Abstracts MAA-SES March 28

Abstracts MAA‐SES March 28 ‐ 29, 2008 The Citadel

The Search for a Perfect Cuboid, Brandon Affenzeller, Auburn University at Montgomery Abstract: A perfect cuboid is a rectangular parallelepiped where the edges, face diagonals, and space diagonal are integral. A lower bound is known for the smallest edge. Since the space diagonal is an upper bound for the other values of interest, I will give an upper bound and have a computer search for a perfect cuboid with the values of interest within the bounds.

Using Technology to Enhance Communication: Tablets, Blackboard/WebCT and Electronic Homework Submission, Shemsi Alhaddad, University of South Carolina Lancaster Abstract: I will briefly demonstrate the following:

• Using the Tablet PC during class as a demonstration tool. • Using the Tablet PC in conjunction with Blackboard/WebCT to give partial notes. • Using electronic submission of homework rather than traditional homework.

I will then discuss some of the pros and cons of using these tools in freshman‐level classes.

Landau’s Problems, Elijah Allen Abstract: At the 1912 International Congress of Mathematicians Edmund Landau mentioned four problems that were“unattackable at the present state of science”. Although there were stated slightly differently back then, these four problems are:

1. The Goldbach conjecture : Can every even integer greater than 2 be written as the sum of two primes? 2. The twin prime conjecture : Are there infinitely many primes p such that p+2 is prime? 3. Legendre’s conjecture: Is there always a prime between n2 and (n+1)2? 4. n2 + 1 conjecture : Does n2 + 1 contain infinatly many primes.

I have created an algorithm that uses Fermat’s little theorem, the Chinese remainder theorem, and Dirichlet’s theorem on arithmetic progressions, to find instances of patterns of primes ,if they exist, according to the parameters used to define it. Using my algorithm it is finally possible to solve the twin prime conjecture and all problems like it (Hardy‐Littlewood, Dickson, Green‐Tao, etc) as well as the n2 +1 conjecture (though at this point other polynomials will require a little more work). In other words, I am announcing my proof of the twin prime and n2 +1 conjectures (though it does still need to be juried to be official).

The Lost Notebook of Ramanujan, George Andrews, The Pennsylvania State University Abstract: In 1976 quite by accident, I stumbled across a collection of about 100 sheets of mathematics in Ramanujan's handwriting; they were stored in a box in the Trinity College Library in Cambridge. I titled this collection "Ramanujan's Lost Notebook" to distinguish it from the famous notebooks that he had prepared earlier in his life. On and off for the past 32 years, I have studied these wild and confusing pages. Some of the weirder results have yielded entirely new lines of research. I will try to provide a gentle account of where these efforts have led. The result that most frightened me (I tried to

ignore it for 26 years) will conclude the presentation.

One Ended 2‐dimensional Cohen‐Macaulay Complexes, Risto Atanasov, Western Carolina University Abstract: A 2‐dimensional simplicial complex is Cohen‐Macaulay if the link of each vertex is connected and the link of each edge is non‐empty. Based on the ideas of Zeeman, we will discuss a combinatorial condition for one ended Cohen‐Macaulay simplicial complexes.

Dumbed‐Down or Real Math?, Paul Baker, Catawba College Abstract: To make mathematics accessible for the general education, liberal arts student, must the math be “dumbed‐down”? For the general student, must you only talk about what math can do rather than actually doing some “real” math? We will consider an alternative approach that was developed and has been successfully used at Catawba College for over a decade.

Teaching a First Year Seminar on Fractals: Frustrations, Fumbles, & Finally Fruition, Julie Barnes, Western Carolina University Abstract: At the 2004 Southeast Section MAA meeting, Dr. Sue Goodman presented information in her invited address about a liberal studies course she had developed on fractals for an honors math class at Chapel Hill. Because I also have an interest in fractals, I talked with her about the course and the possibilities of doing something like that at my school. Since then, I have used her materials, adapted the framework to our liberal studies program, and added several activities. In this talk, I will share how I implemented a fractal course into our curriculum, how I dealt with teaching writing, literature, poetry, music, art, theater, science, speech, and computer science all in a fractal setting, and the benefits of teaching such a course for primarily non‐majors. Although the framework of the course comes from Sue Goodman’s presentation, this talk will emphasize what I’ve added to the course and how I made it work for a non‐honors, liberal studies course that does not fulfill a math requirement.

Minimal laminations containing a rotational polygon, Brandon Barry, Clayton Kelleher, University of Alabama at Birmingham Abstract: The focus of our research has been understanding properties of rotational sets under d‐ tupling on the unit circle S1. This research is a step in gaining a better understanding of the behavior of complex polynomials on their Julia sets. On S1 we label points by their central angle measured counterclock‐wise from the positive x‐axis. The angles are measured in revolutions instead of radians. Thus, a point on S1 is denoted by a real number 1 in [0, 1). The map we consider is σd, or d‐tupling, for d≥2 on S , defined as σd (t)= dt (mod 1). We investigate questions concerning how maximal finite rotational sets act under σd with respect to critical leaves and laminations. A set is rotational under σd if the set is carried onto itself and the points remain in the same consecutive order. A finite rotational set is maximal if we cannot place another rotational orbit in the set without breaking the rotation. This rotational set R will define a rotational polygon P in the closed unit disk, D2, after connecting these points with chords. A lamination is a set L of closed chords, called leaves, in D2 such that: (1) , (2) is closed in We suppose our lamination is invariant under σd. A critical leaf is a leaf so that σd (p) = σd (q). Our previous research has shown that for each d the maximum number of orbits that can be in a finite rotational set for σd is d‐1; however, not all maximal finite rotational sets achieve this maximum. Given a maximal finite rotational polygon P for σd, we will determine what sets of critical leaves guide the pullback lamination containing P and its preimages (where the pullback lamination is the full pre‐image of the rotational polygon P not crossing the guiding critical leaves). After taking the closure of the

pullback of P, we call the resulting lamination the minimal lamination that contains P. We first investigate this question for d = 2, 3.

Monoids for Math Majors, Brian Beasley, Presbyterian College Abstract: Last May, the MAA PREP workshop on "The Art of Factorization in Multiplicative Structures" presented recent and ongoing research in the area of non‐unique factorization. In particular, the workshop gave a variety of results for certain types of monoids, a topic readily accessible to undergraduates. This talk will cover some basic definitions, examples, and theorems involving congruence monoids and arithmetical congruence monoids. In addition, it will outline a possible approach for incorporating factorization in monoids within an undergraduate abstract algebra course.

LEARNING COLLEGE ALGEBRA THROUGH DANCE, Ann D. Bingham, Peace College Abstract: Students in the College Algebra course often struggle. This study examines a method to engage students using visual and kinesthetic methods in addition to the textbook and regular classroom means. The discussion will include the rationale for an attempt to combine dance and college algebra through a pairing of two classes, and qualitative research on the students’ understanding of transformations of functions through this approach.

A Family of Minimization Problems with a Surprising Commonality – Part II, Irl C Bivens, Davidson College Abstract: Using polar coordinates, we consider a collection of optimization problems that include those considered in Part I. Under mild assumptions, this expanded family exhibits a commonality that helps to explain the significance of the value 1/√2. (If time permits, other versions of the problem will be mentioned.)

A wicker basket problem, Bradley Boreing, King College Abstract: Isaac Newton showed that a body falling freely on a homogeneous, spinning earth follows an ellipse (a cos(k t), b sin(k t)) with respect to the background of the stars, where a, b, and k are constants. However, if dropped north of the equator, the body's path with respect to the earth looks like the reeds in a wicker basket. The goal of my talk is to show that the profile of the basket‐‐‐its projection onto the xz‐plane‐‐‐is a hyperbola.

Arithmetic properties of the partition function, Matthew Boylan, University of South Carolina Abstract: A partition of a positive integer is a non‐increasing sequence of positive integers whose sum is n. The partition function, p(n), gives the number of partitions of n. In this talk, we will discuss recent results on the arithmetic of p(n), and some of the ideas used to prove these results.

Service‐learning projects in mathematics courses, Ryan Brown, Georgia College & State University Abstract: Service‐learning and experiential learning projects have become more widely used in mathematics courses. We review two completed service‐learning projects and one in‐progress project. We focus especially on the reflection components of these projects and how (and whether) these contribute to meeting the objectives of the courses.

Rational Residuacity of Prime Numbers, Mark Budden, Armstrong Atlantic State University Abstract: The ``higher'' reciprocity laws of number theory were developed as generalizations of the law of quadratic reciprocity, but they required that both the statements and proofs reside in rings of integers other than $\mathbb{Z}$. In contrast, the rational reciprocity laws attempt to retain a closer

connection with $\mathbb{Z}$ by utilizing rational residue symbols, which only take on the integer unit values $\pm 1$ and are evaluated on integers themselves. A brief survey of the known rational reciprocity laws will be given and we will describe a new generalized law for $2^t$th rational residue symbols.

Engaging Students in Advanced Analysis via Fractal Geometry and the Hausdorff Metric, Doug Burkholder, Lenoir‐Rhyne College Abstract: We shall call the space of all compact subset of R2 the Space of Fractals. By studying this space participants will see an interesting example of, and hence reinforce their understanding of, metrics, compactness, the triangle inequality, Cauchy sequences, and complete metric spaces. Participants will also see how the fixed‐point theorem applied to our space guarantees a unique attractor for any iterated function system. Participants will see how to use this knowledge to supplement an Advanced Analysis course. No prior experience with the Space of Fractals or the Hausdorff Metric is required.

Prime Curios!, Chris Caldwell, University of Tennessee at Martin, G. L., Jr. Honaker Abstract: We are currently finishing a book based on our web site \emph{Prime Curios!} (\href{primes.utm.edu/curios}). This work is essentially a dictionary of prime number trivia. Some of these facts (curios) are directly related to mathematics, others not, but all involve prime numbers. In this talk we will present a small sample, ranging from the sublime to the absurd, of this work’s roughly 2113 items.

Multiple Methods of Assessment for Introductory Mathematics Courses, Lisa Carnell, High Point University Abstract: In introductory math classes for non‐majors, students sometimes come in frustrated and with a sense of helplessness about their (in)ability to do mathematics. To increase student empowerment and facilitate communication between student and instructor, I use multiple methods of assessment. In this talk, I will describe a brief daily assessment technique and a method for getting feedback from students on content knowledge and math confidence that can supplement test grades.

