# The REU Program In Mathematics

# Research and Publications at Previous REU Programs

*This page contains a summary of the research and publications that have resulted from the GVSU REU, from 2000 to 2013. Note that there may also be papers currently submitted or forthcoming that are not listed here.*

2013 ~ 2012 ~ 2011 ~ 2010 ~ 2009 ~ 2008 ~ 2007 ~ 2006 ~ 2005 ~ 2004 ~ 2003 ~ 2002 ~ 2000

**2013**

**Equal Circle Packing:**This team (Madeline Brandt, Hanson Smith and Prof. William Dickinson) found all optimally dense packings of four of equal circles on any flat tori. There turn out to be several two parameter regions in the moduli space of flat tori where there are two or more optimal arrangements (one globally dense and the others locally dense). This is the first example where there have been multiple optimal packings on a single torus with packing graphs that are not homeomorphic as subsets of the torus. The behavior of the optimally dense packings agrees the work of A. Heppes from 1999. A manuscript is under preparation.

**Voting Theory:**This team (Beth Bjorkman, Sean Gravelle, and Prof. Jonathan Hodge) investigated graph theoretic representations of multidimensional binary preferences associated with referendum elections. We specifically studied preferences that can be represented by Hamiltonian paths in cubic graphs with the Gray Code labeling. We characterized the algebraic

structure of the sets of preferences that can be generated in this manner, and we also proved results about the interdependence structures that result from such preferences. Further research is in progress, and we anticipate submitting a manuscript for publication in 2014.

**Mathematics and 3D Printing:**At the beginning of the summer, the mathematics department at GVSU acquired a Makerbot Replicator 2 3D-printer, and research of this team (Melissa Sherman-Bennett, Sylvanna Krawczyk, and Prof. Edward Aboufadel) revolved around this technology. The goal was to develop novel techniques using mathematics to design objects appropriate for the printer. The team developed techniques to “print” algebraically-defined surfaces, manifolds defined from real data (such as elevation data from geography), and friezes based on data collected by the Kinect camera. Using ideas from linear algebra, the team then developed a method to identify depth data for an object based on two photographs, and “print” these objects, such as a human hand. At the end of the summer, the team wrote a primer, “3D Printing for Math Professors and Their Students,” that was made available for free on the Internet. A second manuscript, based on the depth data/photography project, is under preparation.

**Extended Outer Billiards in the Hyperbolic Plane:**This team (Sanjay Kumar, Austin Tuttle and Prof. Filiz Dogru) focused on analyzing the extended outer polygonal billiard map in the hyperbolic plane. We classified polygonal tables with respect to their rotation numbers, whether rational or irrational, and we wrote programs to investigate our conjectures for periodic orbits of this special circle map generated from polygonal tables. A manuscript is under preparation.

**Arrow Path Sudoku:**This team (Ellen Borgeld, Elizabeth Meena, and Professor Shelly Smith) explored a variation of a Sudoku puzzle that uses an arrow in each cell, pointing to the cell containing the subsequent number, in additional to only a small number of numerical clues. We use inclusion-exclusion and computer programs that we wrote to count 2x2, 2x3, and 3x3 number blocks that admit valid arrow paths, Arrow Path blocks that are solvable, and the number of solutions for each block. We developed an equivalence relation on the set of blocks of each size, then partitioned the sets into equivalence classes to facilitate combining blocks to form 4x4, 6x6, and 9x9 Arrow Path Sudoku boards. We described and counted all possible 4x4 boards that are solvable, and determined the maximum number of numerical clues required to create Arrow Path puzzles of each size with a unique solution. A manuscript is under preparation.

**Combinatorial Sums and Identities:**This team (Sean Meehan, Michael Weselcouch and Professor Akalu Tefera) explored, conjectured and formulated various challenging and interesting combinatorial sums and identities. To do this the team spent a great deal of time studying powerful combinatorial (counting) methods and computer assisted proof techniques of Wilf-Zeilberger and other symbolic computation techniques. Using various computer summation algorithms the team was able to discover and prove challenging and interesting old and new combinatorial identities.

