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# Friday, April 7, 2006

### Registration      Carver Second Floor

 2:30-4:30

### Exhibits      Carver

 2:30-4:30

### Concurrent Session 1      Carver 202

 3:00-3:20 Arrow's Hypotheses Irvin Hentzel, Iowa State University We prove three consequences of Arrow's Hypotheses. (1) If some of the ballots put x first and the rest put x last, then x has to be either first or last in the group ranking. (2) If the rankings of a with b match the rankings of c with d on each ballot, then the group ranking must also match the ranking of a with b and c with d. (3) The group ranking must match one of the ballots. This material was taken from "Three Brief Proofs of Arrow's Impossibility Theorem" by John Geanakoplos. The point of the talk is to show that the proofs are very elementary. The various strategies for voting are covered in many very elementary texts. Their discussion is directed towards with of the hypotheses the voting strategies violate. This talk shows how the hypotheses can be combined to directly obtain conclusions that do not seem as fundamentally fair as the original hypotheses.
 3:30-3:50 Decomposing Voters Mariah Birgen, Wartburg College Recent developments in the mathematics of Social Choice by Don Saari, among others, have added an element of geometry and linear algebra to a field that has been dominated by combinatorics. This talk will introduce the linear algebra behind a three-candidate election, including how symmetries underlie traditional voting paradoxes.
 4:00-4:20 Classroom Capsule: Teaching Challenge-Response Arguments Reginald Laursen, Luther College The forward-backward method is a fundamental proof technique for helping students understand how to construct proofs. I will describe my latest variation in the application of this technique for addressing challenge-response arguments in a Real Analysis class. Using this variation my lower ability students have had greater success.

### Concurrent Session 2      Carver 204

 3:00-3:20 Some Hands-on Workshops for Elementary and Intermediate Algebra Courses M. Anne Dow, Maharishi University of Management I found all the topics of my Elementary and Intermediate Algebra courses in the greenhouses we recently built on campus to provide organic vegetables for our campus dining hall. In my talk I will present two workshops on linear functions, one about the amount of broccoli seed needed to produce N thousand pounds of broccoli per week, and one about heat loss to the greenhouse during winter. Both require students to think carefully about what the slope means.
 3:30-3:50 Existence of strong solution for a class of nonlinear parabolic systems ** Kunlun Liu, Iowa State University This paper deals with the local and global existence of the strong solution for a class of nonlinear parabolic PDEs in the domain [0,T]×U. We prove the local existence of strong solution. Formulate a time interval within which the local solution will not blow up. Also, this paper proves a criterion for the global existense of strong solutions.
 4:00-4:20 computing multivalued velocity and electric field of 1D Euler-Poisson equation ** Zhongming WANG, Iowa State University We develop a level set method for the computation of multi-valued velocity and electric fields of one-dimensional Euler-Poisson equations. The sys- tem of these equations arises in the semiclassical approximation of Schrodinger- Poisson equations and semiconductor modeling. This method uses an implicit Eulerian formulation in an extended space | called field space, which incorpo- rates both velocity and electric fields into the configuration space. Multi-valued velocity and electric fields are captured through common zeros of two level set functions, which solve a linear homogeneous transport equation in the field space. Numerical examples are presented to validate the proposed level set method.