Overdetermined Linear Systems: Ideas and Applications, Philip Carroll, King College Abstract: We consider an overdetermined linear input/state/output system, which consists of two partial differential equations defining the state function in two independent directions, as well as an equation describing the output of the system. We describe how to obtain a solution to such a system, as well as discussing two potential applications of such systems.

Mathematical Model of Action Potential Within a Neuron, Javon Carter, Winston Salem State University Abstract: Neurons communicate with cells and other neurons through generating electrochemical impulses and conduct them along membranes. Understanding of the dynamical connectivity of over 4 hundred billion neurons in the brain, each connecting with several thousand others, will, without any doubts open the door to the recognition and discovery of numerous unknown functions and dysfunctions of the nervous system.

In this project, the construction of a mathematical model in the form of a system of differential equations for the sodium‐potassium exchange pump, the mechanism in which information transfers throughout the brain, generating a complex network is presented. This dynamical system is composed of two equations based on the two phases of the mechanism involved in the exchange pump

− KNa ++ . Using qualitative method of solution of ordinary differential equations and technology, a thorough analysis of the solutions for each mechanism and for the whole system will be concluded.

The Mathematics and Computation of Removal Sampling, Mark Cawood, Clemson University Abstract: Estimating the population of a species within a defined sampling region is a difficult problem. The removal method estimates the population using the results of a series of K trappings in which the trapped animals are removed from the population. Accessible to undergraduate math students, this talk will combine elements of calculus, probability, statistics and numerical methods to discuss how to compute the estimate for the population using the removal method and the likelihood‐ratio confidence region for that estimate.

Domains and Extended Calculus in a Quasicontinuous Function Space, Rodica Cazacu, Georgia College & State University Abstract: One of the aims of domain theory is the construction of an embedding of a given structure or data type as the maximal or “ideal" elements of an enveloping domain of “approximations," sometimes called a domain environment. Typically the goal is to provide a computational model or framework for recursive and algorithmic reasoning about the original structure. In this paper we consider the function space of (natural equivalence classes of) quasicontinuous functions from a locally compact space X into L, an n‐fold product of the extended reals [‐∞,∞] (more generally, into a bicontinuous lattice). We show that the domain of all “approximate maps" that assign to each point of X an order interval of L is a domain environment for the quasicontinuous function space. We rely upon the theory of domain environments to introduce an interesting and useful function space topology on the quasicontinuous function space. We then apply this machinery to define an extended differential calculus in the quasicontinuous function space, and draw connections with viscosity solutions of Hamiltonian equations.

A Brief Introduction to Term Rewriting, Jeff Clark, Elon University Abstract: Computer Algebra Systems regularly apply rules for rewriting expressions in simplifying algebraic expressions and computing derivatives and anti‐derivatives. This talk will cover elementary properties of rewriting systems as well as the problem of finding a canonical form for an expression.

The Intersection of a Cone and a Plane, Robert E. Clay, Dalton State College Abstract: We show that the conic section formed by the intersection of a cone and a plane is congruent to a plane figure whose equation corresponds to the analytic definition of a conic section.

This is not new. However, previous proofs involved using a spherical rotation. We show that by a judicious choice of axes we can use a planar (strictly speaking – cylindrical) rotation, thereby giving a simple proof.

A note in Classroom Capsules in Vol.38, No.5, November 2007 of The College Mathematics Journal attempts to give a simple proof, but, in general, the equation does not lead to a congruent figure.

Customizing Technology to Meet the Needs of Your Course, Alex Coleman, Houghton Mifflin Abstract: Discover the benefits of customizing technology to meet the needs of your Math Program. Houghton Mifflin provides everything from tutorials, auto‐graded homework, multi‐media Ebooks, as well as an enormous array of study materials in Mathematics. We also work closely with faculty to customize these materials so that they match your departments syllabi and the unique needs of your

student body. Come see how we have worked with faculty across the country to develop these powerful technology solutions! In addition, we will raffle off an iPod Shuffle!!

Boost for Student Success: MESA at Georgia Perimeter College, Ray E Collings, Alice Eiko Pierce, Georgia Perimeter College Abstract: Now in our fourth year of Mathematics, Engineering, and Science Achievement, we will share student and faculty experiences in this supplemental program to our two‐year college curriculums for these majors. Included will be Academic Excellence Workshop Challenge Problems in Calculus I & II written by GPC students and the speakers.

How to Motivate Students to Learn by Using Software, Emily Cook, Hawkes Learning Systems Abstract: Discover the benefits of using interactive software in teaching and learning mathematics. Hawkes Learning Systems promotes grade improvement and motivates students to learn by providing tutorials, unlimited practice, helpful feedback provided by artificial intelligence, and mastery‐based homework. Come see a demonstration of our state‐of‐the‐art test generator, online gradebook and student courseware! Also, everyone who attends will be entered to win a MP3 multimedia player, which will be raffled at the end of the presentation!

When does k! divide , Joshua Cooper, University of South Carolina Abstract: Motivated by a problem on extremal permutations, we discuss the question: when does k! divide ? The function (mod k!) is periodic modulo k!2, so it suffices to compute it for finitely many n. We show that the smallest possible period is in fact k! lcm(1, . . . , k), which is quite a bit smaller than k!2. The proof is elementary and employs Kummer’s Theorem on binomial coefficients. Joint work with Andrew Petrarca.

An Object Oriented Unit Circle, Chris Corriere, Southern Polytechnic State University Abstract: The Unit Circle is a strictly defined mathematical object. The strict definition of the Unit Circle facilitates the development of its object oriented model. The most direct method of implementing such a model is to hard code the information explicitly, but this is bad programming practice. This presentation compares the hard coded model of the circle to an algorithmic model of the circle. The concepts of software encapsulation and abstraction are used to contrast the two models and explain their strengths and weaknesses.

The Unique Infinity of the Denumerable Reals, Brian L. Crissey, North Greenville University Abstract: Solomon Feferman in his widely acclaimed 1998 treatise In the Light of Logic, defines the reals as those numbers intended for measuring. Max Planck established quantum limits to meaningful measurement. These limits, now called Planck values, are combinations of three fundamental constants: c (the velocity of light), G (the gravitational constant), and h (Planck's constant). Planck values for time, length, area, volume, and mass establish the maximal precision by which meaningful measurements may be taken. Quantum‐limited precision makes it meaningless to try to distinguish two numbers that differ only in the digits beyond the limit. Although no measure can produce an irrational number, irrational numbers are nevertheless traditionally included in the reals. Georg Cantor’s famous diagonal proof of the non‐denumerability of the reals requires the inclusion in the reals of infinitely long digit expansions that derive from irrationals or non‐terminating procedures. Cantor’s mentor Leopold Kronecker vehemently denounced his student’s inclusion of irrationals in the reals. Partitioning the reals into repeating fractions, rationals, and irrationals allows a re‐examination of the cardinality of the reals. A repeating decimal, when converted into the radix of its denominator, becomes a non‐

repeating rational. The rationals in turn are known to be denumerable. Each irrational can be produced by the output of a denumerable procedure which can terminate when the Planck precision limit is reached, at which time a denumerable real has been constructed. Cantor’s diagonal “proof” is such a denumerable procedure. After this procedure terminates at the precision limit, its produced digit string will be enumerated in time without contradiction. Since each partition class of meaningful reals is

denumerable, the complete set is also denumerable, with a cardinality no larger than 0, the cardinality of the integers. An induction proof establishes that the cardinality of power sets of denumerable sets is also denumerable. Higher set theory based upon a presumed progression of ever greater cardinal numbers can be relegated to the realm of speculative mathematics. In the world of meaningful mathematics, there is but one infinity and just one cardinal number. David Hilbert’s

Continuum Hypothesis that c, the cardinality of the reals, is the first cardinal larger than 0, is thus

rejected for meaningful reals, as there is no cardinality above 0.

ROTATIONAL GAPS IN CUBIC LAMINATIONS, Clinton Curry, Andrew McDonald, University of Alabama at Birmingham Abstract: A d‐invariant lamination is an equivalence relation on the circle R/Z which is, in some sense, compatible with the map σd : R/Z → R/Z defined by σd(t) = d ∙ t. There is a broad correspondence between Julia sets of degree d polynomials and d‐invariant laminations, and the lamination can be used to study the Julia set in a combinatorial fashion. Additionally, laminations are themselves interesting as dynamical systems on the unit disk. While 2‐invariant laminations are relatively well‐understood, 3‐ invariant laminations are more mysterious. We take the abstract point of view that some 3‐invariant laminations arise as combinations of 2‐invariant laminations, and that important parts of the dynamics arise this way. For instance, for every number 0,1, there exists a 2‐invariant lamination with a fixed gap whose rotation number is ; we characterize when and how a 3‐invariant lamination contains two rotation gaps with two specified rotation numbers. We also discuss generalizations of this result.

Using Performance Assessments and Number Lines to Build Pre‐Service Teachers’ Knowledge of Fractions and Rational Expressions, Joy W Darley, Georgia Southern University Abstract: Many pre‐service teachers have difficulty transferring their knowledge of arithmetic to related content in algebra, especially in the area of fractions. Descriptions of the tasks used to assess students' initial understanding of fractions as numbers and examples of the performance tasks used to bridge students' understanding of fractions and rational expressions are given.

More Than Meets the Eye: Visualizing Combinatorial Proofs via Graph Theory, Joe DeMaio, Kennesaw State University Abstract: Visual illustrations of proof concepts, known as "Proof Without Words," have become common practice in modern mathematics. In this talk, we establish graph theory as another illustrative tool. Rather than following traditional counting techniques, we couple the visual nature of graphs with the logic of combinatorial proofs. We prove several binomial coefficient identities by analyzing the visual structure of graphs. These results not only demonstrate the understanding such proofs can bring, but how combinatorial methods and the visualizations using graphs can extend identities more easily than traditional proof techniques.

