Pi Mu Epsilon Journal
Abstracts and Authors - Fall 1998

Seeing is Not Always Believing

Loi Ngyen and Tu Tran
students at The University of North Florida

Consider the following conjecture:

Let MN and PQ be two perpendicular diameters in a circle with center O, and A, B be two points on MO and ON such that MA=OB. If line BQ intersects the circle at C, then the angle CAQ is a right angle.

In this note we prove that the measurement of the angle CAQ is between 88.21 and 90 degrees, the conjecture is not true although it looks like true very much.

Integral Functions Whose Right Derivatives
are Average Values of Periodic Functions

Paul Fishback, Nicholas Ceglarek (student), and Tobias Moleski (student)
Grand Valley State University

The First Fundamental Theorem of Calculus asserts that if f is continuous on the unit interval [0,1], then the integral function is right-differentiable at the origin. An interesting class of highly-oscillatory functions that does not satisfy this theorem’s hypothesis consists of functions where g is continuous and periodic. We show that for any member of this class, the corresponding integral function is in fact right-differentiable at the origin, and the value of the right derivative there equals the average value of g over one period. We also construct a counterexample showing how this result does not hold true in general when one replaces g by an arbitrary bounded, continuous function.

On Jacobians in Multiple Integrals

Prem N. Bajaj
Wichita State University

The paper draws attention to the sign of the Jacobian, J, of transformation for multiple integrals. an example is provided where the Jacobian takes both positive and negative signs on the region of integration, and consequently, requiring the integration to be evaluated separately on the regions where the Jacobian takes positive and negative signs.

An Approximation to the Logarithmic Integral

Paul S. Bruckman

This paper gives an explicit approximation to the logarithmic integral function li(x), astonishingly accurate within the range of values explored. The comparison between the true values of li(x) and the approximation formula is made for x=10^m, m=3,4,…,18. It is well-known, of course, that if li(x) is itself a reasonable good approximation to the prime counting function . The approximation formula itself, denoted by f(x), is expressed as x times a polynomial in 1/log(x).

On the Chromatic Number of the Middle Graph of a Graph

Masakazu Nihei
Fujishiro High School
Fujishiro, Ibaraki, 300-1537, Japan

Let G be a graph, let A be the vertex chromatic number of G, and let B be the maximum degree among the vertices of G. The middle graph M(G) of a graph G is the graph obtained from G by inserting a new vertex into every edge of G and by joining by edges those pairs of these new vertices which lie on adjacent edges of G.

Involutions in Permutations

Subhash C. Saxena
Coastal Carolina University

In Geometry, an involution or involutory transformation is a transformation which is its own inverse, and is not the identity. Its period is two. In Abstract Algebra textbooks, this term is not commonly discussed in the context of permutations. In this paper, we examine and explore involutory permutations. Such involutions form building blocks of permutations, in the same manner as line reflections on a plane are building blocks of all Euclidean transformations. A recursive relation is developed and two theorems are proved.

Geometric Interpretations of the Continuous Mean

Arvind Shah and Xin-Min Zhang
University of South Alabama

In this paper, we provide some geometric interpretations for the continuous mean. In particular, for a given family of simple closed plane curves, we discuss the relationship between the average value of their areas and the average value of their isoperimetric quotients.

A note on ‘Analytical Formulas for 

Hans J. H. Tuenter
York University

This note gives a simple derivation of the analytical formulas for the sums   where  p is an arbitrary positive integer.

What is a Proof?

Clayton W. Dodge
University of Maine

Much of our mathematics is taught by rote, with the result that many students have little feeling for the unity and structure of the subject. Even young children, however, can understand and appreciate this structure through proofs, if those proofs are presented at an appropriate level. Some examples are presented in this paper.

Scheduling Round-Robin Tournaments

James Oehmann (student)
University of North Florida

To schedule round-robin tournaments means to arrange N different teams, so that each team plays every other team exactly once. A method of filling numbers in a table is established, which is incredibly simple and quick. No preliminary mathematics knowledge is needed.

Math his passion, teaching his life . . .

Donald E. Miller and Laura Meyers (student)
Saint Mary’s College

Milko Jeglic, a college faculty member and superintendent of Slovenian primary schools before WWII, was forced to flee his native land during the communist overthrow of its government. After spending five years in the refugee camps of Austria, he found his way to Saint Mary’s College in Notre Dame, Indiana. there, he distinguished himself while serving as a member of its Mathematics Department for almost three decades. Following his death in December 1997, professional recognition came for Jeglic with posthumous induction into the Indiana Epsilon Chapter. His story is an inspiration to aspiring mathematicians.