### Concurrent Session 3      Carver 205

 3:00-3:20 Godel Disrobes: a naked approach to incompleteness ** Jeremy Alm, Iowa State University I propose an alternate approach to the incompleteness theorems via the conceptually simpler \emph{abstract provability systems}, due to Raymond Smullyan. These systems have incompleteness theorems that are easy to prove, and whose hypotheses point to the important features of formal arithmetic.
 3:30-3:50 Some Properties of the Smarandache Fitorial and Supplementary Fitorial Functions Charles Ashbacher, Mt. Mercy College The Smarandache Fitorial function FI(N) is defined as the product of all the positive integers less than N that are relatively prime to N and the Smarandache Supplementary Fitorial function SFI(N) as the product of all the positive integers less than or equal to N that are not relatively prime to N. It is clear that FI(N) * SFI(N) = N!. These functions are defined in the book "Generalized Partitions and New Ideas On Number On Number Theory and Smarandache Sequences" by Murthy and Ashbacher. Six open problems are listed in the book and in this presentation, progress made towards their solution will be given.
 4:00-4:20 Vertex identifying codes in graphs: definitions, theorems and open problems Ryan Martin, Iowa State University In 1998, Karpovsky, Chakrabarty and Levitin introduced a new graph invariant called the vertex identification code. If C is a subset of the vertices, then C is a vertex-identifying code if each set N[v]\cap C is distinct and nonempty, where N[v] denotes the closed neighborhood of vertex v. We will discuss a number of results on the size of the smallest code in a graph, particularly on the Erdos-Renyi random graph and we will present open problems.

### Break

 4:25-4:35

### Panel Discussion:  On getting a job for graduate students     Carver 202

 4:35-5:20

### Round Table Discussion:  Section NExT     Carver 204

 4:35-5:20

### Dinner      See Meals Information Section

 5:30-7:30

### Plenary Session 1      Carver 101

 7:30-7:50 Episodes in The Early History of The Lucasian Chair Jim Tattersall, Providence College In 1663, Henry Lucas, the long-time secretary to the Chancellor of the University of Cambridge, made a bequest, subsequently granted by Charles II, to endow a chair in mathematics. A number of conditions were attached to the Chair. Among the more prominent Lucasian professors were Newton, Babbage, Stokes, Dirac, and Hawking. We focus attention on the early Lucasians. Many of whom were very diligent in carrying out their Lucasian responsibilities but as history has shown such was not always the case. In the process, we uncover several untold stories and some interesting mathematics

### Reception      Carver Lobby

 8:30- --

# Saturday, April 8, 2006

### Registration      Carver Lobby

 8:15-10:00

### Plenary Session 2      Carver 101

 8:30-8:50 Vignettes in Number Theory Jim Tattersall, Providence College Properties and the history of several numbers that lend themselves naturally to undergraduate research projects will be discussed. Topics include Demlo numbers, polite numbers, sad numbers, decimal Columbian numbers, Smith numbers, and Niven numbers.

### Break

 9:30-9:45

### Concurrent Session 4      Carver 124

 9:45-10:05 Why does 0! = 1? The evolution of the gamma function Steve Bean, Cornell College The gamma function is typically introduced as an attempt to interpolate the factorial function, but what motivation does one have to do this? After giving a brief overview of the gamma function and its properties--from the modern point of view,--we will talk about the same function from a historical perspective. In particular, we will examine the reasons behind Euler's original formulation of this function.
 10:15-10:35 The Toeplitz-Silverman Theorem Part II Alexander Kleiner, Drake University In the first two decades of the twentieth century summability developed from collection of special results used in other parts of analysis into a full-blown field. One of the main points of this transition was a collection of general results that gave conditions for a method to sum every convergent sequence. Part I of this presentation, which was given last spring, laid out the work that led to the general theory. Papers by Toeplitz, Silverman, Kojima, Schur and others established the theory. This note will look at the development of these conditions and, as time permits, the reoccurrence of these results in the early day of the "Polish" school of functional analysis
 10:45-11:05 Prolongations on a Riemannian Manifold ** Alfredo Villanueva, University of Iowa Traditionally the method of prolongations is carry out by algebraic manipulations which become very complex, especially in cases of partial differential equations on curved spaces, here we are applying some results from representation theory and differential operators to have a systematic method that allow us to close some overdetermined systems on a Riemannian manifold.

