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Iowa Section of the Mathematical Association of America |
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| 2:30-4:30 |
| 3:00-3:20 | Message Authentication Codes and Quasigroups |
| ** Kristen Meyer, Iowa State University | |
| Message Authentication Codes, or MACs, are commonly used cryptographic tools to ensure that a message has not been changed in transit. MACs can be constructed from a variety of mathematical structures and in a variety of ways. In this talk, I will describe a new MAC (called QMAC) which is based on the non-associativity of quasigroups. In order to obtain security against forgeries, quasigroups of large order must be used. I will also discuss how to create and represent such quasigroups. |
| 3:30-3:50 | A Multiple Integral of a Piecewise Algebraic Function. |
| Michael Rieck, Drake University | |
| Fix r>0. Let (x_0, y_0) and (x_n, y_n) be fixed and a distance r apart. Consider the set of all points ( x_1, y_1, x_2, y_2,..., x_{n-1}, y_{n-1} ) in Euclidean (2n-2)-space for which the distance in the plane between (x_{j-1}, y_{j-1}) and (x_j, y_j) never exceeds one (j=1,...,n). The hyper-volume of this set of points in (2n-2)-space can clearly be expressed as a multiple integral, integrating over 2n-2 dimensions, a function that is 1 on the set, but 0 off of the set. Surprisingly, it can also be expressed as a multiple integral over n-1 dimensions, of a piece-wise algebraic function. |
| 4:00-4:20 | Don't Be So Sensitive! --On the Definition(s) of a Group |
| ** Jeremy Alm, Iowa State University | |
| We have all seen different variations on the definition of a group, and we all know that each one admits "the same structures". There are, however, some subtle but important differences among them. The class of groups and the properties that it has are sensitive to the signature (or similarity type) in which the groups are defined. In particular, in some signatures equational definitions are possible and in others they are not. |
| 3:00-3:20 | The "Basic Four" Elementary Functions and Their Applications in College Algebra |
| Scott Herriott, Maharishi University of Management | |
| The "Basic Four" elementary functions are those that result from the association of an additive or multiplicative change in X with an additive or multiplicative change in Y (linear, exponential, logarithmic and power). We consider the importance of these functions in the college algebra course in terms of the breadth of their applications in the fields of study that college algebra students will major in. |
| 3:30-3:50 | A Project Based Finite Math Course |
| Brian Birgen, Wartburg College | |
| In order to breathe new life into a course populated by unenthusiastic non-majors, I have introduced a series of projects which both challenges students and answers the age-old question "When am I ever going to use this stuff?". Successes and failures will both be featured. |
| 4:00-4:20 | A Unified Representation of Function |
| Bernadette Baker, Drake University | |
| The researchers have built a theoretical model of student development of function using the APOS paradigm. Students have difficulty with this concept because of the inability to recognize the common feature of the traditional function representations (analytic, graphical and tabular). By providing techniques for standard operations that focus attention on the defining feature of function (the association of input with output), the researchers hope to rectify this problem in student learning. This representation will be explained; one researcher has piloted this approach successfully and initial results will be reported. |
| 3:00-3:20 | Generalized Arithmetic Triangles via Convolution |
| Sean Bradley, Clarke College | |
| Pascal's Triangle is a convolution triangle; each polynomial that forms a diagonal can be generated by repeatedly convolving the polynomial f(x)=1 with itself. We consider Generalized Pascal Triangles, convolution triangles whose generating polynomials are f(x)=m, where m is a positive integer. We will explore connections to Fibonacci-like sequences and nearly-golden ratios, and consider self-similarity along the lines of C.T. Long's investigation of Pascal's Triangle in his 1981 article in The Fibonacci Quarterly. |
| 3:30-3:50 | The Efficiency of Morse Code as Data Compression. |
| Scott Searcy, Waldorf College | |
| Morse code was invented to allow the efficient transmission of textual data in a digital mode. This talk examines the efficiency in comparison with more modern methods of textual data transmission. |
| 4:00-4:20 | Factoring Trinomials with Less Struggling and More Success! |
| Monica Meissen, Clarke College | |
| This talk will publicize a surprisingly underutilized technique of factoring trinomials which is based on "Factoring by Grouping," a method typically used to factor polynomials with four terms. The speaker discovered this "new" technique seven years ago in a section of a traditional algebra text designated as "Optional." Teaching factoring of trinomials using this method for the past several years has been much more enjoyable than using traditional techniques; students don't struggle as much and are more successful! An additional benefit of this method will also be described; it is useful in more easily verifying that a trinomial with no common factors is prime. |
| 4:25-4:35 |
| 4:35-5:20 | Calvin Van Niewaal, Coe College MAA Strategic Planning - Professional Development |
| 4:35-5:20 | Mariah Birgen, Wartburg College Undergraduate Programs |
| 5:30-7:30 |
| 7:30-7:50 | Planar Linkages: Robot Arms, Carpenters' Rulers, and Other Devices |