Progressive Wavelets on Nonuniform Nodes, Baiqiao Deng, Columbus State University Abstract: A progressive wavelet at time t involves only the wavelets at time before t and some nice compactly supported function at time t. Progressive wavelets can be useful in the analysis of moving images that are received only from the previous time. We construct exponential decayed orthogonal wavelets on the half line using the roof functions on nonuniform nodes, which provides adaptive

hierarchical discretization of signals and moving images.

Using programming to complement the teaching of proofs in a discrete mathematics course, Jeff Denny, Keith Howard, Mercer University Abstract: The discrete mathematics course at Mercer University has come to serve as a mathematics bridge course for computer science and computer engineering majors that is distinct from the bridge course for mathematics majors. Students taking this course tend to possess a degree of aptitude and familiarity with programming but have little knowledge of or inclination toward proofs. By adding a programming component using Python to the course, we invite the students to explore ideas, demonstrate applications of the mathematical topics, and develop thinking and communication habits that are helpful in advancing their understanding of proofs. In this talk, we will discuss the motivation for adding programming to the course, what our projects have been, and how they have impacted the teaching and learning of proofs.

COLLECTIVE BEHAVIORS, Amy Dexter, Austin Peay State University, Anna Devlin, Johns Hopkins University, John Pate, University of Arizona Abstract: The study of collective behaviors is a growing field that observes the dynamics of natural aggregations by monitoring the flow of information through a group's members. One way to effectively study these local interactions and the global consequences they elicit is to model the individual's behavior. We employ an agent‐based model, which creates a set of rules to govern an agent's behavior. Applying these rules to every agent, a simulation is run to study global movement and emergent patterns. Collective movement is then analyzed through group accuracy, elongation and fragmentation by varying certain parameters. For further investigation of collective behaviors we incorporate factors that will produce a more realistic model such as blind spots for individual agents and obstacles in the group’s path. Finally, we are able to create adaptable rules that successfully depict collective behaviors.

Five years of teaching distance learning: What works and what needs work, Lothar Dohse, University of North Carolina at Asheville Abstract: Starting in the spring of 2003 an online version Introductory Statistics has been taught on a regular basis at a small liberal arts college. Since that time various delivery configurations have been attempted and additional courses have been added to the program. Throughout this time events were chronicled. Some of the pedagogical experiments were successful and others were tried only once. The five years of experiences will be used to take a critical view of the distance learning initiative at UNCA's Mathematics department, and to present a vision as to where its distance learning program is heading.

Power Distribution in Weighted Voting Systems: An Application of Integration Which is Appropriate for Single Variable Calculus Courses, Chris Duncan, Lander University Abstract: A weighted voting system is one in which the voters are assigned different numbers of votes to reflect inequality among the voters. Common examples include the Electoral College and shareholders meetings.

The manner in which the power in these systems is shared is of vital interest. Intuitively, one might suspect that the power of a voter is equal to the proportion of the total number of votes that the voter controls. Several interesting examples of the fallacy of this intuition will be given. The power distribution in these examples will be calculated with an integral version the Shapley‐Shubik power index. Since the integrals that arise only involve polynomials, it is an appropriate application for an introductory integration course.

Examining the error of linearization for an duality‐based a‐posteriori error estimate, Sean Eastman, Armstrong Atlantic State University Abstract: Error estimates for nonlinear elliptic two‐point boundary value problems typically involve linearization around the approximate solution. In practice, this presents no real problem, but the true effect of this linearization on error estimates is not well known. One approach is to study the derivative of the mapping between the approximate solution and the corresponding linearized dual solution. In this talk, we'll discuss an a‐posteriori error estimate based on duality, and utilize the Green's function to derive a formula for the derivative of the forward to dual map.

Mentoring as a vital component to a graduate teaching assistant teaching development program, Carrie D Eaton, University of Tennessee at Knoxville Abstract: Our "new‐and‐improved" GTA teaching development program is in its second year at our University. From the beginning, we instituted a formal mentoring component which has continued to develop. This year, the mentors were offered the option of enrolling in a mentoring discussion seminar to complement and enhance their mentoring of the first‐year GTAs. I will share how we worked to address various mentoring needs, a goal which led us to discuss a variety of areas of professional development, not just "teaching."

Prime Numbers: Finding and Applying Them, Jeffrey Ehme, Spelman College Abstract: Since the advent of public key cryptography, the prime numbers and their properties have been an active area of interest. We begin this short course by reviewing some cryptosystems that require large prime numbers. Then for the remainder of the course, we consider different types of approaches to finding large prime numbers. Mathematicians would prefer methods that yield numbers that are unambiguously prime, but these deterministic methods are slow. Probabilistic methods are fast and yield “industrial grade” prime numbers. That is, numbers that are extremely likely to be prime and will work in the context where they are used, but we can’t be sure if they are really primes. Examples of the later methods include the Miller‐Rabin test and a test involving Lucas sequences. No previous experience with these topics is assumed.

Topologies of complex networks: Understanding a GRAPH of graphs, Chinwendu Enyioha, Gardner‐ Webb University Abstract: There has been an increasing interest amongst researchers in comprehending the topology of complex networks, particularly the internet and biological networks. Understanding the interplay between their evolution and robustness particularly makes studies in this area fascinating. As technology continues to advance, a proper understanding of complex network topology is imperative to design and optimize performance of the next generation complex networks, subject to technological and economic constraints. This work was primarily aimed at finding the right approach to understand (and visualize) the topology of a particular class of complex networks using concepts from graph theory. We considered tree graphs as nodes interconnected, according to a local rule referred to as a ‘general flip’, in a larger space of graphs (a GRAPH of graphs). The diameter of the GRAPH of graphs comprising trees was shown to be of order n, where n is the number of nodes of the node‐graphs within the larger GRAPH. An algorithm to generate the nearest neighbor for arbitrary graphs using a general flip was developed and implemented in Matlab; we are in the phase of making the program enumerate the graphs generated. Algorithms for finding minimal paths between two canonical graphs were also developed and successfully implemented. The established bound on the diameter of the GRAPH of graphs comprising trees, O(n), down from O(n2) for arbitrary graphs, has motivated us to attempt to establish a tighter bound for the diameter of the GRAPH of graphs comprising arbitrary

graphs.

Coloring the USA Map and Other Graphs, Gilbert Eyabi, Anderson University Abstract: In this talk we present some work being done by the author with his undergraduate students in his senior research class in Graph Theory. A graph $G=(V,E)$ is a triple consisting of a vertex set $V(G)$, and an edge set $E(G)$, and a relation that associates with each edge two vertices called its endpoints. We use vertex coloring to demonstrate how the map of USA (and any other map) can be colored with at most four colors such that no two neighboring states can have the same color. Some Traffic problems would be presented and we shall conclude with a proof to a problem we found in a text by Chartrand and Zhang.

A Linearization of a Numerical Scheme for the Saturation Equation: Regularity Results and Consistency, Koffi B. Fadimba, University of South Carolina Aiken Abstract: We consider a linearization of a numerical scheme for the saturation equation (or porous medium equation) ·u ·S 0 through first order expansions of the fractional function f and the inverse of the function , after a regularization of the porous medium equation. We establish some regularity results for the Continuous Galerkin Method and the linearized scheme. We use these regularity results to show the consistency of the linearized scheme.

Steady‐State Distributions of Closed Asset Systems, Maria Fedore, Elon University Abstract: The purpose of this research is to compare certain asset‐exchange models to the distribution of household incomes in the U.S.A. Two new exchange rules are shown to have strong promise in overcoming weaknesses previously noted in the literature for such models.

To Steal or Not to Steal: A Game Theoretical Model of Brood Parasitism in the dung beetle Onthophagus taurus., Meghan Fitzgerald, University of North Carolina Greensboro Abstract: We have adapted previously‐developed game theory models of kleptoparasitism to model brood parasitism in the female paracoprid dung beetle Onthophagus taurus. O. taurus is known to find existing brood balls, destroy the egg, and use the prepared brood chamber to lay her own egg. Using existing literature and our own field and lab studies, we gathered empirical data to estimate parameters of the model, incorporating search and preparation times. We used this data to form a model that can predict the conditions under which a beetle is likely to steal the brood mass verses preparing her own, as well as when guarding the brood mass is optimal. We concluded that if it takes less time to kleptoparasitize then it does to prepare a brood ball for herself then it is advantageous to steal whenever a vulnerable brood ball is found. We also concluded that if she cannot produce a new egg faster than kleptoparasites can find the old one, it is better to guard for the entire time the egg is vulnerable. If the opposite is true, it is better not to guard the egg at all. In addition, we can use the model to predict the proportion of beetles in the population performing certain tasks, such as: resting, searching for a dung pat or an existing brood ball, preparing, kleptoparasitizing, laying the egg, and guarding. We are currently using these proportions to determine the Evolutionary Stable Strategies (ESSs) of the beetles under varying population densities.

Packed and Monadic Balanced Ternary Designs, Margaret Francel, The Citadel Abstract: A balanced ternary design, or BTD, with parameters (V,B,R,K,_) is a collection of B blocks on V elements such that (1) each element occurs R times in the design, (2) each pair of distinct elements occurs Λ times in the design, and (3) each block contains K elements, where an element may occur 0, 1, or 2 times in a block (i.e., multiset). For example the collection of subsets {1,1,2,2}, {1,1,3,3}, {1,1,4,4},

{2,2,3,3}, {2,2,4,4}, {3,3,4,4}, {1,2,3,4} and {1,2,3,4} is a BTD with parameters (4,8,8,4,6). Our talk will investigate BTDs with the added property that all blocks contain the same number of doubletons. Two cases will be considered. The case where each block contains as many doubletons as possible (called packed BTDs), and the case where each block contains exactly one doubleton (called monadic BTDs). Both existence and non‐existence results will be given. Emphasis will be on useful construction methods.

Factorization Lengths in Multiplicative Monoids, Michael Freeze, University of North Carolina Wilmington Abstract: Let M be a multiplicative monoid. For a positive integer k, let V(k,H) denote the set of all positive integers n such that there exist irreducible elements u_1, … ,u_k, v_1, … , v_n in M with u_1 … u_k = v_1 … v_n. We consider sufficient conditions on M for V(k,H) to be an interval.