### Concurrent Session 5      Carver 128

 9:45-10:05 Model Fitting and Selection for County-Level Depression Hospitalization Rates Using Bayesian Statistical Methods ** Scott Wood, University of Iowa Researchers in the health sciences are interested in identifying and modeling the risk factors that are associated with high rates of hospitalization for depression. Being able to identify U.S. counties with high standardized hospitalization rates (SHR) would be useful in allocating federal resources. This project analyzes and critiques three potential Bayesian statistical models that can be implemented using WinBUGS software. Ordinary least squares, Poisson regression, and Bayesian conditional autoregressive (CAR) models are considered in detail. Though each has its advantages and disadvantages, qualitative and quantitative evidence suggest that the Bayesian CAR model is the optimal choice for this data. While a Bayesian CAR model will be shown to account for spatial autocorrelation and Poisson response variables, it was not as reliable as hoped for making accurate predictions at the county level.
 10:15-10:35 Teaching Mathematical Probability and Statistics with Internet Applications and R Michael Larsen, Iowa State University Courses in mathematical statistics can use Internet applications and simulation using the R statistical package to enhance the learning experience. Internet material has been developed for introductory probability and statistics courses and for teaching mathematics at the level of calculus. In order to adapt this material to an intermediate undergraduate probability course, it is necessary to select material to use and incorporate it into lecture, homework assignments, and study problems. The R statistical package is a free software package that can be used for simulation, includes functions related to many probability distributions, and can be used to produce nice graphical displays. Using R in a calculus-based probability course requires writing problems for homework assignments, in-class use, and review that make substantial use of simulation and R's other relevant capacities. Funds were received from ISU's Computer Advisory Committee to support a special teaching assistant to help with development of Internet and R resources for instruction in ISU's Statistics 341: Introduction to the Theory of Probability and Statistics I in the spring of 2006. This talk will describe some developments and give practical demonstrations.
 10:45-11:05 Protein Structure Determination: A Rigid Geometric Build-up Algorithm for Solving a Distance Geometry Problem with Sparse Exact Distance Data ** Di Wu, Iowa State University Protein Structure Determination: A Rigid Geometric Build-up Algorithm for Solving a Distance Geometry Problem with Sparse Exact Distance Data Di Wu and Zhijun Wu Program on Bioinformatics and Computational Biology Department of Mathematics Iowa State University Ames, Iowa 50011 Abstract. Given a set of distances for certain pairs of atoms in a protein, the coordinates of the atoms and hence the protein structure can then be determined through solving a so-called distance geometry problem. However, it has been proved to be a NP hard problem when only a set of partial distances given. Previously, we used a so-called geometric build-up approach to develop several algorithms for solving the distance geometry problem with a set of sparse distance data. In this method, the coordinates of the atoms in a protein are determined as one atom at a time, with the distances from four base atoms to the atom to be determined. However, the requirement for four base atoms for the unique determination of each atom is sufficient, but unnecessary and even redundant for rigid structural determination. Here we investigate a rigid geometric build-up algorithm, which requires three base atoms instead of four base atoms for the determination of each atom. It could generate rigid structures, even a unique structure for very sparse distance data of a protein eventually. Due to the reflection in the determination for some atoms, this algorithm may also produce multiple structures satisfying given distances. We present the results obtained by using the algorithm for the determination of the structures, which suggests the potential of applying the algorithm to the distance based protein structural modeling.