**Equal Circle Packing on a Flat Square Klein Bottle:**This team (Matthew Brehms, Alexander Wagner and Professor William Dickinson) explored one and two equal circle packings on a square flat Klein bottle. To do this using the same methods as in packings on flat tori, we had to explicitly calculate the identity component of isometry group of any flat Klein bottle and we rewrote a program to compute the possible packing graph structures on a Klein bottle. Along the way, we proved a theorem that every flat Klein bottle is isometric to a flat Klein bottle where the generating vectors are orthogonal (i.e. a rectangular flat Klein bottle). This enabled us to discover and prove the optimality of the one and two equal circle packings on a square flat Klein bottle and to conjecture the optimal packing arrangement for three equal circles on a square flat Klein bottle. A manuscript is planned in the future.

**Single-Peaked Preferences in Multiple-Question Elections:**This team (Lindsey Brown, Hoang Ha, and Professor Jonathan Hodge) applied the concept of single-peaked preferences to the multidimensional binary alternative spaces associated with a variety of multiple-criteria decision-making problems, including referendum elections. They generalized prior work on cost-conscious preferences in referendum elections, showing that single-peaked binary preferences are nonseparable except in the most trivial cases, and that electorates defined by single-peaked preferences always contain weak Condorcet winning and losing outcomes. They also developed a general method for enumerating single-peaked binary preference orders, finding exact counts for 2, 3, and 4-dimensional alternative spaces. A manuscript, "Single-peaked preferences over multidimensional binary alternatives," has been accepted for publication in

*Discrete Applied Mathematics*.

**Applications of Wavelets:**This team (Nathan Marculis, SaraJane Parsons, and Professor Edward Aboufadel, with help from Clark Bowman) worked on the following problem: using accelerometer, GPS, and other data collected by smartphones while driving, how can this data be used to identify the location and severity of potholes? The team made use of wavelet filters, Kruskal’s algorithm, and other mathematical tools to develop an algorithm to solve the problem. In February 2012, the City of Boston announced that they would be using the wavelet-Kruskal solution, along with algorithms from other researchers, in their Street Bump app. A manuscript is under preparation.

**Equal Circle Packing:**This team (AnnaVictoria Ellsworth, Jennifer Kenkel and Professor William Dickinson) found all optimally dense packings of three of equal circles on any flat tori. For all but a two parameter region in a moduli space of tori there is exactly one optimally dense arrangement. Inside this region there are two optimally dense packings (one globally dense and the other locally dense). The behavior of the optimally dense packings agrees with the previous summer's work and the work of Heppes. A manuscript is under preparation.

**Outer Billiards in the Hyperbolic Plane:**This team (Neil Deboer, Daniel Hast, and Professor Filiz Dogru) analyzed the orbit structures and the geometric properties of the outer (dual) billiard map in the hyperbolic plane. We geometrically constructed the 3-periodic orbit for small triangles. This construction led us to define a new term to describe the strong criterion, "triangle-small polygon" to classify polygons in the hyperbolic plane. As a result, we have discovered the special class of polygons which have at least one 3-periodic orbit inside the hyperbolic plane.

**Voting Theory:**This team (Clark Bowman, Ada Yu, and Professor Jonathan Hodge) developed an iterative voting method for referendum elections. Our method allows voters to revise their votes as often as they would like during a fixed voting period, with the current results of the election displayed in real time. Through extensive computer simulation, we showed that our method yields significant improvements from standard simultaneous voting and in many cases solves the separability problem, a phenomenon that is known to yield undesirable and even paradoxical outcomes in referendum elections. A paper based on this work, "The potential of iterative voting to solve the separability problem in referendum elections," was published in

*Theory and Decision*.

**2010**

**Higher Dimensional Rook Polynomials**: This team (Professor Alayont, Moger-Reischer, Swift) focused on generalizations of 2-dimensional rook polynomials to three and higher dimensions. The theory of 2-dimensional rook polynomials is concerned with counting the number of ways of placing non-attacking rooks (no two in a row or a column) on a 2-dimensional board. The theory can be generalized to three and higher dimensions by letting rooks attack along hyperplanes. In 2 dimensions, the rook numbers of certain families of boards correspond to known number sequences, including Stirling numbers, number of derangements, number of Latin rectangles and binomial coefficients, and provide other combinatorial interpretations of these sequences. Our focus this summer was exploring similar correspondences for three and higher dimensional rook numbers. Building upon research conducted in 2009 funded by a GVSU S3 Grant, we found a family of boards in higher dimensions generalizing the 2-dimensional boards with Stirling numbers as their rook numbers. The rook numbers of these higher dimensional boards and those of their complements resulted in generalized central factorial numbers and the generalized Genocchi numbers. We also found a family of boards in higher dimensions that generalize the staircase boards in 2 dimensions. The rook numbers of these boards are binomial coefficients as are those of the 2-dimensional staircase boards. These examples provide new combinatorial interpretations of these sequences. An manuscript based on this work has been submitted for publication.