| Nancy Hagelgans, Ursinus College | |
| A planar linkage is constructed in the plane from rigid links or rods that are connected with movable joints. Robot arms and carpenters' rulers are examples of planar linkages in which the links are connected to form a chain. We will examine the reachability region of robot arms, which are chains with one end fixed. Then we will go on to solve the minimal folding problem of carpenters' rulers with links of different lengths. Finally we will address some planar linkages that can be used to convert one type of motion to another type of motion. |
| 8:30--- |
| 8:45-9:45 | Undergraduate Programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide 2004 |
| David Bressoud, Macalester College | |
| The MAA's Committee on the Undergraduate Program in Mathematics (CUPM) is charged with making recommendations to guide mathematics departments in designing curricula for their undergraduate students. The CUPM Curriculum Guide 2004, published last Fall provides an up-to-date perspective on the mathematics curriculum for many different student audiences, including of course our own majors. This session will be a presentation followed by a question and answer session with committee member David Bressoud from Macalester College. Free copies of the Guide and Curriculum Foundations Project will be available for those who come to the session. |
| 9:45-10:00 |
| 10:00-10:20 | The Henrici Harmonic Analyser |
| Joel Haack, University of Northern Iowa | |
| The mathematical basis for analysis of a periodic function was provided by J. B. J. Fourier in Paris in 1822 in the form of a series expansion. Calculations, however, were very tedious. Several ingenious mechanical devices to perform the analysis were devised in the late 19th Century and perfected in the early 20th Century. An important example is the Henrici analyzer, a working version of which is housed in the Department of Speech and Audiology at The University of Iowa. The mathematical background of the device will be described, and videos will be shown of the authors using the Iowa Henrici to analyze a waveform. Pictures and descriptions of another device at the Science Museum in South Kensington, London will also be presented. |
| 10:30-10:50 | Axioms of Kinship |
| * Cindee Calton, University of Northern Iowa | |
| Throughout the world, there are many different ways of defining our relationships with our family members. Who we choose to group together with the same kinship term reveals much about how we view those relatives. Throughout the world, there are only 6 basic ways of grouping relatives, despite the many possible ways of doing so. This talk explores how to think about human kinship axiomatically and discover why only certain patterns appear, using both mathematical and anthropological thinking. Interesting case studies of elaborate patterns of marriage are also explored briefly at the end of the talk. |
| 11:00-11:20 | Rigor in Analysis: From Newton to Cauchy |
| * James Collingwood, Drake University | |
| This paper seeks to examine the level of rigor which was present in work with the calculus from the time of its invention through Cauchy's work in the 1820s. The paper traces the changing standards of rigor which marked the work with the calculus during this period. Newton's and Leibniz's approaches to rigor are compared and it is shown how each affected the level of rigor in mathematics present in the British Isles and on the Continent, respectively. The general level of rigor used by eighteenth century mathematicians will be compared to the changing standard of rigor used in the nineteenth century by mathematicians such as Cauchy. Also, forces which may have impeded early attempts to make the calculus rigorous and forces which may have provided the impetus for later work with the calculus to become more rigorous will be examined. |
| 10:00-10:20 | The Mathematics of Common Consent |
| Ronald Smith, Graceland University | |
| Many religious traditions, including my own, value "common consent" as a means of group decisionmaking, although there is no agreed on definition of this principle. Arrow's impossibility theorem is a tool to help understand why attempts to equate common consent with any particular voting scheme fail. We use it to analyze what alternatives are possible, and we show how understanding a restriction on voting called the Black-Arrow condition may help groups who wish to come to "common consent." |
| 10:30-10:50 | Searching For Images Embedded in Mathematics |
| Charles Ashbacher | |
| In the science fiction book "Contact" by Carl Sagan, investigators discovered a message inside the digits of PI. We can take the digits of PI or any other sequence of numbers and split them into groups of threes. For example, the first group of three for PI would be (3,1,4). By mapping 0 to 0 and 9 to 255 and interpolating in between, we will have a sequence of potential triplets starting at (0,0,0) and going through (255, 255, 255), progressing through discrete steps. These triplets can be converted into pixel colors using the (R,G,B) transformation done in computer programming and the pixels generated by the digits used to create images. In this presentation, a Java program that will perform this operation for a set of initial digits and some preliminary images will be displayed and discussed. |
| 11:00-11:20 | The Evolution of Cooperation |
| Rick Spellerberg, Simpson College | |
| This talk will present some of the basic concepts of Evolutionary Game Theory as we discuss models related to the evolution of cooperation. This talk should be of special interest to students or faculty interested in undergraduate research in mathematics. Included will be a preview of a few of the student presentations related to the topic that will be presented at the second annual Midwest Undergraduate Mathematics Symposium held at Simpson College April 9th. |