### Concurrent Session 6      Carver 132

 9:45-10:05 "Convergence depth": proof of the nonrotation and nontranslation of galaxies ** Joseph Keller, Iowa State University HC Arp (Max Planck Inst.) amassed evidence that most large redshift is intrinsic, not due to motion or expansion. WG Tifft (Univ. of Arizona) says that redshift periods, large and small, suggest abandoning the motion/expansion hypothesis altogether. "Convergence depth", a phenomenon studied by this author since 2002, means that the average velocity over successive shells of galaxies, converges in a mere 400 M lt yr, to the apparent velocity ("anisotropy") of the sources of the cosmic microwave background ("CMB"). The shape of the convergence depth curve, and the observed 400 M lt yr period of galaxy distribution, suggest that Hubble's parameter varies sinusoidally along the axis of the CMB anisotropy, with half-period 400 M lt yr. Taylor series extrapolation of the convergence depth curve to the origin, then shows that the velocity of the sun relative to distant galaxies is about equal to its velocity relative to nearby stars. Galaxies neither rotate nor translate. "Dark matter" need not exist. Oort's law is not due to motion. An absolute frame of reference (Maxwell/FitzGerald ether?) is supported. DC Miller (Case Univ.) found that apparent "ether drift" agrees, in its component parallel to Earth's axis, with the solar apex motion, i.e., motion in the extragalactic frame.
 10:15-10:35 The Remarkable Equation tan(x) = x Dave L. Renfro, ACT Inc. Although tan(x) = x is virtually the prototypical example for solving an equation by graphical methods, and this equation frequently appears in calculus texts as an example of Newton's method, there seems to be nothing in the literature that surveys what is known about its solutions. In this talk I will look at some appearances of this equation in elementary calculus, some appearances of this equation in more advanced areas (quantum mechanics, heat conduction, etc.), the fact that this equation has no nonreal solutions and that all of its nonzero solutions are transcendental, and some curious infinite sums involving its solutions. In addition, I will discuss some of the history behind this equation, including contributions by Euler (1748), Fourier (1807), Cauchy (1827), and Rayleigh (1874, 1877).
 10:45-11:05 Hey, Kids! Improve Your Theorems! Add Superfluous Hypotheses! ** Jacob Manske, Iowa State University In spite of the fact that we tell students not to assume what they are trying to prove, we all must do precisely that. The interesting theorems, then, turn out to be the ones whose tautologous nature is elusive. This will be a philosophical discussion; bellicose debate is encouraged.

### Break

 11:15-11:25

### Iowa Section Business Meeting      Carver 101

 11:25-12:10

### Lunch together or on your own

 12:10-1:40

### Concurrent Session 7      Carver 124

 1:40-2:00 Subgroups of the Rubik's Group Brian Birgen, Wartburg College The set of possible arrangements of the Rubik's Cube forms a group with 4*10^19 elements. We will locate some well known groups which occur as subgroups of the Rubik's group and begin to understand the source of some of the complexities in understanding the Rubik's group.
 2:10-2:30 Mathematics by Experiment Marc Chamberland, Grinnell College The use of computer packages has brought us to a point where the computer can be used for many tasks: discover new mathematical patterns and relationships, create impressive graphics to expose mathematical structure, falsify conjectures, confirm analytically derived results, and perhaps most impressively for the purist, suggest approaches for formal proofs. This is the thrust of experimental mathematics. This talk will give some examples to discover or prove results concerning goemetry, integrals, binomial sums, and infinite series.