**Orthogonality in the space of compact sets**: This team (Professor Schlicker, Sanchez, Jon VerWys) focused on the topic of Pythagorean orthogonality in the space H of all nonempty compact subsets of n dimensional real space. Our ultimate goal was to develop a trigonometry on H. The space H is a metric space using the Hausdorff metric h, and previous REU groups have learned much about line segments in H. If A, B, and C are elements of H, we defined the segments AB and AC to be orthogonal if their lengths satisfy the Pythagorean identity, that is the square of h(B,C) is the sum of the squares of h(A,B) and h(A,C). When this happens we say that A, B, and C form the vertices of a right triangle in H with segment BC as hypotenuse and segments AB and AC as legs. This group made progress on a characterization of exactly when a segment BC can be the hypotenuse of a right triangle in H ( this is not always possible) and when a segment AB can be a leg. This group also discovered many different ways that orthogonality in H is different than orthogonality in n dimensional real space. Progress was made on defining the concept of spread in H which may lead us to an interesting and useful notion of trigonometry in H.2-dimensional staircase boards. These examples provide new combinatorial interpretations of these sequences. We continue to work on these problems to complete some of our conjectures and, if successful, will submit a paper for publication in the future.

**Equal Circle Packing on Flat Tori**: This team (Professor Dickinson, Tries, Watson) focused on equal circle packings of small numbers of circles in a one-parameter family of flat tori. The family of tori that we worked on were those that are the quotient of the plane by a lattice generated by two unit vectors with an angle of between 60 and 90 degrees. (A packing of circles on a torus is an arrangement of circles that do not overlap and are contained in the torus. The density of a packing is the ratio of the area covered by the circles divided by the area of the torus.) We focused on finding all the optimally dense (both locally and globally) arrangements of 1 to 4 circles on this one-parameter family of tori. Roughly speaking, a packing is locally optimally dense if it has equal or greater density than all near by arrangements. A packing is globally maximally dense if it is the most dense locally optimally dense packing. For 1 and 2 circles packed on any torus in the one-parameter family of tori, we proved there is a unique optimally dense arrangement for each torus. For 3 circles packed on a torus, the number of and type of arrangement depend on the angle of the torus. For any angle strictly between 60 and 90 degrees, we proved that there are exactly two locally maximally dense arrangements. When the angle is 90 degrees the two arrangements are identical and at 60 degrees, the one of the arrangements is no longer locally maximally dense. The behavior of the optimally dense packings at the extreme angles of 60 and 90 degrees agrees with previous REU [2007 - DMS 0451254] and other research.

**Rank Disequilibrium in Multiple-Criteria Evaluation Schemes**: This group (Professor Hodge, Stevens, Woelk) developed a mathematical model of the concept of rank disequilibrium, which occurs when individuals are evaluated over multiple criteria and have different perceptions of the relative value of each criterion. Rank disequilibrium has been shown to be a significant source of organizational conflict, and this project was the first attempt to formally model and investigate rank disequilibrium from within a mathematical framework. The main objects of study were rank aggregation functions, which assign overall rankings to combinations of rankings on individual criteria. The group defined several desirable properties for rank aggregation functions to satisfy and proved necessary and sufficient conditions for the existence of rank aggregation functions with these properties. The group proved that while it is nearly impossible to avoid all forms of inequity, certain forms of inequity can be avoided by limiting the number of possible rankings on each criterion, restricting the set of possible ranking profiles, or by exploiting known information about the evaluees’ preferences. A manuscript based on this work is in preparation.

**Wavelets and Diabetes**: This group (Aboufadel, Olsen, Castellano) created a new measurement developed to quantify the variability or predictability of blood glucose in type 1 diabetics. Using continuous glucose monitors (CGMs), this measurement -- called a PLA index -- is a new tool to classify diabetics based on their blood glucose behavior and may become a new method in the management of diabetes. The PLA index was discovered while taking a wavelet-based approach to study the CGM data. This wavelet-based approach emphasizes the shape of a blood glucose graph. Their article, "Quantification of the variability of continuous glucose monitoring data algorithms," appeared in

*Algorithms*.