| 10:00-10:20 | The Graph Traces of Finite Graphs and Applications to Tracial States of C*-Algebras |
| ** Matthew Johnson, University of Iowa | |
| We determine the extreme points of the set of graph traces of norm one for any finite graph E satisfying Condition (K). We also describe and application to the space of tracial states on the graph C*-algebra. |
| 10:30-10:50 | On the Negative Mass Assigned By the Univariate Zao-Tsiatis and Wang Estimators |
| Mahmoud Almanassra, Wartburg College | |
| The Zhao-Tsiatis estimator, for the restricted quality adjusted lifetime (RQAL), is not a monotonic estimator and hence it is not a proper survival function. The Wang estimator, which is a modified version of the ZT-estimator, is also not a monotonic estimator. Both the ZT-estimator and the W-estimator are consistent and reasonably efficient estimators. The simple weighted estimator is monotonic and consistent, but it is less efficient than the other two estimators mentioned above. I will identify the jump points of the simple weighted estimator, the ZT-estimator and the W-estimator. I will also identify which of these points are assigned a negative mass by the estimator. Moreover, I will propose two new consistent estimators for the survival functions of the RQAL. |
| 11:00-11:20 | Explanation of Lepton and Meson Masses |
| ** Joseph Keller, Iowa State University | |
| Let the muon be a gaussian distribution of electric charge, as small as the Heisenberg uncertainty principle allows. Using G.D. Birkhoff's theorem, apply the Schwarzschild metric as if electric force were the same as gravitational force. Hawking's theorem says that the entropy of a black hole is proportional to its area. Choosing the mass to maximize entropy per unit mass, gives the muon mass within about 1%. There will be an inner infinite redshift surface also. This surface encloses a "core". Choosing the mass just large enough to trap the "core", gives the mass of the tauon to better than 1%. Two quarks, one inside the other, give a model of mesons. Similar considerations give the charged pion, K, B and D meson masses to within about 1% or better. |
| 11:25-11:30 |
| 11:30-12:15 | Joel Haack, University of Northern Iowa, Past Chair of the Iowa Section of the MAA |
| 12:15-1:45 | (Those who are interested in Section NeXT should gather at a table in the Central Market.) |
| 1:45-2:05 | The Toeplitz-Silverman Theorem |
| 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. Papers by Toeplitz, Silverman, Kojima, Schur and others established these theorems. 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. |
| 2:15-2:35 | Prime divisors of Mersenne numbers and Dirichlet series |
| Christian Roettger, Iowa State University | |
| Mersenne numbers are the numbers 1, 3, 7, 15, ... 2^n - 1, ... It is a long-standing conjecture that this sequence contains infinitely many primes. We show how to get some asymptotic results on the 'average' prime divisor of Mersenne numbers using Dirichlet series. These series are useful for asymptotic counting, because there is a close link between their domain of convergence and the growth of their coefficients. Do not expect a big breakthrough, but a pretty result, few technicalities, and some exciting open questions. |
| 1:45-2:05 | Knots of Constant Curvature |
| ** Jenelle McAtee, University of Iowa | |
| In this paper, we use the method of Richard Koch and Christoph Engelhardt to construct many knots of constant curvature. |
| 2:15-2:35 | Equidissections of Trapezoids |
| Charles Jepsen, Grinnell College | |
| Denote by T(a) the trapezoid with vertices (0,0), (1,0), (0,1), (a,1). We are interested in dissections of T(a) into triangles of equal areas (i.e., equidissections of T(a)). What numbers of triangles are possible? We answer this question for certain infinite collections of trapezoids where a has the form a = p + q*sqrt(d). These results lead to a conjecture as to what might be true for all such values of a . |
| 1:45-2:05 | Mathematics and Civic Engagement |
| Mariah Birgen, Wartburg College | |
| Science Education for New Civic Engagements and Responsibilities (SENCER) is a comprehensive national dissemination project funded by the National Science Foundation. This presentation will be an introduction to SENCER followed by some discussion issues specific to mathematics. The goal of SENCER is to engage student interest in the sciences and mathematics by supporting the development of undergraduate courses and academic programs that teach "to" basic science and mathematics "through" complex, capacious, and unsolved public issues. For the past five years SENCER has promoted many science courses, but has found a dearth of undergraduate mathematics courses that work on the same model. The most promising (non-statistics) public issue that we have found is that of Democracy. I am in the process of developing a mathematics course for non-majors focused on issues of democracy including: voting, apportionment, polling, electronic security, and others. |
| 2:15-2:35 | Simple Teaching of Differential Calculus |
| Phil Wood | |
| Calculus may be taught more understandably by first describing its practical uses and then presenting it as simple algebra and geometry. In doing this all mention of infinitesimals, increments, theory of limits and formal proofs has been eliminated. |
| 2:40-2:45 |
| 2:45-3:05 | The Strange Case of Shapiro's Inequality |
| A.M. Fink, Iowa State University | |
