Digital Commons Network^™

On Polynomial Solutions Of A Class Of Linear Differential Equations Of The Second Order, W. C. Brenke

Department of Mathematics: Faculty Publications

Certain well known polynomials have a number of common properties. They arise as coefficients of tⁿ in the expansion of a generating function ; they may be obtained by means of orthogonalization of a set of functions xⁿg(x) when the function ρ(x) = g²(x) and the interval are properly chosen ; they may be regarded as polynomials which become orthogonal when multiplied by a proper factor g(x) ; they satisfy a certain type of difference equation ; they satisfy a certain type of differential equation. The results …

Matrices Whose Characteristic Equations Are Cyclic, T. A. Pierce

Department of Mathematics: Faculty Publications

One of Sylvester's theorems f on matrices states that if the characteristic equation
(1) | M - λI| = f(λ) = 0
of a square matrix M has the roots λ₁, λ₂, … , λ_n, then the characteristic equation
(2) | φM - ρI| = = g(ρ) = 0
of any integral function of M, namely, φM, has the roots ρ_i = φ (λ_i), i = 1, 2, … , n. In this note an isomorphism is shown to exist between …

History Of Applied Geometry, Evelyn Jackson

Electronic Theses and Dissertations

Mathematics: Just what does the word mean to us? After a moment of thought many different meanings may present themselves to our minds. At first we are inclined to say that the word mathematics covers a vast field. We are justified in so thinking because mathematics embraces a wide scope of study. Were we to say that it is a science we should place it in its proper genius, for it is truly a science of numbers and space. However, could not the science be the art of calculation or the art of computation?

History Of Applied Geometry, Evelyn Jackson

Electronic Theses and Dissertations

No abstract provided.

The Utility Of Mathematics, Jack R. Dunn

Electronic Theses and Dissertations

No abstract provided.

A Syllabus Of Line Geometry, Alice Willmarth

University of the Pacific Theses and Dissertations

In the study of advanced geometry, we shall deal with a certain important relation between pairs of figures in space, and also between their properties. There are two distinct parts to analytic geometry, the analytic work and the geometric interpretation. Two systems of geometry, depending upon different elements with the same number of coordinates, will have the same analytic expressions and will differ only in the interpretation of the analysis. In such a case it is often sufficient to know the meaning of the coordination and the interpretation of a few fundamental relations in each system in order to find …

The Cycloid, Some Related Curves And Their Derivation, Ann Downer

Bachelors’ Theses

In studying the history of the cycloid and the related curves, it is interesting to note that the earliest notations and explanations were given not by a mathematician but by an artist. The description and instrumental construction of the epicycloid curve was presented in about 1525 by Albrecht Durer (1471-1528) a celebrated sculptor and painter of Nurnberg in his book "Underweysund der Messung mit dem Zyrkenund Rychtsceyd." The original idea, however, goes as far back as the time of Hipparchus, an astronomer, who used it in his astronomical theory of epicycles. This curve was now neglected until G. Desargues and …

The Study Of The Curves Of The Equations Having The Type Form X^N + Y^N = A^N, Lena S. Reif

Bachelors’ Theses

The object of this thesis is to determine the general properties of the family of curves, x^n + y^n = a^n from some of the specific curves in that family. The first curve to be considered is the one in which a = 4 and n =1. x+y = 4

Curve Tracing In Polar Coordinates, Virginia Higgins

Bachelors’ Theses

No abstract provided.

The Cubic Equation In One Variable, Ella M. Horst

Bachelors’ Theses

No abstract provided.

Quadratic Equations, Floyd George Hydar

Bachelors’ Theses

In algebra, an equality which exists only for particular values of certain letters representing the unknown quantities is called an equation. These particular values are called the roots of the equation, and the determination of these roots is known as the solution of the equation.

Modern Geometry Of The Triangle, Irene Nolan

Bachelors’ Theses

It has been the object of t he writer to present in this thesis theorems and proofs of modern geometry which seemed most useful and most interesting to her. Concepts of modern geometry have been presented and developed with reference to the triangle.

Periodic Functions, W. Kenneth Koehler

Bachelors’ Theses

The first known publications on the periodic functions were those of Fantet de Lagnv, a French mathematician born at Lyons in the year 1660. Although previous to this, mathematicians, notably Galilio in his analysis of the cycloid, had studied functions and curves whose properties necessarily involved periodic functions, no specific mention of this property was made.

Mathematical Induction, Lucille A. Mccann

Bachelors’ Theses

No abstract provided.

Symmetric Functions Of N-Ic Residues (Mod P), T. A. Pierce

Department of Mathematics: Faculty Publications

If p be an odd prime, q is said to be an n- ic residue of p if the congruence xⁿ = q (mod p) has solutions; otherwise q is an n-ic non-residue of p. A necessary and sufficient condition that q be an n-ic residue of p is that
(1) q ^{(p- 1)/δ} ≡ 1, (mod p),
where δ = g.c.d. (p - 1, n). The numberf of n-ic residues of a given prime p is (p-1)/δ.
It is with the symmetric functions of these …

Graphical Calculation And Computation, Alice Hudson Vallier

Bachelors’ Theses

The word ’’graph” has lately become a part of the active vocabulary of the ordinarily educated person of today. We may say lately because even fifty years ago the use of the graph to represent data was limited to the knowledge of a few theorizing mathematicians and the miscellaneous disorganized usage of some practical mechanics, engineers, business men and statisticians. The situation is not that fifty years ago the rudiments of graphing were not identical with the bases of graphing data today, at least by such simple methods as are commonly employed in periodicals and newspapers for the enlightenment of …

The Theory Of Limits, Molly E. Davis

Bachelors’ Theses

The theory of limits is among the most important theories in mathematics as mensuration of curves, and surfaces, the treatment of series and the foundation of calculus rest upon this theory.