**Greater Than Sudoku**: This group (Smith, Burgers, Varga) investigated aspects of a variation of a Sudoku puzzle that uses inequalities between adjacent cells rather than numerical clues. They showed that the cells of an m by n inequality block are a partially ordered set, and that an inequality block is solvable if and only if it is acyclic. They also defined an equivalence relation on the set of solvable 2 by 2 inequality blocks and proved that there are 224 Greater Than Shidoku (4 by 4) boards with unique solutions. A manuscript is in preparation.

**Hypergeometric sums and identities**: This group (Tefera, Dahlberg, Ferdinands) studied computerized proof techniques, specifically Gosper's and Zeilberger's algorithms and the Wilf-Zeilberger proof method, and counting (combinatorial) proof techniques. Using computerized and counting proof techniques the team investigated several hypergeometric sums, found closed forms of various challenging and interesting hypergeometric sums and proved identities involving binomial coefficients. Using counting techniques, the team was able to find a beautiful proof for one of the challenging problems proposed in the June 2009 issue of *Mathematics Magazine* (the solution is submitted for publication). The team also gave an elementary and elegant approach, using the WZ method, for sums of Choi, Zoring and Rathie. Their article, "A Wilf-Zeilberger Approach to Sums of Choi, Zornig and Rathie," appeared in *Quaestiones Mathematicae.*

**Cost-conscious voters**: This group (Hodge, Golenbiewski, Moats) developed an axiomatic model of cost-conscious voters in referendum elections. They used this model to prove a variety of useful results, including: (i) that the probability of cost-consciousness approaches zero as the number of questions grows without bound; (ii) that, under certain conditions, elections in which voters are cost-conscious will always contain at least a weak Condorcet winning outcome (and a weak Condorcet losing outcome); and (iii) that the interdependence structures within the preferences of cost-conscious voters can be varied and unpredictable. Their article, "Cost-conscious voters in referendum elections," appeared in *Involve*.

**Wavelets**: This group (Aboufadel, Boyenger, Madsen) developed a new method to create portraits that imitate the style of artist Chuck Close. Wavelets were used for detecting and classifying edges. While Chuck Close uses a diamond tiling of the plane for his portraits, this new method can use regular tilings by triangles and other objects. Their article, "Digital creation of Chuck Close block-style portraits using wavelet filters", appeared in 2010 in the Journal of Mathematics and the Arts. In addition, the research group created a digital "Chuck Close-like" print of Ramanujan that was featured at the art exhibition at the 2011 Joint Mathematics Meetings.

**Hausdorff Metric Geometry**: This group (Schlicker, Montague) obtained some fascinating results on finite betweenness in the Hausdorff metric geometry. More specifically, they found necessary and sufficient conditions under which there can be a finite set between two sets A and B. One unexpected consequence of this characterization is that for some sets A and B, there can be finite sets at some locations between A and B but not at others. This group also found some interesting preliminary results on convexity in this geometry. Their paper, "Betweenness of Compact Sets"

*is in preparation.*

**The Geometry of Polynomials**: This group (Schlicker, Nuchi, Shatzer) investigated several problems related to the Sendov conjecture: a two-circle type theorem for polynomials with all real zeros, finding polynomials with minimal deviation from their roots to their critical points, and maximizing the deviation from the roots of a polynomial with real roots (and its derivative) to their centroids. The most interesting results were obtained in the latter problem where the group has made significant progress in proving its conjecture of the polynomials with maximal deviation.

**Gerrymandering**: This group (Hodge, Marshall, Patterson) explored the notion of convexity as it relates to the problem of detecting gerrymandering and producing optimal congressional redistrictings. They defined the convexity coefficient of a district or region, and used Monte Carlo simulation to approximate the convexity coefficient of each of the 435 congressional districts in the United States. They explored several theoretical questions pertaining to approximation of convexity coefficients and the effect of subdividing regions using straight-line cuts. They also explored ways to modify the convexity coefficient to account for population density and irregularities in state boundaries. Their article,"Gerrymandering and Convexity" appeared in the September 2010 issue of the

*College Mathematics Journal*, and the convexity coefficient introduced therein has its own entry on Wolfram Mathworld.