Generalized Mazurkiewicz’s Theorem, Kailash Ghimire, Georgia Southwestern State Univesity Abstract: A generalized version of Mazurkiewicz’s theorem is proved in the Hilbert cube by using a homological approach and finite codimension in the Hilbert cube. Which gives a necessary condition for a closed subset of the Hilbert cube to separate the Hilbert cube .

Using Web‐based Multimedia Applications to Enhance Calculus Instruction, Greg Gibson, North Carolina A & T State University, Neil P Sigmon, Radford University Abstract: In many introductory and upper level college mathematics courses, due to time constraints, students are often not exposed to many real‐life applications of the material. As a result, students can fail to see the importance of the topics covered. However, the use of technological tools can quickly bring practical applications of mathematics to life. Using these tools, students can quickly see how the mathematical concepts they cover have great value without the cumbersome background that would normally be needed in a more traditional setting. This presentation will discuss web‐based multimedia modules designed for calculus. The applications presented include circuits, orbits of satellites and Reed Solomon Codes. The classroom implementation of the modules and the student response, as measured by surveys, will be discussed.

Redesigning the Major: One Departments Story, Mark Ginn, Appalachian State University Abstract: This talk will describe the process our department went through in revamping all of our majors. In 2003 we decided to take a fresh look at the curriculum we offered to our majors, which at that time consisted of three tracks, Mathematics, Mathematics Secondary Education, and Statistics, with two concentrations on the Mathematics major, General or Applied. Starting in the fall of 2008 we will be offering four majors, Mathematical Sciences, Mathematics Secondary Education, Statistics, and Actuarial Sciences, with six concentrations on the Mathematics option, Computation, Life Sciences, Physical Sciences, Business, Statistics, and Mathematics. In addition there were substantial changes in the existing curricula within each major. We will discuss both the process and the rationale behind these changes.

Some Integral Inequalities for Entire Functions of Exponential Type, Narendra K Govil, Auburn University Abstract: An entire function f is said to be of exponential type τ if it is of order ρ < 1 and of any type, and if of order 1 then of type T ≤ τ, that is, f is of exponential type τ if for every ε > 0, there exists a number K(ε) such that || || . It was proved by Bernstein that if f(z) is an entire function of exponential type τ, then

sup || sup ||. 1 Also, it was proved by R. P. Boas (Illinois J. Math. 1957) that if f(z) is an asymmetric entire function of type τ, then sup || sup ||. 2 2 Both the above inequalities are best possible. In this talk we present some integral inequalities for entire functions of exponential type.

Matter Gravitates, But Does Gravity Matter?, Charles Groetsch, The Citadel Abstract: Fill a tank with water, punch a hole in its side, and see how far the spurt reaches. The simplest mathematical model (one that neglects air resistance) gives a surprising result: the answer is independent of the strength of gravity. The range is the same on Jupiter as it is on Earth. But what if resistance is taken into account? We discuss some mathematical features of this problem for a simple resistance model. An easy fixed point theorem, the implicit function theorem, some gnarly analysis, and a little computing all have parts to play.

Study abroad for math majors ‐ visiting students welcome!, Jane Hartsfield, University of North Carolina Ashville Abstract: Why should humanities students have all the fun? Why can't there be study abroad options for math students? I will outline the process involved in developing a study abroad in the history of science and math. An overview of our itinerary and coursework will be provided along with a suggested timeline for putting together your own program. Visiting students are welcome to participate in our program, so check it out and see if it fits the needs of your students.

Negative Dependence and the Bivariate Normal Distribution, Denise Haynes, Carver High School, Ronald F Patterson, Wanda M Patterson, Winston Salem State University Abstract: We consider negative dependence for certain bivariate distributions. Two random variables are negatively dependent if their joint distribution function is less than or equal to the product of their marginal distribution functions; that is ≤ ≤ ≤ ≤ ⋅ ≤ yYPxXPyYxXP )()(),( We will show that the condition (XP >> ≤ > ⋅ > yYPxXPyYx )()(), is also sufficient for negative dependence. We will also prove that the random variables X and Y whose joint distributions is the bivariate normal distribution is negatively dependent if the covariance of X and Y is negative.

A Combinatorial Interpretation of a Modified Version of the Fibonacci Polynomials, Curtis Herink, Mercer University Abstract: A representation of the Fibonacci numbers using binary strings, inspired by the fact that every positive integer can be uniquely represented as the sum of nonconsecutive Fibonacci numbers (excluding F_1), is generalized to give a sequence of polynomials. The terms of these polynomials have a natural combinatorial interpretation.

Lucas Primality Test, Karen Hicklin, Spelman College Abstract: Prime numbers have always been an interesting field of study. How they are used, how can they be found, and how to check a number for primeness are just a few questions many researchers have asked. Prime numbers are widely used in areas of cryptography. In cryptography prime numbers of about 100 digits or more are used. In an attempt to understand how to test these large numbers for primeness, I have studied and implemented the Lucas Probabilistic Primality Test. This test is well known for its application in many computer software programs. The main topic of this presentation

will be how to efficiently test a number for primeness.

Nearly Almost Perfect Numbers, Ryan Hill, Wofford College Abstract: Let σ(M) denote the sum of the divisors of a positive integer M and define a function F(M) = 2M – σ(M). If F(M) > 0, M is perfect‐plus. If F(M) < 0, M is perfect‐minus. In particular, if F(M) = 2, M is perfect‐plus‐two (PP2). For example, 215(216+1) is PP2. This paper corroborates and extends the work of K. Inkeri in the April 1977 edition of The American Mathematical Monthly. Specifically, we show that a number of the form M = 2npk is PP2 if and only if k = 1. Furthermore, if p and q are odd primes such that p < q and q ≥ p2 + 1, then no number of the form M = 2npq is PP2.

Generalizations of a lamination lemma by Thurston, Jeffrey Houghton, University of Alabama at Birmingham Abstract: A lamination is a set, L, of closed chords, called leaves, in the closed unit disk, D, such that for every ℓ1 ≠ ℓ2 L, ℓ1 ∩ ℓ2 Bd(D) = S = R/Z and (L) S is closed in D. We say that a lamination is leaf invariant under the map σd : S → S defined by σd(t) = td (mod 1) if for every leaf , . The length of a leaf is defined as the length of the shortest arc of S connecting p and q. Thurston proved that if a leaf ℓ is longer than 1/3, and C denotes the region in D bounded by ℓ and ℓ′ = ℓ + 1/2, then the first return of ℓ to C under σ2 always connects the two components of C∩S. We will explore generalizations of this result to σd for d > 2.

SOCAMATYC and the Role of Two Year College Mathematics in SC, Laura Hoye, Trident Technical College. Abstract: Mathematics in the two year college is almost exclusively a service program for other areas. Participants will discuss ways that professional organizations such as SOCAMATYC, AMATYC, and, MAA can better support the needs of faculty and students.

5‐divisibility of 5‐regular partitions and the eta function, Michael Hull, Furman University Abstract: In this talk, we will introduce the notion of partitions and the l‐regular partition function. By using generating functions, we will show how Dedekind's eta function can be used to study the l‐ regular partition function mod l. We will then use the special properties of the eta function to give exact criteria for when the 5‐regular partition function is divisible by 5.

Podcasts, Video “Tutors” and More in Introductory Statistics, Patricia Humphrey, Georgia Southern University Abstract: How many times have you heard a student complain “I understood it perfectly in class, but when I got home, I was lost!” This type of comment may become a relic of the past. For many years, we who teach Introductory Statistics thought we were “high tech” because we were using a computer package or a graphing calculator. This is no longer state‐of‐the‐art.

Many publishers are now going “all out” with electronic ancillaries through their websites. I will discuss my (and my students’) experiences with some of these, including StatsPortal, a website companion to David Moore’s text “The Basic Practice of Statistics.” This website includes not only an interactive complete text, quizzing and homework capabilities, applets and statistical software, but the “Stat Tutor” which can be called up at many points in perusing the e‐book (or by itself if students “forget part of a lecture”), and podcasts of the summary material for each chapter. One advantage for the instructor is that the site is easy to use, and you don’t have to invent everything yourself.

Convergence of Approximate Solutions to Scalar Conservation Laws by degenerate diffusion, Simon Seok Hwang, LaGrange College Abstract: In this talk we consider the convective porous media equation. We will show that the approximate solutions of the convective porous media equation converge to the entropy solution of a scalar conservation law using the methodology developed by Hwang and Tzavaras. The proof relies on the kinetic formulation of conservation laws and the averaging lemma.

SOME TOPICS IN GORENSTEIN HOMOLOGICAL ALGEBRA, Alina Iacob, Georgia Southern University Abstract: Using the class of finitely generated Gorenstein projective modules, Avramov and Martsinkovsky defined Gorenstein cohomology modules for finitely generated modules over noetherian rings. They also extended the definition of Tate cohomology and they showed that the Tate cohomology measures the ”difference” between the absolute and the relative Gorenstein cohomology.We extend their ideas: given two classes of modules P and C such that P ⊂ C, we define generalized Tate cohomology modules with respect to these classes and show that there is an exact sequence connecting these modules and the relative cohomology modules computed by means of P and respectively C resolutions. We prove that the generalized Tate cohomology with respect to the class of projective and that of Gorenstein projective modules is the usual Tate cohomology and that our exact sequence becomes Avramov‐Martsinkovsky’s exact sequence in this case. We also show that we have balance in generalized Tate cohomology.

An introduction to two alternative styles of teaching introductory statistics, Keshav Jagannathan, Coastal Carolina University Abstract: This talk will focus on research related to the effectiveness of two different pedagogies related to teaching Introductory Statistics courses. The first approach uses visual interactives to help students comprehend concepts in a statistics course rather than use rote memorization. The second approach involves the use of ”guided notes” that allow students to take better notes in class during a lecture and summarize the course material in a clear and concise manner. This talk will provide examples of the two pedagogies and a few results obtained from a dry run of the project.

The parity of the 5‐regular and 13‐regular partition functions and related results, Kevin James, Clemson University Abstract: In this talk we will give a characterization of the parity of the 5‐regular and 13‐regular partition functions. We will exhibit some Ramanujan‐type congruences modulo 2 for the 5‐regular partition function. We will discuss a conjecture for infinitely many such congruences modulo 3 for the 13‐regular partition function and present some partial results.

