One of the nation's top undergraduate    engineering, science, and mathematics colleges About Rose-Math  |  Programs  |  Faculty/Staff  |  News  |  Facilities  |  Alumni |  Contact |  Site Index  |  Search

## The Rose Mathematics Seminar - History

 1994-95  |  1995-96  |  1996-97  |  1997-98  |  1998-99 1999-00  |  2000-01  |  2001-02  |  2002-03  |  2003-04 2004-05  |  2005-06  |  2006-07  |  2007-08  |  2008-09 2009-10  |  2010-11  |  2011-12  |  2012-13  |  2013-14

### Overview

This page give a history of the talks in the Rose Mathematics Seminar, which was started in 1994-95 under the name of the Applied Math seminar. Some years later it was expanded to include all mathematics with a suitable name change. The speakers, titles and abstracts are listed below with the later years first. The current year's schedule of talks in given on the current seminar page.

### 2012-13 (latest first)

• Topic:  Building a Better Runge-Kutta Method
• Speaker:  Dr. David Goulet
• Date:  15 May 2013
• Abstract:  Runge-Kutta methods are a common and effective means of numerically approximating the solutions to ordinary differential equations. The idea underlying these methods was spawned at the end of the 19th century, but high quality methods have emerged only in the past few decades. To build a high quality RK method, researchers have drawn on ideas from Calculus, Differential Equations, Graph Theory, Combinatorics, Complex Analysis, Geometry, Continuous Optimization, Control Theory, and other areas of mathematics and computer science. This talk will review basic RK methods, often taught in Calculus and DE classes, before delving into the study of more sophisticated methods. Many areas of mathematics will be applied to create methods which have been optimized for different classes of ODE problems. In this context, practical implementations, such as those used by Matlab and Mathematica, will be discussed. Throughout the talk, RK method design principles will be highlighted, culminating in the creation of pedagogical examples of simple but accurate methods with good stability properties, and built-in error controls. The talk will conclude with contemporary research directions in the design of RK methods, including parallelized methods, methods optimized for advection equation solvers, and generalizations known as Rosenbrock and W methods.

• Topic:  Graphs and DNA
• Speaker:  Larry Langley, University of the Pacific
• Date:  08 May 2013
• Abstract:  During his early work into microbiology in the 50s, Seymour Benzer examined the topology of DNA through mutant bacteriophages. The data gained through his experiments was consistent with a hypothesis that genetic material was linearly structured. The results and analysis of his experiments provide the foundation for much of the research in graph theory regarding interval graphs. This talk will look at Benzer's work, the corresponding theory of interval graphs as well as present some more recent work on generalizations of interval graphs.

• Topic:  A New Information-Splitting Image Analysis Technique
• Speaker:  Mark Inlow
• Date:  01 May 2013
• Abstract:  : Important insights into various brain diseases, Alzheimer’s for example, can be obtained by correlating changes in the brain with genetic information. Detecting such correlations is complicated by the size and complexity of the data. An MRI image of a subject’s brain may consist of over 200,000 picture elements and his/her genetic data may consist of 500,000 or more pieces of information (single nucleotide polymorphisms). For such data current brain image analysis methods based on Gaussian random field theory are inadequate for various reasons. We present research on new methods which are based on a simple geometric property of the t-statistic. Thus although preliminary results indicate these methods are superior to random field methods, the theory behind them is straightforward, requiring little beyond introductory probability and statistics.

• Topic:  Robust Analysis of Metabolic Pathways: Engineering, Biology, and Math
• Speaker:  Allen Holder
• Date:  27 Mar 2013
• Abstract:  We show how topics in engineering design can aid problems in the biological sciences, and in reverse, how the engineering fields can gain from the biological application. We particularly focus on robust optimization, which has been used in several engineering fields to support optimal designs in which parameters are uncertain. We review a couple of classic examples to highlight the central modeling themes. We then adapt the robust paradigm to a popular problem in computational biology called flux balance analysis (FBA). Previous FBA models have been linear or quadratic and have assumed a static relationship between a cell's environment and its growth rate. This assumption is doubtful, and we extend the static model to a robust counterpart that accounts for the inherit uncertainty in individual variation. The robust model advances traditional FBA's validity with regard to its scientific goals since it removes the menacing shortcoming of ignoring dynamics. The biological setting leads naturally to questions about if, and if so how, solutions to robust models converge to their static counterparts as uncertain parameters become certain. One of these results argues that static solutions are robust solutions if the variation is appropriately restricted. With regard to engineering designs, this means that optimal designs created under static conditions are indeed robust under some restricted set of parametric variation. Many of the engineering models are solved efficiently with modern second-order cone solvers. However, these solvers have been unsuccessful at solving the robust FBA models. The exact reason for this failure is unknown, and we are working to enhance the numerical stability of the optimizers. We will point to some of our suspicions about why the solvers have been unfaithful in the biological setting. If we are successful in rectifying the numerics, then the engineering applications will gain more trustworthy solvers. Thankfully, we can re-model the robust FBA problem to make use of a different solver, which has proven itself worthy of the computational task.

