E. L. McDowell and Sam B. Nadler, Jr., Absolute Fixed
Point Sets for Continuum-Valued Maps, Proc. Amer. Math. Soc. 124
(1996), pp. 1271-1276. (Read Abstract)
Journal Acceptance Rate: unknown
Description: Pick a point
on the table in front of you. Now, spill some coffee on the table so
that the stain forms a single splotch and note whether or not the stain
from that spill covers the point. Perform a new spill for each point on
the table in such a way that the stains from the spills are nearly
identical when the points are close together. Each time, note whether or
not the spill covers the point that you're working with. The collection
of all points that are covered by their associated stains is called the
fixed point set of this particular "spill function". This
paper determines exactly those sets of points that are the fixed point
set of some spill function, regardless of the shape of the
Joe Kennedy and Eric McDowell, Geoboard Quadrilaterals,
Mathematics Teacher, 91 (1998), pp. 288-290. (Read Intro)
Journal Acceptance Rate: 20%
Description: A geoboard
is an square array of regularly spaced
dots. A quadrilateral can be drawn by selecting any four of these dots
and connecting them with line segments. Sometimes, two or more different
(non-congruent) quadrilaterals can be drawn by connecting the four
given dots. This paper examines the number of different quadrilaterals
that can be drawn on an geoboard for various numbers, n, and encourages students to conduct such explorations on their
Notes: This paper
originally appeared in 1996 in a regional journal subscribed to by Ohio
educators. Since being published in the Mathematics Teacher, this
article has received more citations than all of my other papers
Eric L. McDowell, Absolute Fixed Point Sets for
Multi-Valued Maps, Proc. Amer. Math. Soc. 126
(1998), pp. 3733-3741. (Read Abstract)
Journal Acceptance Rate: unknown
Description: This paper
extends the results of (1) by allowing each coffee stain to be
B. E. Wilder and Eric L. McDowell, Components and
Quasicomponents of Subsets of Continua, JP Journal of Geometry and
Topology, 1 (2001), pp. 163-171. (Read Abstract)
Journal Acceptance Rate: 45%
A sheet of notebook paper is a connected space. You can create a
disconnected space by feeding that sheet of paper into a shredder. Pick
any point of this disconnected space. The component of that
point is the shred of paper that contains it. In other words, the
component of a point is the largest connected subset that contains the
point. Quasicomponents are the same as components in this
shredded paper space, but can often differ in more abstract examples.
This paper considers when components and quasicomponents of subsets of
spaces are identical and when they are not.
Eric L. McDowell and B.E. Wilder, Boundary Bumping in
Connected Topological Spaces, Continuum Theory: In Honor of the 60
th Birthday of Sam Nadler, Lecture Notes in Pure and Applied
Mathematics, Marcel Dekker, Inc., New York (2002), pp. 237-244.
Description: It seems
intuitive that if a part of a connected space is removed from the space,
then the pieces (components) that remain should be close to the part
that was removed. We give examples in this paper that reveal the flaws
of this intuition, and investigate similar consequences of removing
particular parts of a connected space.
Notes: The results in this
book chapter provided answers to a question that I posed in graduate
school regarding the flaws in my intuition described above.
Eric L. McDowell, A Numerical
Introduction to Partial Fractions, The College Mathematics
Journal, 33 (2002), pp. 400-403. (Read Intro)
Journal Acceptance Rate: 25%
Description: It is
occasionally useful to write a ratio of two polynomials as a sum of
ratios of polynomials with low-degree denominators: for example, the
members of the tenure review committee may wish to verify that is equal to. There are techniques to
perform such decompositions, but students often become lost in the
details. In this paper, a similar decomposition idea is introduced that
applies to numerical fractions. It is shown that any fraction can be
decomposed into a sum of fractions whose denominators are all powers of
prime numbers; for example. The procedures for performing
the algebraic and the numeric decompositions are nearly identical.
Therefore, the numeric decompositions provide an excellent introduction
to the algebraic ones.