Probability 1/e, Martin L. Jones, College of Charleston, Reginald Koo, University of South Carolina Aiken Abstract: In this talk we will present three non‐trivial problems in probability in each of which the event of interest has probability 1/e. We will attempt to show that there is a common structure to all three problems and that there is a reason that events of interest often have probability 1/e. This talk should be of interest to anyone who teaches probability and/or statistics.

Domain decomposition method for solving partial differential equations, Younbae Jun, University of West Alabama Abstract: Many physical problems can be described by mathematical models that involve partial differential equations. They describe phenomena in some important fields such as fluid flow, ground water contamination and transport, heat transfer, and many others. In most cases, it is difficult, or

infeasible, to find the analytic solution of the problem. In such cases, numerical methods are required. Domain decomposition methods are known as very efficient methods to solve large systems of equations arising from the discretization of partial differential equations by the Finite Difference Method. They are based on a decomposition of the entire domain of the partial differential equation into several subdomains. The original problem is divided into small sub‐problems that correspond to the subdomains. Since these sub‐problems can be solved simultaneously in parallel, the solution process can have a considerable speed‐up over classical methods when they are implemented on parallel computers. Many domain decomposition methods have been developed for the parabolic partial differential equations in two‐dimensional case. In this research three‐dimensional non‐overlapping domain decomposition method is proposed and it is analyzed in terms of the stability and efficiency. It was observed that the method is unconditionally stable and very efficient and the prediction phase is no longer negligible even if the number of unknowns on the interface plane is much less than the number of the interior points for the three‐dimensional case.

Illustrating Volume of Revolutions via Maple, Lyndell Kerley, East Tennessee State University Abstract: The concept of an object in C++ is an important part of object orientated programming. Maple implements the idea of an object by using a module. Maple also has recently introduced a nice interface which allows an user to supply a function such as y = x^2 on [0,1]. Then the user can revolve it about either the x‐axis or y‐axis obtaining a nice 3d plot of the resulting solid. A student can set up such an interface. In the process, one needs to understand how to supply Maple commands to perform the needed integration for the volume of the resulting solid. An understanding of the disk and shell method are required. Once the object is displayed, one can rotate it about either the x‐axis, y‐ axis, or z‐axis in 3‐space as well as animating the movement about one of the 3 axes.

NONLINEAR DISSIPATIVE WAVE EQUATIONS WITH SPACE‐TIME DEPENDENT POTENTIAL, Maisa Khader, University of Tennessee at Knoxville Abstract: We studied the long time behavior of solutions of wave equations with absorbtion |, | , and damping with space‐time dependent potential , , , where , ~1|| 1 for large |x| and t; a0 > 0. For ∞, 1, 1,1 and 1 2/ 2 we establish decay estimates for the energy, L2 and Lp+1 norms of solutions. We used the new technique developed by Todorova and Yordanov, which is able to capture the exact decay of the wave equations with space dependent co effcients. The presence of a space‐time dependent potential, as in our case, requires modifcations of this technique.

(1) For exponents , such that 0,1 1,1 and 0 1 we found three different regimes for the decay of solutions dependent on the exponent of the absorbtion term. More precisely we found two thresholds p1(n,α,β) and p2(n,α,β) such that: • For , , 2/2 the decay of solutions of the nonlinear equation coincides with the decay of the corresponding linear problem.

• For 1 < p < p2(n,α,β) the decay is independent of α. • For p2(n,α,β) < p < p1(n,α,β) the decay is very fast, almost exponential. (2) In the case ∞, 0 and 1,1 we found one threshold and correspondingly two different regimes for the decay of solutions‐fast decay for the subcritical case and slow decay, coinciding with the decay of the linear problem, for the supercritical case.

A Family of Minimization Problems with a Surprising Commonality – Part I, Benjamin G Klein, Davidson College 1 Motivated by the problem of choosing m to minimize the integral sin()x − mx dx , we consider the ∫0 1 more general problem of minimizing fx()− mx dx for several classes of functions f. We show, ∫0

under fairly general hypotheses, that if m0 is the value of m that minimizes the integral, then the

graphs of y = m0 x and y = fx()meet at a point with x coordinate 12.

Some observations about the generating function for C4(n), Louis Kolitsch, University of Tennessee at Martin Abstract: In this talk I will present some observations about the generating function presented in Theorem 11 of George Andrews’ paper, A survey of multipartitions: Congruences and identities. This generating function can be used to calculate C4(n), the number of multipartitions of n satisfying certain component restrictions.

Reverse Sierpinski Number Problem, Daniel Krywaruczenko, University of Tennessee at Martin Abstract: A generalized Sierpinski number base $b$ is an integer $k>1$ for which the $\gcd(k+1,b‐ 1)=1$, $k$ is not a rational power of $b$, and $k\cdot b^{n}+1$ is composite for all $n>0$. Given an integer $k>0$, we will seek a base $b$ for which $k$ is a generalized Sierpinski number base $b$. We will show that this is not possible if $k$ is a Mersenne number. We will give an algorithm which will work for all other $k$ provided that there exists a composite in the sequence $\{(k^{2^m}{+}1)/\gcd(k{+}1,2)\}_{m=0}^\infty$.

The Hitting Time for the Height of a Random Recursive Tree, Thomas M Lewis, Furman University Abstract: A random recursive tree is a tree that evolves according to the following probabilistic rule: at each stage, an existing vertex of the tree is selected at random and a new vertex is attached to this selected vertex as a descendant. Random recursive trees and related structures have been used as models for pyramid schemes, chain letters, and family trees of ancient manuscripts. In this talk we provide a simple formula for the expected time for a random recursive tree to grow to a given height.

Generating Functions of Symplectic Rook Monoids, Zhenheng Li, University of South Carolina Aiken Abstract: This talk concerns the generating functions of symplectic rook monoids. The symplectic rook monoids are submonoids of rook monoids. They first appeared in linear algebraic monoid theory. Their algebraic structures have been studied well. However, their connection to combinatorics and differential equation is still a mystery. This talk will calculate the generating functions induced from these monoids by solving some interesting differential equations.

Competitive Events for Fun, Excitement and Curriculum Development, John Long, Midlands Technical College. Abstract: Competitions can be used to highlight desired changes and the excitement of trying to win can motivate people to incorporate change. Teamwork and mathematical modeling of real world situations are the focus of several competitive events that will be described.

SMART Notebook and the Sympodium: Before, During and After Class, Laura Lundy, Georgia Perimeter College Abstract: Using SMART Notebook to prepare class notes allows for a presentation that can be easily

added to during class and then posted for student review. This talk will offer examples of incorporating text, graphs, images from the book, and clip art into a SMART notebook which is then used to direct classroom lecture and discussion, much as a PowerPoint document would be. The prepared notes can then be amended during class using the Sympodium, highlighting important points and connections and working examples, and the complete notebook saved as a record of the class session.

A Generalized Inverse Representation of solutions to the primal and dual linear programming problems, Shan Manickam, Western Carolina University Abstract: If the primal and dual linear programming problems to optimize c.x subject to Ax ≤ b, b ≥ o and b/.y subject to A′ y ≥ c, c≥ o respectively have feasible vectors, it is shown that they have optimal vectors x, y ≥ o respectively, given by ∧ ∧ (y x x y)′ = A− (− c, b, 0) for some {1, 2, 3} - inverse A- of a matrix A defining the primal and dual problems.

Science , Technology, Engin eering, and Mathematics (STEM) Education in the Upstate: Communication, Cooperation, and Collaboration, Gerald L. Marshall, Tri‐County Technical College Abstract: This presentation will cover recent accomplishments and future plans related to the activities of a group of STEM Education Partners. This group includes educational leaders from the seven (7) school districts of Anderson, Oconee, and Pickens counties; faculty from Tri‐County Technical College and Clemson University; and business leaders from local industries in the Upstate of South Carolina.

Partitions and Compositions, Sarah Mason, Davidson College Abstract: A partition of an integer n is an set of positive integers that sum to n. A composition is an ordered sequence of positive integers that sum to n. We begin this talk with several classical results in partition theory. Next we describe analogous results for compositions. Finally, we explore further directions and open questions with the goal of applying the insights gained through the study of compositions to unsolved problems in partition theory.

Laminations of the unit disk and Julia sets, John C Mayer, University of Alabama at Birmingham Abstract: In a program begun in the early '80's William Thurston proposed to study the Julia sets of complex polynomials through a combinatorial approach using laminations of the unit disk. Thurston's work was never formally published, but his notes circulated widely. Laminations are topological/combinatorial, rather than analytic, tools. In a Circle Dynamics Seminar extending back to the late 90's, a group at UAB, faculty and a changing cast of students, graduate and undergraduate, have been working on understanding Thurston's work on quadratic laminations and extending it to laminations of higher degree. In this talk we will survey preliminarily some of this work currently being carried out by undergraduate and graduate students attending this meeting and giving student presentations or posters.

A lamination L of the unit disk D2 is a closed collection of chords of D2 that intersect, if at all, in an endpoint of each on the boundary circle S1. The chords in L are called leaves. A gap G of L is the closure of a component of D2 \ UL. The boundary Bd(G) of G is composed of leaves and points of S1. We 1 1 1 parameterize S by [0, 1) in the natural way. Consider the d‐tupling map σd : S → S defined by σd (t) = dt (mod 1). The map σd can be extended to leaves linearly and continuously on UL. A lamination is d‐ invariant if, under σd, it is fully invariant (forward and backward) on the leaves of L and is gap invariant (which has a long technical, but natural, definition). A leaf of L is a critical leaf if the images of its

endpoints are the same point.

Understanding the types of gaps in an invariant lamination, and what leaves and other types of gaps they may force or permit to co‐exist is critical in understanding the corresponding Julia sets. We will survey some of the questions along these lines that we are (partially) answering. More hands lighten the work.

Using Fathom to Investigate Sampling Distributions in an Introductory Statistics Class, Erin McNelis, Western Carolina University Abstract: By their nature, the more challenging concepts in an introductory statistics course are visited at the end of the semester when students' motivation, interest, and stamina are at their lowest. One of those concepts is the sampling distribution of a sample mean. The sampling features in Fathom, a statistical software from Key Curriculum, allow the instructor to extend a good hands‐on experiment on sampling distributions to an excellent lesson that enables students to derive the Central Limit Theorem and the relationships between the means ( and ) and standard deviations ( and ) of the sampling distribution and the population. Handouts on the data, the hands‐on experiment, and Fathom activity will be provided.

