of the curve equal numerically to one-half the abscissa, or some similar problem); motion of a projectile; motion in a resisting medium; finding areas and volumes of revolution; length of a curve; areas …

The Asymptotic Development Of A Special Type Of Power Series, David A. Lawson

Mathematics & Statistics ETDs

It is the purpose of this study to consider certain aspects of the theorem due to Walter B. Ford, may be considered as a function g(w) of the complex variable w=x+iy and as such satisfies two conditions chosen by the author.

Unicursal Bicircular Quartic Studied By Inversion, William A. Uporsky

Bachelors’ Theses

The quartic equation was studied by Arab scholars after their attention to the results of their study of a cubic had not proved very significant. Abu'l-Faradsh completed the Fihrist (c, 987), and in this he refers to a problem by Abu’l-Wefa (c, 980). The Arabs were Interested in the quartic equation only as they were in the cubic, i.e., from the standpoint of the intersection of two conics.