• Topic:  Pairs of Pants and the Congruence Laws of Geometry
• Speaker:  Allen Broughton
• Date:  30 Jan 2013
• Abstract:  Many of us know that a torus can be constructed by gluing together the opposite ends of a parallelogram. Different parallelograms yield geometrically different surfaces. For surfaces of higher genus, with more holes, the surface can be constructed by gluing together hexagons with six right angles (yes that can happen in hyperbolic geometry). Then all possible surfaces arise from the gluing of some set of hexagons. The hexagons are from a "pairs of pants" decomposition of the surface which is the big idea of this talk. Understanding the possible constructions depends on the following proposed Congruence Law in Hyperbolic Geometry: If two right-angled hexagons have three corresponding sides of equal length then they are congruent. The talk will explain all concepts from the ground up. The proposed congruence law will be related to the familiar side-side-side and side-angle side congruence theorems from high school geometry.

• Topic:  Network-based Quantitative Analysis of Crossword Puzzle Difficulty
• Speaker:  John McSweeney
• Date:  16 Jan 2013
• Abstract:  What distinguishes a crossword puzzle from a simple list of trivia questions is the interlocking nature of the answers in the grid -- one solution can promote further ones in a cascading fashion. To model this mathematically, we build a network object from a puzzle: answers in the puzzle are nodes in the network, and nodes are linked via an edge if the corresponding answers cross. Each node also has a state, "solved" or "unsolved", that depends dynamically on the states of its neighbors. Motivated by analogous issues which arise in epidemiological analyses of structured populations, we consider the following general questions: what features of the distribution of the difficulties of the clues, and of the structure of the crossword network, determine whether a puzzle can be fully (or nearly fully) solved? Are impediments to full solution typically due to puzzle structure or clue difficulty? I will present rigorous results for certain puzzles with a high degree of symmetry, as well as simulation-based analyses of "real-world" puzzles from the Sunday New York Times.

• Topic:  Challenging Problems in Computational Biochemistry
• Speaker:  Yosi Shibberu
• Date:  19 Dec 2012
• Abstract:  Biochemistry is a rich source of important computational problems that should be of interest to mathematicians, computer scientists and engineers. The dramatic drop in the cost of sequencing DNA as well as progress in several structural genomics initiatives have created many new and exciting opportunities. I will begin with a review of elementary concepts in biology and biochemistry and then describe recent progress reported in the literature on solving the grand challenge problem of biochemistry - the protein folding problem. Protein molecules are the workhorses of life. An efficient solution to the protein folding problem will dramatically improve our ability to identify the function of individual proteins and go a long way towards enabling us to design proteins with new functions. I conclude with an update of ongoing research conducted in collaboration with Mark Brandt in Chemistry and Biochemistry on characterizing structural changes believed to occur in the estrogen receptor, a protein that plays an important role in breast cancer.

• Topic:  Discrete Optimization Problems at NASA Langley Research Center
• Speaker:  Rex Kincaid, Visiting Scholar from College of William and Mary
• Date:  12 Sep 2012
• Abstract:  An overview of several discrete optimization problems of interest to NASA Langley Research Center will be presented. Applications include placement of actuators for noise control in turboprops, locating truss elements for vibration control in space structures, optimizing network metrics for air transportation, and scheduling runway configuration changes at airports.

### 2011-12 (latest first)

• Topic:  Synthetic Biology: Collaboration Required
• Speaker:  Ric Anthony, Applied Biology and Biomedical Engineering
• Date:  09 May 2012
• Abstract:  Synthetic biology is, essentially, the engineering of life. Ultimately, the discipline aims to utilize engineering principles and practices to rationally and systematically design and build novel biological systems. This emerging discipline has the potential to provide solutions to diverse problems in the areas of medicine, energy, materials, security and sustainability, among others. The purpose of this presentation is to explore some current applications of synthetic biology and to elucidate opportunities for interdisciplinary collaboration in this exciting new field.

• Topic:  Projective Planes and Graph Theory
• Speaker:  Gabriela Araujo-Pardo, National University of Mexico
• Date:  02 May 2012
• Abstract:  We will discuss relationships between two areas of discrete mathematics: finite geometry and graph theory. Specifically, we investigate properties of the projective planes and their impact on the solution of problems in extremal graph theory and graph colorings.

• Topic:  Who Painted this Painting - A Project in Image Registration?
• Speaker:  Allen Broughton
• Date:  25 Apr 2012
• Abstract:  An art collector possesses a "painting of unknown origin" which he suspects was painted by a "very famous painter". The only possible evidence is a portion of the painting in an old photograph taken in the painter's studio. Is there a way to match the two, such as matching fingerprints? A proposed technique is to match the portion to a modern image of the painting. Since the the two images are taken by two different cameras, you just can't compare the two photos. This problem is called image registration which is an optimization problem in multi-dimensional camera orientation space. This problem will be discussed in two parts. First we set up the problem mathematically in mathematical terms. In a later talk we discuss the computer implementation of the comparison.