Application of Mathematics in Biology and Medicine, Akongnwi C. Mformbele, Georgia Gwinnett College Abstract: A mathematical method is presented for analyzing the exchange of substrates and oxygen from a two‐dimensional array of parallel capillaries arranged in a manner characteristic of skeletal muscle. In general, the tissue is non‐uniformly perfused by these capillaries and large scale diffusion occurs from regions that are richly perfused to the regions that are poorly perfused. The methods developed here lead to coupled systems of non‐linear ordinary differential equations equations for the substrate and oxygen concentrations within the capillaries. Interaction among the capillaries is examined when the flow is in the same direction, co‐current, opposite direction, counter‐current and when the capillaries are staggered so that both co‐current and counter current occur. This illustrates the use of mathematics in solving Biological problems. Extension of these results are very useful in cancer research (chemo‐therapy).

Mathematics and Social Justice, Andy Miller, Belmont University, Sheila Weaver, University of Vermont Abstract: Mathematicians are well aware of the wide variety of applications of interesting mathematics to real‐world problems. One of our challenges as educators is to effectively communicate the power of mathematics to students, many of whom have little interest in mathematics beyond its ability to meet a graduation requirement. Many applications are either artificial (“Train A approaches Train B …”) or use tools that are beyond the scope of a general education course. Attempting to bridge this gap, a group of mathematicians have been developing course materials for use in entry‐level and general education courses that teach mathematics through social justice applications. Intriguing, accessible mathematics can be applied to understand and attempt to remedy compelling social issues. In this short course, we will examine some of these materials and discuss how I and others have used them in class. Attendees will also be invited to join the community working on these materials.

Critical Leaf Configurations for σ3, Debra L Mimbs, Univeristy of Alabama at Birmingham Abstract: A lamination L of the unit disk D2 is a closed collection of chords of D2 that intersect, if at all, in an endpoint of each on the boundary circle S1. The chords in L are called leaves. We parameterize S1 1 1 by [0, 1) in the natural way. Consider the map σ3 : S → S defined by σ3 (t) = 3t (mod 1). The map σ3 can

be extended to leaves linearly and continuously on UL. A leaf of L is a critical leaf if the images of its endpoints are the same point. Under the map σ3 there are seven basic configurations for critical leaves with respect to the fixed points 0 and 1/2. I will investigate the general structure of laminations invariant under σ3 resulting from each type of basic configuration. Further, I will explore the leaves which are forced or permitted by each basic configuration.

Inferring Linguistic Leadership Structure, Garrett Mitchener, College of Charleston Abstract: Sociolinguistics suggests that language change is driven by a leading subset of a community. I consider the problem of inferring the size of this leadership core from historical linguistics data. The underlying model is a Markov chain in which each individual has a factor b less influence than the next most influential individual on children as they learn language. The available data, collected by Ellegard, comes from a study of a change in English syntax. Using a Monte Carlo approach, it is possible to estimate the likelihood of the data for various values of b. A maximum‐likelihood estimate for b places it at 0.85 or less. That value suggests that the 20 most influential people account for 95% of the total influence.

Two Theorems of Zellbergers, Shatina Morgan, Winston Salem State University Abstract: Mills, Robbins and Rumsey conjectured and Zeilberger proved the formula known as the alternating sign matrix conjecture which finds the total number of n by n alternating sign matrices. Zeilberger also proved after the work of Izergin, Korepin, and Kuperberg that the set of alternating sign matrices can be partitioned into n subsets with a known cardinality. We examine these two theorems of Zellbergers and work with different expressions of the results and look at some of the mathematics used to prove them.

Hyperbolic Functions, Daudi Muhamed, South Carolina State University Abstract: This paper analysis to the introduction of Hyperbolic Functions and how integration and derivative drive from these functions. I have defined the identities involving in hyperbolic function, and how to find derivative and integral of hyperbolic function through differentiation and derivative of inverse of hyperbolic functions. The domain and range of some hyperbolic functions are also defined. There is analysis on relationship between inverse of hyperbolic function and logarithmic function. In summary, this paper will emphasize the importance of Hyperbolic function and their application in Calculus.

Mathematics Exams in the CLEP Program: Their Relevance to the College Curriculum, Robin O'Callaghan, The College Board Abstract: This paper will discuss the test development process for the four CLEP mathematics exams (College Algebra, College Mathematics, Precalculus, and Calculus). The presenter will describe the curriculum surveys that are designed to keep the exams relevant to current classroom practices, the setting of test specifications and standards, and the work of college faculty committees in guiding and reviewing the assembly of the exams. A brief discussion of the role of the online calculator in the Precalculus exam will also be included.

Refocusing College Algebra for Student Success, Zephyrinus C Okonkwo, Albany State University Abstract: At many colleges and universities, the College Algebra course attracts students who intend to major in the social science disciplines, nursing, business, and education. Many of these students are under‐prepared for the rigors of college education. National data indicates high failure rate in the College Algebra course. This paper presents action steps which have been put in place to enhance

student learning and student success in the College Algebra course at Albany State University.

Learning Geometry in the Dance Studio, Jason Parsley, Christina Soriano, Wake Forest University Abstract: What's that math class doing in the dance studio? We, a dance professor and a math professor, brought together each of our introductory classes to study geometric objects (Platonic solids and ellipses) through human forms. Come see how they realized a dual octahedron lying inside a cube. We investigate how this physical learning environment molded our students' spatial reasoning and analyze the effects of this cross‐disciplinary pedagogical exercise.

Topology and Prime Numbers: Are You Kidding?, Andrew Penland, Western Carolina University Abstract: The infinitude of prime numbers has been known to the mathematical community since the time of Euclid, but it is still a delight to see another proof of the fact. Hillel Furstenburg's 20th century proof is notable for its creative use of topology in approaching the topic. This presentation will explain the methods and significance of Furstenburg's proof, and also serve as an introduction to some basic ideas in topology.

Guiding critical leaves for pullback laminations of the unit disk with irrational rotation gaps, Ross Ptacek, William Bond, University of Alabama at Birmingham Abstract: A lamination L of the unit disk D2 is a closed collection of closed chords of D2 that may only intersect at endpoints. Each chord is referred to as a leaf. For a lamination to be invariant leaves must 1 1 1 be invariant under the d‐tupling map, σd. For points on S , parameterized by [0, 1), σd : S → S defined by σd (t) = dt (mod 1). This map may also be extended continuously to leaves. A gap of L is is the closure of a component of D2 \ UL. In particular an irrational rotation gap is a gap whose boundary is a rotational Cantor set. Gaps too must be invariant in some sense under σd. A critical leaf is a leaf whose endpoints both map to the same point under σd.

Pullback laminations are a certain type of invariant lamination in which a collection of leaves are used to generate the lamination by taking successive pre‐images of those leaves. Since σd is d‐to‐one, there are d pre‐images of each endpoint of a leaf. Critical leaves can be used to guide the pullback by determining which endpoint pre‐images may be connected to form a new leaf by stipulating that these leaves may not cross the critical leaves.

In this talk, we will investigate the effect that choice of guiding critical leaves has on the resulting pullback laminations of irrational rotational gaps. In particular, we want to discuss what choices of guiding critical leaves result in a lamination with only the gap and its pre‐images.

Kinematics and Dynamics of the Straight Lead, Blake L Queen, Western Carolina University, Jeffrey K Lawson, Western Carolina University Abstract: The straight lead, or lead jab, is an offensive technique that is the foundation of many martial arts and combat sports. Although a relatively simple maneuver, its intricacies are often overlooked or misunderstood. We provide a working mathematical model of the kinematics and dynamics associated with the generalized movements of the straight lead in two dimensions (the “Rock ‘em Sock ‘em” model). The model consists of an unconstrained system of linked rigid bodies with uniaxial (SO(2)) Lie groups representing the joints. We will derive the equations of motion, solve numerically the resulting boundary value problem, and simulate the solution. We also mention how to extend the planar case to three dimensions as well as how to include constraints.

Active Learning Strategies for the College Liberal Arts Mathematics Course, Nell Rayburn, Austin Peay State University Abstract: We will model the active learning instructional strategies which we employ in our “liberal arts mathematics” course at Austin Peay State University. Each semester the course consists of the instructor’s choice of two modules. Currently available modules are mathematics of politics, sound and music, cryptanalysis, and art. Each of the modules is organized around the application, and the relevant mathematics is studied as it naturally arises. Some of the mathematical concepts which are involved are modeling, geometry (similarity, scale, isometries, projections), elementary group theory, impossibility theorems, trigonometric and logarithmic functions, and problem solving strategies.

Numeration Systems Based on a Real Number, Alan Reece, Samford University Abstract: This presentation discusses the study of β‐ expansions, a generalization that considers a numeration system with a real number base. We will write a positive real number x in an infinite β‐ representation as follows: where each xi is an element of the alphabet 0,1, … , . In this case a number may have several different β‐ re presentations. A particular such β‐ representation, playing an important role, is obtained using a greedy algorithm, and is called the β‐ expansion of x. If we let 1 1 √5, 2 the golden ratio, then many interesting properties come out when we work with τ‐ expansions of real numbers. We find that the τ‐ expansion of x > 0 is an infinite word s using the alphabet {0,1} so that the word 11 does not appear in the word s. Any positive number that can be written as a finite sum of distinct powers of τ is called a τ‐ rational number. The above expansion shows that the positive τ‐ rational numbers are dense in 0, ∞. If the powers are non‐negative then the numbers are called τ‐ integers; we include 0 as a τ‐integer. We will show that the difference between consecutive τ‐ integers is either 1 or . The sequence of non‐ negative τ‐ integers gives rise to a sequence of elements of 1, by simply writing the differences between successive terms. If we identify 1 with a and with b, then the sequence of successive differences will be shown to be the well‐known Fibonacci word f = abaababaabaa. This provides an effective way of computing τ‐ integers. τ‐ integers come up in the study of diffraction patterns in quasicrystals.