### Concurrent Session 8      Carver 128

 1:40-2:00 Global Dynamics and Chaos Wolfgang Kliemann, Iowa State University Global Dynamics and Chaos Wolfgang Kliemann Department of Mathematics, Iowa State University, Ames, Iowa 50011, U.S.A. February 27, 2006 Abstract We discuss dynamical systems given by  a time set - in our case the real line R,  a state space M - a compact subset of Rd or a compact metric space,  a continuous map  : R M ��! M with two properties (0; x) = x for all x 2 M (t + s; x) = (t; (s; x)) for all x 2 M, all t; s 2 R. Typical examples are solutions of (time-homogeneous) di¤erential equations or iterations of maps. In dimension 1 the system behavior is described by …xed points and trajectories connecting the …xed points. In dimension 2 the system may exhibit …xed points, periodic orbits, and connecting orbits. A linear system, i.e. a system arising from a linear di¤erential equation x_ = Ax, always has a …xed point at the origin - it may display additional …xed points (in the kernel of A) or periodic orbits (in the eigenspaces corresponding to eigenvalues with real part 0). For general dynamical systems we obtain a picture of its global behavior using the following approach: We de…ne the system’s !��limit sets (and !��limit sets) as the points that are approximated by trajectories as t ! 1 (and t ! ��1, respectively). An attractor A  M is then a set that has a neighborhood whose !��limit set is A, and similarly for repellers R  M and !��limit sets. Considering …nite sequences of attractors ? = A0  A1  :::  An = M and their associated repellers M = R0  R1  :::  Rn = ? we obtain as intersections Mn��i := Ai+1 \ Ri a so-called Morse decomposition fM1; :::;Mng. For all x 2 M there are indexes i; i such that !(x)  Mi and !(x)  Mi , de…ning an order  on a Morse decomposition. If a system has …nitely many attractors, it admits a (unique) …nest Morse decomposition, which describes exactly the global behavior of the system. On the Morse sets Mi of a …nest Morse decomposition the system is chain transitive, i.e. any two points inMi are connected by chains of trajectories with arbitrarily small jumps. This behavior can be fairly complicated, so we de…ne "chaos" as a more ’regular’behavior on Morse sets: A Morse set M is chaotic, if (i) it does not consist of a single …xed point, (ii) it does not consist of a single periodic trajectory, (iii) there is a point x 2 M with !(x) = M, and (iv) M has a dense set of periodic points. Systems that satisfy (iii) are necessarily chain transitive. The properties (i) - (iv) imply that the system on M exhibits ’sensitive dependence on initial conditions’, which is often taken as one of the de…ning properties of chaos. We will discuss the following questions in some detail:  Is the weather chaotic?  Is the human genome chaotic?  Is a random system chaotic?  Is a controllable system chaotic?
 2:10-2:30 The Polynomial Root Squeezing Theorem * Justin From, Central College Polynomials are one of the most widely used functions in mathematics, yet there are surprisingly many unanswered questions about their properties. This talk will present an innovative new idea referred to as the Polynomial Root Squeezing Theorem which shows that squeezing two of a polynomial's roots towards one another causes the polynomial's critical points to also squeeze towards one another. This talk will also include an explanation of how this new theorem can be utilized to prove the Span Minimization Conjecture. (Note: This research was completed at the Grand Valley State University REU program.)

### Concurrent Session 9      Carver 132

 1:40-2:00 The stable endomorphism group of non-simple string modules over a very particular finite dimensional algebra. ** Giovanna Llosent, University of Iowa Let A be a finite dimensional algebra over an algebraic closed field k of characteristic 2 with a quiver representation and relations. Consider all non-simple string modules for this algebra which do not lie in the Auslander-Reiten component of the simple modules. Is there a non-simple string module M for which the group of stable endomorphisms is isomorphic to k? Under the hypothesis above we were able to prove that the underlying string S of the string module M has a substring S' and there is an endomorphism that does not factor through a projective A-module and lies in S'. The maximun lenght of the underlying string of a string module needed for completing the study of all stable endomorphism groups of non-simple string modules was 17. In particular, the cases needed for complete generalization are 54.
 2:10-2:30 Primitive prime divisors of Mersenne numbers via Uniform Distribution Christian Roettger, Iowa State University Given a sequence a of integers, a primitive divisor of a(n) is an integer which divides a(n) but no earlier term of the sequence. Last year, we presented a result about a weighted average of primitive prime divisors of the well-known Mersenne numbers M(n) = 2^n-1. This year, we have an entirely different, simple proof of the same result, using cyclotomic polynomials and uniform distribution. We are indebted to Carl Pomerance for helpful insights. We will also mention possible applications to other sequences like the Fibonacci numbers.