The Study Of The Curves Of Equations Having The Type Form Of Y=XN, Frank Kohn

Bachelors’ Theses

The purpose of this thesis is to investigate those properties of curves having the general type form of y = xⁿ that will enable us to determine the graph of a particular curve of this type. The investigations will not only determine the graph as it appears in the finite region of the curve but its appearance in infinite regions as well.

Applications Of Infinite Products, Lillian Schnell

Bachelors’ Theses

Jacob Nicolaus Bernoulli (1654-1705) wrote a work on "Ars Conjectandi." It was published eight years after his death. The work consists of four parts. The second part is on permutations and combinations, which Bernoulli used in a proof of the binomial theorem for the case of positive integral exponents. The book contains a formula for the sum of the rth powers of the first n integers. This involves the famous Bernouillian numbers. Bernouilli boasted that by means of his formula he could calculate "Intra semi-quadrantem Horae" the sum of the 10th power: the first thousand integers.

Experimental And Observational Geometry, Albert D. Field

University of the Pacific Theses and Dissertations

Geometry has the distinction of being one of the oldest subjects given in the high-school.

Its subject-matter was formulated and organized by the Greeks into a fine system of thought before the time of Christ. Since leaving the hands of the Greeks, geometry has received only a few minor changes, and these largely in recent years.

Heretofore, the study of geometry has been made almost entirely dependent upon memory and reasoning. Geometricians have been slow in adopting the laboratory and observational methods.

This thesis has been written to encourage the student in his work of observing geometrical forms, and in …

Irrational Numbers, Paul O. Smith

Bachelors’ Theses

By an irrational number is meant a number which cannot be expressed by an Integer or by a fraction. The definition in itself is rather vague and to many would not mean very much but by a definite geometric example the idea of just what an irrational number is can be much more clearly shown. Suppose we are given a line of a definite length of two inches. It can easily be seen how this line can be divided into a great number of smaller parts which can be expressed by integers and fractions.

History And Proof Of The Binomial Theorem, Ethel B. Ray

Bachelors’ Theses

Everyone realizes what a wonderful place the world really is. Yet, we of this day little think of the many generations that have gone to make our universe as it is. We think we have done wonders. We pride ourselves on our modern way. There is a tendency in man to "pat himself on the back" and boast of how well he has done. Little do we realize that millions and millions of people have given their whole lives so that we might be as we are. Of course the soldier gives his life for his country; but what of …

Methods Of Tracing Curves, Stephen Lewandowski

Bachelors’ Theses

There are two kinds of equations, algebraic equations and transcendental equations. An algebraic equation is one that involves a finite number of ordinary operations of algebra i.e. addition, multiplication, subtraction, division, extraction of roots, and raising to powers denoted by constant exponents.

Hyperbolic Functions, Margaret S. Johnston

Bachelors’ Theses

A Study of hyperbolic functions is both interesting and fruitful because of their important and historical relation to the more elementary branches of Mathematics: Trigonometry, Analytical Geometry and Differential and Integral Calculus. Historically the development of hyperbolic functions has been closely associated with the development of the properties of the circle, ellipse, and hyperbola; the earliest knowledge of the functions rising out of a comparative study of these curves, and resting on more analogy rather than clear definition and proof.

Indeterminate Coefficients, Armond William Bear

Bachelors’ Theses

No abstract provided.

Summable Types Of Series, Dorothea Maria Baumann

Bachelors’ Theses

By an infinite series we mean a polynomial, or a sequence of terms, each of which derives from its predecessor according to a fixed formula or process; there is a one to one correspondence, consequently between the terms of the series and the set of positive integers 1,2,----,n,----. Summation means the finding of a finite expression equivalent numerically to the given series; known functions only may occur, connected by a fixed number of steps; such a closed expression is called the "sum" of the series.

An Approximation To The Least Root Of A Cubic Equation With Application To The Determination Of Units In Pure Cubic Fields, T. A. Pierce

Department of Mathematics: Faculty Publications

Approximation to the Least Root of a Cubic Bernoulli's method of approximating the largest root of an equation
(1) x³ = ax² + bx + c,
with real coefficients, is to use (1) as a scale of relation for the recursion formula A_n = aA_n-1 + bA_n-2 + cA_n-3. Successive A’s are calculated starting from any initial values. Then A_n+i/A_n for increasing values of n approximates that root of (1) which has the greatest absolute value if that root is real. The method here given for …

Graphic Methods For Solving Algebraic Equations, L. Tremaine Dunlap

Bachelors’ Theses

No abstract provided.

Ua94/6/1 English & Grammar Notebook, Ruth Whitlow

Student Organizations

Grammar and English notebook used by Ruth Whitlow, includes diagrammed sentences and some math problems on the back page.

Ua94/6/1 English & Grammar Notebook, Ruth Whitlow

Student/Alumni Personal Papers

Grammar and English notebook used by Ruth Whitlow, includes diagrammed sentences and some math problems on the back page.