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 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. A manuscript is being written.
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.
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 is also gave an elementary and elegant approach, using the WZ method, for sums of Choi, Zoring and Rathie.
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.
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. A manuscript has been submitted and accepted, pending revisions.
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.
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.
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.
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" has been accepted to appear 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.
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," has been accepted by 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, has been accepted by the Journal of Geometry.
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.
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.
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