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Eric McDowell

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

Category: Research

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 table.

Joe Kennedy and Eric McDowell, Geoboard Quadrilaterals, Mathematics Teacher, 91 (1998), pp. 288-290. (Read Intro)
Journal Acceptance Rate: 20%

Category: Pedagogy

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 own.

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 combined.

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

Category: Research

Description: This paper extends the results of (1) by allowing each coffee stain to be disconnected.

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. (Read Abstract)

Category: Research

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%

Category: Pedagogy

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 anticipated.
http://maa.org/pubs/cmj_nov02.html 

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%

Category: Research

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

Category: Pedagogy

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.

Category: Research

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.

Category: Pedagogy

Eric L. McDowell. A Correction to: The Connectivity Structure of the Hyperspaces Cε(X), Topology Proceedings 34 (2009) pp. 47-51.

Category: Research

Description: We demonstrate that proposition 3.1 of [2] 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.

Category: Pedagogy

Description: Coincidence 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 current interest.

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