Notes: I have learned that
the ideas from this paper have been used in the middle school classroom
to broaden students' experience with prime numbers and addition of
fractions. This is an application of my efforts that I had not
Eric L. McDowell, Fixed Point Set
Characterizations of Peano Continua and Absolute Retracts,
Topology and Its Applications, 128 (2003), pp.
123-134. (Read Abstract)
Journal Acceptance Rate: 40%
Description: As in (1),
pick a point on the table in front of you. Now, put a mark on the table
with a ball-point pen rather than a coffee stain and observe whether or
not the mark contains the point that you picked. Do this for each point
on the table in such a way that the ink blotches are nearly identical
when the points are close together. The set of points that are contained
in their associated blotches is called the fixed point set of this
particular "blotch function". In this article, we determine exactly what
sets of points can be fixed point sets of some blotch function.
Eric L. McDowell, A Development of
Simpson's Rule for the Classroom, The International Journal of
Computational and Numerical Analysis and Applications, 3
(2003), pp. 9-15. (Read Abstract)
Journal Acceptance Rate: unknown
Description: One of the
oldest and most important problems of mathematics is the determination
of the area of a flat shape. For certain types of shapes, Calculus
provides a way to notationally express an answer to the
question, but the expression cannot always be evaluated to provide an
exact numeric answer. Simpson's Rule provides a means of
approximating numeric answers in such cases. A proof of Simpson's Rule
is often provided for mathematics majors, but following the proof
requires a great deal of background material. This paper provides a new
proof that is accessible to students in a first-year Calculus course.
Notes: The ideas in this
paper were realized during a very special numerical analysis class in
the spring of 2001. The epiphany that my students and I experienced as
we discovered this proof together was one of the highlights of my
teaching experience. I have continued to present Simpson's Rule to
students in the manner described above ever since.
Eric L. McDowell and B.E. Wilder, The
Connectivity Structure of the Hyperspaces Cε(X), Topology
Proceedings 27 (2003), pp. 223-232. (Read Abstract)
Journal Acceptance Rate: 75%
Note: The name of this journal is misleading.
Rather than a proceedings in the usual sense of the word,
this is a well-regarded, refereed journal in the field.
Description: Identifying a
point on a piece of paper with a ball-point pen is impossible; the mark
that you make will always be too large. We think of such a mark as an
approximation of the point that you're trying to identify. Considering
all such approximations as points in their own right produces a new set
that looks much different from the original piece of paper; in fact,
this new set requires infinitely many spatial dimensions to describe it
properly. In this paper, we investigate some of the properties of these
sets of ink-blotch approximations.
Notes: The space of
approximations described above appeared in the literature for the first
time with (7). However, (9) is the first paper in which these spaces are
the focus of investigation. It turns out that these hyperspaces
have many interesting properties, and are beginning to attract the
attention of other mathematicians. They continue to steer the direction
of my current research.
Eric L. McDowell and B. E. Wilder. Invited
paper. Small-point Hyperspaces of the Arc, Circle, and Simple
Triod. Continuum Theory: In Honor of Professor David P. Bellamy on
the Occasion of His 60th Birthday, Aportaciones Matemáticas, Serie
Investigacion #19. Eds. I. Wayne Lewis, Sergio Macias, Sam B. Nadler,
Jr. Mexico: Sociedad Matemática Mexicana, 2007. 91-96.
Eric L. McDowell. A Correction to: The Connectivity Structure
of the Hyperspaces Cε(X), Topology Proceedings 34
(2009) pp. 47-51.
Description: We demonstrate that proposition 3.1 of
 is false by constructing a locally connected metric continuum which
admists a non-locally connected small-point hyperspace.
Eric L. McDowell. Coincidence values of Commuting Functions,
Topology Proceedings 34 (2009), pp. 365-384.
values of commuting functions from a topological space into itself have
been investigated for more than the past 60 years. We survey some of
the key results of these investigations in the context of questions of