Developing a Website for Mathematics History in Calculus 3 and Differential Geometry (Preliminary Report), Gregory Rhoads, Sarah Greenwald, Appalachian State University Abstract: The authors are developing a searchable website of new and existing curricular materials in the form of formatted activities for incorporating history in Calculus 3 and Differential Geometry. Users will be able to search via a standard syllabus for these courses, specific mathematical topics, or using keywords (i.e. women). The database will contain activities of various lengths with specific goals, instructions on how to incorporate the history in the course, and assessment strategies.

My Experiences on Teaching PEMDAS in Pre‐Algebra Classes, Shumei Richman, Columbia College Abstract: After teaching college algebra classes for more than ten years, this semester for the first time I have the chance to teach pre‐algebra arithmetic classes, in which PEMADAS is the focus in teaching order of operations. Seeing students numerous mistakes, I have the following three questions: (1). Is it good for students to always use PEMDAS in simplifying an arithmetic expression with mixed

operations? (2). What is the impact of the memory device PEMDAS in learning algebra later? (3). Is there anything else more important than PEMDAS to teach in these arithmetic classes prior algebra? In this presentation, we will discuss the above three questions, beginning with the analysis of the students’ mistakes.

Arrow's Impossibility Theorem, Brevin S Rock, University of North Carolina Wilmington Abstract: Arrow’s Impossibility Theorem is notably one of the most significant controversial theories ever posed to Economists. Published by Kenneth Arrow in the book Social Choice and Individual Values in 1951 when he was achieving his PhD. The theorem states that for a voting system to be fair that certain criterion must be met. The fairness criteria consist of: non‐dictatorship, unrestricted domain, independence of irrelevant alternatives, monotocity, and non‐imposition. The proof leads to the conclusion that it is impossible for all the criteria to be satisfied in ranked voting systems.

A Constraint Language Approach to Constructing Certain Cyclic 2‐class Association Schemes, George Rudolph, The Citadel Abstract: Consider cyclic partially‐balanced incomplete block designs with 2 associate classes, denoted {PBIBD(v = 4t+1; 3; 2; 1) : t > 0; v prime}, with distinct indices 1 and 2. Embedded within the problem of generating designs is the NP‐hard problem of generating 2‐class association schemes for those designs. We propose a finite domain constraint algorithm for constructing these association schemes and explore their properties, with a view toward generating very long sequences, and ultimately very large designs. The algorithm uses v‐length periodic autocorrelation sequences to construct valid association schemes. We derive this property directly from algebraic manipulation of association scheme matrices. We conjecture, based on empirical evidence, that for every value v, there are exactly two (complementary) schemes. If true, this conjecture is of computational interest, because it implies that as v grows, the number of solutions stays constant, therefore it is the number of non‐solutions that grows combinatorially. We further conjecture that techniques from parameterized complexity may lead to better‐performing algorithms for this particular problem, following from the first conjecture.

Computers and Calculators Are Not Always Right, Iason Rusodimos, Barrett Walls, Georgia Perimeter College Abstract: Calculators and Computers are invaluable tools in Mathematics but can not solve every problem flawlessly. Our talk discusses several calculations which are problematic with current technology. Our talk also discusses ways to identify when problems occur and how to find the correct solutions, primarily with Mathematica but also other kinds of technology.

Are Solitary Waves Color Blind to Noise?, Herman Russell, University of North Carolina at Wilmington Abstract: Solitary waves are traveling wave solutions of partial differential equations. We know how to simulate the evolution of these solutions. How is the evolution of special solutions, such as solitons and solitary waves, affected by colored noise? We review the known behavior of stochastic solitons under white noise and describe preliminary results for colored, or exponential, noise.

Evolution of kleptoparasitic behavior, Jan Rychtar, University of North Carolina Greensboro, Mark Broom, University of Sussex, England, Christian Sykes, University of North Carolina Greensboro Abstract: Kleptoparasitism, the stealing of food items, is a common biological phenomenon which has been modeled mathematically in a series of recent papers. In this talk we consider the evolution of a population under adaptive dynamics. We show that under some conditions on strategies, mixed strategies can be stable. On the other hand, if there are no restrictions on the strategy set, no mixed strategy is stable; and the dynamics has only two stable attractors ‐ the kleptoparasitic strategies Hawk

(always steal, always defend) and Marauder (always steal, never defend). Moreover, for some parameter values and with proper initial conditions, the population evolves in cycles without any spatial or temporal patterns; each such a cycle involves unpredictably long periods of kleptoparasitism free polymorphic mixtures; each such a period ends by an invasion of a single (unpredictable) kleptoparasitic strategy after which the population becomes monomorphic and evolves back to the beginning of the cycle.

Mathematics in the Movies and Television, Hugh Sanders, Georgia College & State University Abstract: Mathematics has played a role in movies and television from their early days through the present. Sometimes this has been a prominent role and other times merely as a side issue. Many times computations have been accurately portrayed while many real mistakes have been shown as well as some nonsensical math has appeared. This session will give a sampling of these and give ideas about their use in the classroom.

Teaching Introductory Statistics Using Course Compass, Joel Sanqui, Appalachian State University Abstract: This semester, several statistics instructors at Appalachian State University started using Course Compass in our Introduction to Statistics courses. In this talk we will discuss the main features of Course Compass, an online dynamic teaching and learning system that allows easy integration of textbook materials and instructor generated materials. We will also present some preliminary data on the effectiveness of using this tool in teaching and learning basic statistics. This talk is accessible to anyone teaching or wishing to teach college or high school level Introductory Statistics.

Reinforce rigor in College Geometry with Sketchpad, Cabri, or Cinderella, Subhash C. Saxena, Coastal Carolina University Abstract: Rigor and usage of technology in College geometry are not mutually exclusive. Cleverly designed continuous motion on figures of Sketchpad, Cabri, or Cinderella can produce convincing confirmation of theorems, leaving lasting impressions. Such practices produce ‘Aha’ experiences. This presentation will deal with the strength of each of these three technologies in instruction of geometry at the college level.

Using the Great Wall of China Myth in a Trigonometry Course, George E. Schnibben, Francis Marion University Abstract: The Great Wall of China Myth states that the Great Wall is the only human made object that is visible from the moon. The first part of the talk discusses how old this myth is and the second part gives a trigonometric resolution (I hope) to it. This is a topic that I have used to generate student interest in presenting trigonometric ideas.

A Measuring Device Used in the Leather Tanning Industry and Riemann Sums, George E. Schnibben, Francis Marion University Abstract: On the television series “Dirty Jobs” the host, Mike Rowe, visited a leather tanning factory. In the course of the show he was taken to a machine that measures the area of the hide. It was obvious that the idea behind the machine was the Riemann Sum from calculus. This give a real world answer to a math student’s ever present question, “But what is this good for?” Using patents from two different machines, the mechanisms that calculate the areas are discussed.

The Qualified Quantifier: A Very Handy Logical Gadget, Damon Scott, Francis Marion University Abstract: The qualified quantifer is a very simple device and has been formalized before. Here, though, it is formalized with “first class” syntax and phrase structure. The new, improved version is

remarkably useful to any working mathematician. In small‐scale usage, a qualified quantifier is something like “For all real a and b with a < b, . . .”. In middle‐scale usage, it is the context a professor sets up in a room after saying “Let a and b be real numbers with a < b.” There is a complementary qualified quantifer for the “there‐exists” construction. In this talk, as time permits, we shall present the new formalization and give examples of its use making mathematical exposition cleaner both in the classroom and in one’s research. The phrase structure of expressions using the new device will be discussed, especially how it allows proper and formal incorporation of details without causing the rest of the expression to gain a nest level, again making for more user‐friendly exposition. A bit of calculus for the qualified quantifiers will be shown, together with “how to prove it” rules for mathematical statements that contain them. Finally, the two qualified quantifiers, together with negation, form a sufficient language for first‐order logic (!). One never had any idea how much logical power one was tapping into when saying such things as “Let a and b be real numbers with a < b” in a lecture.

Computational Science Coursework and Internships: Applying Mathematics and Computer Science to Important Scientific Problems, Angela Shiflet, Wofford College Abstract: Many significant scientific research questions are interdisciplinary in nature, involving biological and/or physical sciences, mathematics, and computer science in an area called "computational science"; and much scientific investigation now involves computing as well as theory and experiment. Consequently, a critical need exists for scientists to know how to use computation in their work. With an appropriate foundation in mathematics and computer science, science majors can perform meaningful interdisciplinary research in internships, graduate school, and post‐graduate positions. Internships involving computation in the sciences can expose undergraduates to many new ideas, techniques, and applications that can greatly enhance their knowledge, make their classroom education more meaningful, involve them in research on significant scientific problems, and expand their opportunities. Working at various laboratories, students have applied techniques of modeling and simulation to significant scientific problems, such as determining biochemical pathways associated with vascular disease, correlating birth defects to diet, discovering heart mechanics in order to treat cardiac disease, tracking asteroids, and developing strategies to combat Chagas’ disease. Besides citing particular student experiences, this talk will include coursework and internship recommendations from "Undergraduate Computational Science and Engineering Education," a report from a Society for Industrial and Applied Mathematics (SIAM) Working Group of which Dr. Shiflet is a member.

The Trochoid as a Precessed Ellipse, Andrew J. Simoson, King College cos sin Abstract: We show that trochoids are precessed ellipses. That is, let , sin sin be a 2×2 rotation matrix and , cos ,sinT be an ellipse with semi‐minor axial length a, where T is the transpose operator, then , , parametrizes a trochoid. As a nice application, we show that the path of a pebble falling freely through a rotating, homogeneous earth is a hypocycloid, and we demonstrate how any trochoid can be perceived as the path of a pebble falling through the earth.

Hurricane Evacuation: An analysis of evacuation models on the Houston / Galveston area, Laura Sinden, Elon University Abstract: This research analyzes different evacuation models to study the effect and impact of different variables in the models on the traffic flow and speed of an evacuation. The models that will be studied in this research are: steady‐state model, one‐dimensional cellular automata model, and the space‐speed curve. While studying these models, this research computes evacuation statistics for the

Harris and Galveston county area keeping in mind any suggestions for future evacuations.