| An old Monthly problem aroused the interest of 2 people with F.R. S. behind their name, spawned a Princeton thesis, but remains partly unsolved today. It is an interesting story about the culture of the mathematical community. |
| 3:15-3:35 | The History of Complex Dynamics, Part II |
| Dan Alexander, Drake University | |
| Part I was given to the Iowa section in 1994 and focused on the 1918 papers of Pierre Fatou and Gaston Julia on which the contemporary study of complex dynamics is based. In part II I will talk about both prior and subsequent developments in an attempt to put these 1918 papers in a more complete historical context. In particular, I will discuss some "new" influences on their works that I have recently been made aware of as well as discuss contemporaneous (that is, immediately following World War I) studies of complex dynamics around the world. I will also gladly review the works of Fatou and Julia for those who were not present for (or can't quite recall) Part I. This talk is based on collaborative research by Felice Iavernaro, Alessandro Rosa, and me. |
| 3:45-4:05 | Prince Rupert's Rectangles |
| * Erika Hartung, Central College | |
| How would you like to win a bet? Could your skills in mathematics help you? Over 300 years ago this was the case for Prince Rupert. He won a wager that given two equal cubes, a hole can be cut in one that is large enough to pass the second through it. Since Prince Rupert's time, the idea of fitting a cube into another cube has been examined extensively. Using geometry and the rectangular cross-section of a cube or box, we will discover the largest such box that can be passed through the unit cube. This process examines multiple symmetries such as rotations and reflections along with algebraic manipulations to find the dimensions of the box. |
| 2:45-3:05 | Escher's World and Green Jello World - A Concrete Introduction to Hyperbolic Geometry |
| Ruth Berger, Luther College | |
| Understanding theorems in non-Euclidean Geometry can be challenging to people who live in a Euclidean World. Since we do live on a sphere, Elliptic geometry makes some sense, but Hyperbolic geometry completely defies all our intuition. I will present two concrete examples of Poincare's models, which in class I refer to as "Escher's World" and the "Green Jello World". Thinking about what the inhabitants of these worlds might consider to be a straight line and other geometric concepts lets students accept the fact that Hyperbolic geometry is in fact just as natural as Euclidean Geometry. |
| 3:15-3:35 | Equidistant Sets and Similarity Transformations |
| Eugene Herman, Grinnell College | |
| The main result to be presented is the following: If f is a nonconstant function from R^n to R^n that preserves equality of distances, then f is a similarity transformation. A key concept in the proof is a special type of affinely independent set of points -- a set of points that are equidistant from one another. The proof uses elementary linear algebra and geometric reasoning and little else. Much of the emphasis in the presentation will be on the interplay of algebra and geometry. Also, there will be some remarks on the connections with classical geometry, including the Fundamental Theorem of Affine Geometry. |
| 2:45-3:05 | Getting Students to Read a Linear Algebra Text--Methods and Reactions |
| Karen Shuman, Grinnell College | |
| Linear algebra may be the first undergraduate course in which is it crucial for students to understand definitions, theorems, and special examples. Exposing students to new material for the first time in class can take up a lot of time and prevent other, deeper material from being covered. This talk will focus on how I have gotten students to write and think about new material that they read on their own, how I have responded to them, and how students have reacted to the experience. |
| 3:15-3:35 | Retinal disparity via computer |
| K Stroyan, University of Iowa | |
| The horizontal separation of our eyes causes the image each eye receives to fall on a slightly different portion of the retina. This difference is called "retinal disparity" and has been studied extensively for its relation to depth perception. (This kind of depth perception is called stereopsis. Helmholtz' book in 1910 is an old "standard" reference to this) Recently a psychologist friend mentioned that he was studying how retinal disparity changes as a driver views two objects off to the side of the road. He also mentioned that most of his colleagues are "math-o-phobic" and used rather coarse approximations to retinal disparity. I wrote a Mathematica animation to show the motion of the eyes of a driver and compute the time derivative of retinal disparity. We corresponded sending graphs via email until I had a start at what interests the scientists. The math is simple vector geometry with some arc tangents, but it is a little messy, so I didn't immediately look at the formulas. When I did, I had a surprise. And I believe the surprise means we could train better users of mathematics if we worked towards better integration of modern computing in basic math. We hope to build a web-Mathematica site for psychologists to use for their computations. |
| 3:45-4:05 | A tour of the new website for the IA section of the MAA |
| Al Hibbard, Central College | |
| This will be an overview of some of the new features that are available for members of the IA section of the MAA including looking at accessing the section database. Part of the talk will be particularly relevant for officers and liaisons. |
* denotes an undergraduate speaker and ** indicates a graduate student speaker
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