Convexity and Gerrymandering, from the 2008 REU

**Group A**(Schlicker, Honigs, Martinez) defined and investigated three different types of connectedness in the geometry of the Hausdorff metric and obtained some preliminary results about sets that satisfy each type of connectedness. Each finite configuration in this geometry has an associated bipartite graph. Several results obtained about edge covers of graphs by this group gave insight into the possible number of sets at each location between the end sets in any configuration. A paper based on this research, "Missing edge coverings of bipartite graphs and the geometry of the Hausdorff metric," has been accepted for publication by the

*Journal of Geometry*.

**Group B**(Hodge, Lahr, Krines) explored questions pertaining to the notion of separability in voter preferences over multiple questions. Specifically, they generalized methods for counting preseparable extensions, which are used to build larger preference orders from smaller ones, and developed a characterization of the types of preferences that can be constructed via preseparable extensions. They also characterized the algebraic structure of the sets of permutations that preserve the separability of a given separable preference order, and discovered a connection between separable preference orders and Boolean term orders, which have applications to abstract algebra and comparative probability theory. Their article, "Preseparable extensions of multidimensional preferences" appeared in 2009 in the journal

*Order*.

**Group C**(Dickinson, Guillot, Castelaz) focused on finding all the locally and globally dense packings of 1 to 5 circles on the standard triangular torus. Roughly, a packing of

*n*equal circles is locally maximally dense if there exists a positive epsilon, so that if each circle center is moved by less than epsilon, the density must decrease (i.e. the smallest pairwise distance between the centers must decrease). A packing of

*n*equal circles is globally maximally dense if it is the most dense locally maximally dense packing. Two published articles resulted from this research: "Optimal Packings of up to Five Equal Circles on a Square Flat Torus" (with Sandi Xhumari), which appeared in

*Beiträge zur Algebra und Geometrie*; and "Optimal Packings of up to Six Equal Circles on a Triangular Flat Torus" (with Sandi Xhumari), which appeared in the

*Journal of Geometry*.

**Group D**(Dogru, Komlos, Gorski) investigated the dynamics of the dual billiard map in the Euclidean plane and the hyperbolic plane. The orbit behavior of the map changes with respect to the shape and size of the table. Their work concentrated on regular polygonal tables which tessellate the Euclidean or hyperbolic plane.

**Group A**(Aboufadel, Armstrong, Smietana) investigated position coding and invented two new position codes -- one based on binary wavelets, while the other uses a base-12 system combined with binary matrices. A manuscript has been posted to the ArXiV.

**Group B**(Wells, Lerch, Leshin) addressed the problem of finding an optimal seating strategy to maximize acquaintances made at successive events.

**Group C**(Schlicker, Schultheis, Morales) created a computer program to automate the computation of the number of elements at each location on a Hausdorff segment between two finite sets. They also created new integer sequences which arise from special types of Hausdorff configurations. Their paper, "Polygonal chain sequences in the space of compact sets," appeared in 2008 in the

*Journal of Integer Sequences*

**.**

**Group D**(Dickinson, Hediger, Taylor) solved San Gaku geometry problems from Japan, ones that involved parallel lines. They generalized these problems and solutions to spherical and hyperbolic geometry.

**Group A**(Boelkins, From and Kolins) investigated several problems in the geometry of polynomials focused on the relationship between the set of zeros and the set of critical numbers. The article, "Polynomial Root Squeezing" appeared in

*Mathematics Magazine*in February 2008.

**Group B**(Fishback, DeMore, Bachman) investigated various problems involving "least squares derivatives" associated with various random variables and corresponding families of orthogonal polynomials.

**Group C**(Aboufadel, Lytle, and Yang) applied wavelets and statistics to match handwriting samples and determine forgeries. The article, "Detecting Forged Handwriting with Wavelets and Statistics", written by these researchers, appeared in the Spring 2006 issue of the

*Rose-Hulman Undergraduate Mathematics Journal*.