A Baseball‐Themed Statistics Course, Neal Smith, Augusta State University Abstract: Abstract. With vast amounts of data readily available on the internet, the game of baseball makes excellent fodder for statistics courses. In this talk, we will outline an elementary statistics course where all of the topics are motivated by the study of baseball. We will give sample problems ranging from the most elementary concepts (descriptive statistics, probability distributions, linear regression) to some mildly advanced ideas (hypothesis tests, goodness‐of‐fit tests, Markov chains). We will also briefly mention where those who are interested in developing such a course may find resources (textbooks, other readings, and online baseball databases).

Central Limit Theorems for Distributions With Heavy Tails, Alicia Smith, Winston Salem State University, Evelyn J Patterson, Penn State Abstract: A central limit theorem for independent random variables with heavy tails is presented here by first obtaining a large random sample and then resampling (bootstrapping). It will be demonstrated that although the distribution of the sample means is not normal, the distribution of the trimmed means is. We will emphasize the use of SAS in obtaining and illustrating the distribution of the trimmed means.

Frames and Equiangular Lines, Jim Solazzo, Coastal Carolina University Abstract: In this talk we will explore the connections between graph theory and frame theory. In particular, we will discuss equiangular lines and two‐uniform frames as well as some applications.

Calculus without a Textbook?, Cornelius Stallmann, Augusta State University Abstract: Do your students really make use of that expensive (and heavy) calculus textbook? If so, do they use it for good or for evil purposes? Do you seek an alternative to the reliance on a textbook? I describe an approach to teaching calculus that doesn’t rely on a textbook. My approach is student centered, suited for small classes (less than about 25), and keeps students engaged even at 7:00 am. I will give a brief description of what I do and why I believe that it works. I will also give some guidelines to those contemplating a similar approach.

Should Normal Data Have Outliers?, David Steele, University of North Carolina at Asheville Abstract: After noticing that the boxplots of simulated normal data revealed several "outliers," I began to wonder about the relevance and applicability of the generally accepted notion of outliers. Should we taylor the outlier formula to generate a smaller number of expected normal outliers, or should we just take the concept of outlier less seriously?

On A = mP Heronian Triangles and V = mA Cones, David Stone, John Hawkins, Georgia Southern University Abstract: A Heronian triangle has integer sides and area. For a fixed m, we consider the problem of determining all such triangles with area = m (perimeter). We also consider the analogous problem of finding cones with integer radius and height satisfying volume = m (total surface area). Recent activity and results on both problems have appeared recently in the School Science and Mathematics Problem Section and in the Mathematics Magazine, where our conjecture that the largest A = mP triangle is 2 (4m2 + 2,() 4m2 +1 ,16m4 +12m2 +1) was settled affirmatively.

Jumping at the Chance, David Sumner, University of South Carolina Abstract: Several elegant and surprising examples of puzzles and games using elementary mathematics, but illustrating the important ideas of invariance and evaluation functions.

Solution Techniques for Optimal Control of Projectile Motion, David C. Szurley, Francis Marion University Abstract: Many problems may be formulated as optimal control problems. In these, we would like to control some aspect of a system in order to optimize an objective function. The motion of a projectile (ignoring air resistance) yields an example. We want to determine the angle at which to shoot the projectile for it to land a given distance away. Thus, there is a need to understand how to solve optimal control problems. We will consider three techniques to solve an optimal control problem involving projectile motion. We begin by considering projectile motion with one control, and will discuss sensitivity‐based optimization, which may be used to solve optimal control problems involving any number of controls. The advantages and disadvantages of each method will be presented.

Using the Limit Process and Numerical Sequences to Enhance Student Understanding of the Concept of the Derivative, Mohammed Talukder, Zephyrinus C Okonkwo, Albany State University Abstract: Student in‐depth understanding of the Definition of the Derivative lays a strong foundation for subsequent topics encountered in Calculus, Real Analysis, Numerical Analysis, Mathematical Statistics, and other mathematics courses. Action research indicates that lack of deep understanding of the use of the limit process to determine derivatives negatively impacts on student success in many mathematics courses. This paper presents innovative instructional and assessment techniques which have been used to increase student in‐depth understanding of the concept of Derivative.

A powerful test for testing order‐restricted hypothesis in longitudinal Data, M. Hanif Talukder, Albany State University Abstract: In many biomedical and health science research, we are interested in comparing treatment effects with an existing ordering. In this paper, we proposed a restricted LRT for detecting ordered treatment effects for correlated data. This test is powerful than conventional one‐sided t‐test for comparing two treatments. The cut‐off values were computed for test statistics by using simulation. Furthermore, we also proposed an asymptotic distribution of the LRT statistic for testing ordered treatment effects. Its null limiting distribution is shown to be mixture of chi‐squares distribution. A power comparison study with one‐sided t‐test also provided.

ON THE GALOIS MODULE STRUCTURE OF SQUARE CLASSES OF MAXIMAL ELEMENTARY ABELIAN, Adam Topaz, Davidson College Abstract: Let E/F be the maximal elementary abelian 2‐extension of a field F of characteristic not 2, x x2 with Galois group G, and let J = E /E be the F2G‐module of square classes of the multiplicative group of E. Denote by Jk the kth element in the socle series for J as an F2G ‐module. Adem, Gao,Karagueuzian, and Mináč determined necessary and sufficient conditions for the existence of an element in J2 in terms of the existence of elements in J1 and a class in H3(G, F2) expressed in terms of cup products and the G transgression map on H1(E, F2) . We produce a formula for Jk for all k ≥ 1.

Discovering the Curve‐Creating Black Box, Amy Valentine, Sarah Claiborne, Nicole Finuf, Megan Hamilton, Belmont University Abstract: The Belmont Undergraduate Research Student Team (BURST) is a group of undergraduate students who have been working on a research project for a class this semester. We are studying an equation that can be symbolized as a black box with functions as input and different curves as the

output. The focus of the study has been to input a ‘random’ function to see what the output may look like through a computer generated picture. The presentation will concentrate on some of the math and programming behind the project as well as our progress and goals.

Partially Orthogonal Sequences of Functions, William Ward, Samford University Abstract: We develop an economical method based on the classic Gram – Schmidt process for orthogonalizing certain sequences of functions. When considering polynomials on [‐1, 1] a classic orthogonal sequence is defined by 1 1, 2! which is the famous Rodrigues formula for the Legendre polynomials. We consider a modification of the Rodrigues formula which generates a sequence

1 of polynomials that spans the set of all polynomials that vanish at x = ‐1, 1 and for which Qk(x) is orthogonal to all other Ql(x) except l = k – 2 or k + 2. In general a set in a Hilbert space is partially orthogonal if every element of the set is orthogonal to all but a finite number of other elements in the set; thus the sequence {Qn(x)} is partially orthogonal. We consider a general set in a Hilbert space which is partially orthogonal, and show that in such a case the process of orthogonalization may be significantly truncated. When this method is applied to the sequence {Qn(x)}, the resulting orthogonal sequence can be used to approximate functions that vanish at x = ‐1, 1:We briefly describe other examples of partially orthogonal sequences of functions and their applications.

ELLIPTIC CURVE CRYPTOGRAPHY WITH THE TI‐83, Bill Yankosky, North Carolina Wesleyan College, Blake Rice Abstract: It is possible to implement the ElGamal cryptosystem using elliptic curves over finite fields, thereby creating the potential for an extremely secure public‐key cryptosystem. One common finite field used when generating elliptic curves is the group of units modulo p, Zn, where p is prime. Since it is very time consuming to perform the computations necessary to generate an elliptic curve by hand, even when p is small, it is common to use computers to do this work. Perhaps somewhat surprisingly, the algorithms necessary to perform the computations required to generate elliptic curves over finite fields can also easily be programmed into a TI‐83 graphing calculator. Furthermore, other TI‐83 programs can be written to aid in the implementation of basic elliptic curve cryptosystems.

On the 2‐adic digits of Bernoulli numbers of the second kind, Paul Young, College of Charleston Abstract: The Bernoulli numbers of the second kind play many roles in combinatorics and number theory. In 1997 A. Adelberg described a pattern of gaps in the digits of the 2‐adic expansions of these numbers. He proved a theorem describing the behavior of the first gap and observed a mysterious second gap by numerical computation. In this talk we’ll show how an expression for the Bernoulli numbers of the second kind involving traces of algebraic integers can explain these gaps. We will conclude with some other applications of this formula.

Counting Threshold Graphs with Young Diagrams, Laurie Zack, High Point University Abstract: Young diagrams and Young tableaux were first applied to the study of representations of the symmetric group, but have since been used to study different classical groups as well as graph theory, combinatorics, and even physics. Typically they are introduced in a representation theory class, however they have appealing applications suitable for earlier introduction. This talk will present one elementary application of how Young diagrams are can be used to count the number of threshold

graphs.

Introducing the new Academic Systems Algebra, a cutting‐edge mathematics solution from PLATO Learning, Mike Zak, PLATO Learning, Inc. Abstract: PLATO Learning is proud to introduce the new Academic Systems Algebra! This solution was designed for students entering college under‐prepared for college‐level math courses, providing comprehensive, self‐paced instruction that actively engages college students in learning and applying mathematics. AS Algebra leverages state‐of‐the‐art technology to deliver top‐quality math instruction in an interactive, engaging environment that addresses multiple learning styles and eases math phobia, allowing students to advance to credit courses more quickly while increasing overall pass rates, retention, and persistence. And it provides consistent instruction by faculty and adjuncts alike, while allowing for on‐campus, distance learning, or hybrid programs to accommodate your institution’s needs. I hope you will join me for a look at this exciting new program from PLATO Learning!

New iterative methods for solving nonlinear equations and multiple roots, Yilian Zhang, University of South Carolina, Aiken Abstract: Newton‐Raphson method is a well known effective procedure used for solving nonlinear equation in the form of f(x)= 0. Recently there has been some progress on Newton type methods with at least cubic convergence. The methods are based on the proposals of Abbasbandy on improving the order of accuracy of Newton–Raphson method [S. Abbasbandy, Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Applied Mathematics and Computation 145 (2003) 887–893]. This talk contains a brief survey of the methods and discussion of the performance when we have multiple roots. A similar method for multiple roots is also presented.