### Break

 2:35-2:45

### Concurrent Session 10      Carver 124

 2:45-3:05 Comparison of P-matrix completions with Q-matrix completions. Luz De Alba, Drake University A P-matrix is a real square matrix, in which the determinant of every principal submatrix is positive. A Q-matrix is one in which the sum of the determinants of principal submatrices of the same size is positive. Clearly every P-matrix is a Q-matrix. A partial P-matrix is a matrix in which some entries are specified while others are not known, and every fully specified principal submatrix has positive determinant. The P-matrix completion problem asks the question: "Which partial P-matrices can be completed to a P-matrix?" In this talk we give the definition of partial Q-matrix, and compare the Q-matrix completion problem to the P-matrix completion problem. We also discuss some partial answers to the Q-completion problem.
 3:15-3:35 Visualizing Linear Algebra with Geometer's Sketchpad Catherine Gorini, Maharishi University of Management I will present Sketchpad labs for visualizing the following concepts in linear algebra: Linear transformations and image, range, kernel, and projection. The determinant of a matrix and the orientation-preserving or-reversing property of the corresponding linear transformation. The determinant a matrix to the area of the image of a unit area under the corresponding linear transformation. Eigenvectors and eigenvalues
 3:45-4:05 Using NASCAR to Illustrate Optimization Mark Mills, Central College Over the past several years, the authors have taken part in a Fantasy NASCAR league. For each race, we were required to “purchase” five drivers using a “pot” of imaginary money. In this talk, we will discuss several ways that we used optimization techniques to choose the five drivers, as well as talk about our (disappointing) results. While neither of us has used this in a class, we feel that it could be an interesting project for a finite mathematics class, and we will discuss how it could be implemented as such. We also talk about other ways that NASCAR could be brought into the classroom to illustrate various mathematical topics.

### Concurrent Session 11      Carver 128

 2:45-3:05 Reconstructing genomes in the presence of hybridizations Stephen Willson, Iowa State University A homoplasy at a site in the DNA occurs when the value of a character (A, C, G, or T) changes more than once in the evolutionary history. Homoplasies create extra difficulties for reconstructing the evolutionary history of a collection of taxa. Recent interest has grown concerning evolutionary histories that are not described by trees but rather by more general networks that allow for hybridization events. A natural question is, in an idealized situation where homoplasies occur only at hybridization events, whether the characters at the leaves and the root of the network determine the characters at the internal vertices. Mathematically, one has a directed rooted acyclic graph in which the vertices correspond to taxa. At each vertex there is a set of genes. Under appropriate assumptions, the genes at all vertices are determined by the genes at the root and at the leaves.
 3:15-3:35 Sperm Competition Games Rick Spellerberg, Simpson College Sperm Competition occurs when ejaculates of multiple males compete to fertilize the eggs of one female. In this talk we will discuss the work of G.A. Parker in his paper; Sperm Competition: sneaks and extra-pair copulations. In this paper, Parker examines ejaculation strategies for cases when an opportunist male "steals" a mating with the female of a paired male.
 3:45-4:05 Chemotactic Collapse in Keller-Segel Equation ** WEN ZHOU, Iowa State University Chemotaxis phenomenon is one of the most fundamental phenomenons in the biology field. In 1970s, Keller and Segel characterize this phenomenon with two coupled equations. Study on the blow up of the solutions of the this equation is one of the key part of the research on this equation. This short talk will briefly introduce some recent results of the study on this equation, including Nagai, Velazquez, Stevens, Levine, and Hortsman's work, etc.

### Concurrent Session 12      Carver 132

 2:45-3:05 The Dynamics of a Planar Two Link Chain and Some Applications to Human Motion Kenneth Driessel, #non-IA section Try the following 'acceleration experiment': Stand balanced with your legs straight and a slight forward bend at the waist. Then step backwards. Consider the following 'acceleration question': How do humans initiate this motion? Or more generally: How do humans usually initiate horizontal motion from a balanced position? (I first met this question when thinking about cross country skiing.) We analyze the acceleration question by analogy. In particular, we study the classical dynamics of a mechanical system consisting of two linked rods. We assume that the first rod is connected to the ground by a hinge. (The first rod corresponds to the human legs. The ground hinge corresponds to the human ankles.) We assume that the second rod is connected to the first one by another hinge. (The second rod corresponds to the human torso. The second hinge corresponds to the human hips.) We derive the equations of motion for this mechanical system. We prove that if the system is initially at rest in a balanced position then gravity causes the center of mass to accelerate in the horizontal direction toward which the system is 'pointed'. We infer that the step backwards in the acceleration experiment is initiated by a relaxation of the muscles at the hips. Reference: Kenneth R. Driessel and Irvin R. Hentzel, 'Dynamics of a Planar Two Link Chain', http://www.fiberpipe.net/~driessel/2-links.pdf

* denotes an undergraduate speaker and ** indicates a graduate student speaker