**Group D**(Schlicker, Blackburn, Zupan) investigated the geometry the Hausdorff metric imposes on the collection of non-empty compact subsets of

*n*-dimensional real space. A major finding for this group was that, although there are configurations in this space that allow

*k*elements at each location between two sets for infinitely many different values of

*k*(including examples for

*k*between 1 and 18), there is NO possible configuration that allows exactly 19 elements at each location. The paper “A Missing Prime Configuration in the Hausdorff Metric Geometry,” with Chantel Blackburn, Kristina Lund, Patrick Sigmon, and Alex Zupan, appeared in 2009 in the

*Journal of Geometry*. A second paper is in preparation.

**Group A**(Sorensen, Morris, and VanHouten) created "Bubble Bifurcations" in dynamical systems.

**Group B**(Dickinson, Katschke, and Simons) solved San Gaku geometry problems from Japan, and generalized these problems and solutions to spherical and hyperbolic geometry.

**Group C**(Aboufadel, Brink, and Colthorp) applied wavelets and other tools to the problem of finding airplanes in aerial photographs.

**Group D**(Schlicker, Lund, and Sigmon) investigated the geometry the Hausdorff metric imposes on the collection of non-empty compact subsets of n-dimensional real space. In particular, they found many interesting occurrences of Fibonacci and Lucas numbers in this geometry. The paper “Fibonacci sequences in the space of compact sets,” with Kris Lund and Patrick Sigmon, appears in

*Involve*, Vol. 1 (2008), No. 2, 197-215. The paper “A Missing Prime Configuration in the Hausdorff Metric Geometry,” with Chantel Blackburn, Kristina Lund, Patrick Sigmon, and Alex Zupan, has been accepted in the

*Journal of Geometry*.

**Group A** (Schlicker, Bay and Lembcke) investigated the geometry of the Hausdorff space, in particular the lines of that space. The article "When Lines Go Bad in Hyperspace," written by these researchers, appeared in *Demonstratio Mathematica* in 2005.

**Group B** (Aboufadel, Olsen and Windle) applied wavelets and other tools to the problem of breaking CAPTCHAs. The article "Breaking the Holiday Inn Priority Club CAPTCHA," written by these researchers, appeared in the March 2005 *College Mathematics Journal*.

**Group C** (Boelkins, Miller and Vugteveen) investigated polynomial root dragging. The article, "From Chebyshev to Bernstein: A Tour of Polynomials Large and Small", written by these researchers, appeared in the May 2006 *College Mathematics Journal*.

**Group D** (Wells, Bromenshenkel and Hogg) conducted work in Lie Theory and its relations to rotations in 3-dimensional space.

**2002**

**Group A** (Schlicker, Mayberry and Powers) investigated the geometry of the collection of non-empty compact subsets of n-dimensional real space. The article, "A Singular Introduction to the Hausdorff Metric Geometry", written by these researchers and D. Braun from the 2000 REU, appeared in 2005 in the *Pi Mu Epsilon Journal*.

**Group B** (Aboufadel, Driskell and Dailey) worked on problems involving wavelets and steganography. The article "Wavelet-Based Steganography," written by Lisa Driskell, appeared in the *Cryptologia* in 2004.

**Group C** (Sorensen, Mikkelson and Armel) investigated the complete bifurcation diagram in dynamical systems. This work was noted in the article, "Sprinkler Bifurcations and Stability," by Jody Sorensen and Elyn Rykken, which appeared in the November 2010 issue of the *College Mathematics Journal*.

**Group D** (Wells, Fagerstrom and DeLong) conducted work in Lie Theory and its relations to rotations in 3-dimensional space.

**2001**

There was no REU program during 2001.

**Group A**(Schlicker and Braun) investigated the geometry of the Hausdorff space.

**Group B**(Aboufadel, Cox and Oostdyk) developed bivariate Daubechies scaling functions.

**Group C**(Sorensen, Ashley and Van Spronsen) conducted work in "The Real Bifurcation Diagram." The article "Symmetry in Bifurcation Diagrams," written by these researchers, appeared in the Fall 2002

*Pi Mu Epsilon Journal*.

**Group D**(Fishback and Horton) researched Mandlebrot sets for ternary number systems. The article "Quadratic Dynamics in Matrix Rings: Tales of Ternary Number Systems," written by these researchers, appeared in

*Fractals*, Volume 13, 2005.

**Group E**(Wells, Pierce and Taylor) investigated rotations that arise from chemistry.

*Page last modified March 21, 2014*