The REU Program In Mathematics

Research and Publications at Previous REU Programs

Note that there are other papers that are in preparation or have been submitted for publication.

REU 2011 at GVSU
 
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.  Our solution was submitted in August 2011 to the City of Boston.
 
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.
 
REU 2010 at GVSU

Higher Dimensional Rook Polynomials: This team (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.  We are planning to write an article based on our results on the generalized central factorial and Genocchi numbers to be submitted to an undergraduate mathematics research journal.
 
Orthogonality in the space of compact sets: This team (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 (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 (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.


REU 2009 at GVSU

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.  A manuscript has been accepted for publication by 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," has been accepted by Involve.


REU 2008 at GVSU

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.

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.

CMJ
 

Convexity and Gerrymandering, from the 2008 REU


REU 2007 at GVSU

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 has been submitted.

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. Their article "Optimal Packings of up to 5 Equal Circles on Square Flat Torus", with Sandi Xhumari, has been published in Beiträge zur Algebra und Geometrie.
 

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.
 

REU 2006 at GVSU

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.


REU 2005 at GVSU

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 varialbles 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.

REU 2004 at GVSU

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.


REU 2003 at GVSU

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.


REU 2002 at GVSU

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.


There was no REU program at GVSU in 2001


REU 2000 at GVSU


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.

These publications are based upon work supported by the National Science Foundation under Grants No. DMS-1003993, DMS-0451254, DMS-0137264, and DMS-9820221.  Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).