• Topic:  What's a data algebra and how do you build one?"
• Speaker:  Gary Sherman, Algebraix Data
• Date:  18 Apr 2012
• Abstract:  Ask n people the question "What's data?" and the cardinality of the set of responses is better approximated by n than by one. Any self respecting mathematician is puzzled by this --- denizens of data-world, not so much. Indeed, ever since E. F. Codd's 1970 paper, A Relational Model of Data for Large Shared Data Banks (Comm. ACM, Vol. 13, No. 6, pp. 377-387) gave rise to the Relational Data Model (RDM), the data-world's solution to this congenital ambiguity has been to exacerbate it by conflating the data, whatever it is, with some prejudicial visual artifice (tables in the case of the RDM); i.e., by confusing the message with the paper it's written on --- so to speak. What is worse, each new artifice comes equipped with a brief, supposedly-mathematical incantation to justify the trip down a new rabbit hole. This talk discusses Algebraic Data Corporation's approach to knowing data in the context of Zemelo-Frankel set theory, the foundation for all modern mathematics and, therefore, the only legitimate incantation to use when invoking the good name of mathematics. Indeed, our incantation births a rigorous notion of data algebra in plain sight of the RDM and its mongrel spawn, Structured Query Language (SQL).

• Topic:  Domination Densities of Stacks and Generalized Stacks ?
• Speaker:  Jake Wildstrom, University of Louisville Mathematics
• Date:  11 Apr 2012
• Abstract:  The Cartesian product of a graph G and a large path Pn can be thought of as a "stack" of copies of G. The domination numbers of such stacks are asymptotically linear in n, and the coefficient of linearity can be thought of as the "density" of the domination. This parameter will be explored in this talk, with bounds developed in terms of the traditional and total domination numbers, with emphasis on calculating the domination densities of specific graphs and bounding the densities of graph Mycielskians. A generalized variant of the stack domination density will also be presented, where the underlying stack topology is nonlinear.

• Topic:  A Mathematical Model for the Baking Process – A Phenomenological Approach
• Speaker:  Andrew Harris, Rose Student, Rose Mathemtics Major
• Date:  28 Mar 2012
• Abstract:  In this talk, I will present a mathematical model for the baking process of a cake and/or bread. The model is based on basic physical principles including diffusion, elasticity, and thermodynamics. I will explain the modeling process from these first principles to partial differential equations. The final model then consists of a coupled system of seven nonlinear partial differential equations that specify the temperature, moisture content, vapor content, pressure, and deformation in the dough. This is solved numerically to produce a reasonable representation of the baking process.

• Topic:  Word Graphs
• Speaker:  Tanya Jajcay
• Date:  21 Mar 2012
• Abstract:  In the talk, we will introduce graphs (automata) that are closely related to inverse semigroups (as Cayley graphs are related to groups). Inverse semigroups can be viewed as natural generalization of groups: As every group can be represented via permutations (one-to-one transformations), inverse semigroups can be represented via partial one-to-one transformations. We will consider actions of special groups on the graph related to an inverse semi-group, which will allow us to answer some structural questions about the original semigroup. In specific cases, automata are in general infinite, but they contain a very strong finiteness - finite core - in the sense that all important information about the automaton is encoded in a "finite" subgraph. This will allow us to answer some algorithmic questions (the Word Problem, for instance) as well. We will also address the question of languages recognized by these automata.

• Topic:  Singular Perturbations and Reductive Asymptotics via the Renormalization Group - Part II
• Speaker:  Vin Isaia
• Date:  14 Mar 2012
• Abstract:  Forced oscillators (like those describing a musical instrument: simple versions are the Van Der Pol oscillator and Duffing's equation) and fluid boundary interactions (e.g. Stokes-Oseen) give rise to ODEs (and PDEs) which exhibit behavior that may require care in establishing. Such problems share the distinction of having a regularization parameter epsilon which tends to be small in practice, and interesting behavior can arise in the limit as epsilon approaches zero. This talk will show how RG ideas sidestep the inherent difficulties of approximating solutions to these problems with a systematic approach rather than ad hoc methods. Two examples with boundary layers and a WKB problem will all be handled by the same RG approach, and it will capture a transcendentally small term ("beyond all orders"), correctly generate an asymptotic expansion without a priori information about the form of the solution and derive a WKB approximation. The viewpoint will then shift in an attempt to completely bypass the solution, so that the correct asymptotic expansion can be generated directly from the ODE itself, which tends to be a much simpler process. The result is a proof of asymptotic validity of the RG expansion for sufficiently general ODEs on infinite domains, which will make use of the notion of almost periodic functions.

• Topic:  Optimization Problems in Computational Biology
• Speaker:  Allen Holder
• Date:  07 Mar 2012
• Abstract:  Computational Biology is a burgeoning field of study that applies mathematics and computer science to answer questions in biology. We will discuss recent work on two problems. 1) One of the primary studies in computational biology lies with identifying protein structure and function, and one way to assess a protein's function is to associate it with similar proteins whose function is already known. We develop a model for pairwise comparisons using fundamental topics in linear algebra. We show that we can solve our optimization model in polynomial time with a single application of dynamic programming. Recent computational results show that our mathematical model is stable with regards to data perturbations due to experimental errors and/or protein dynamics. 2) Another common study in computational science is the study of whole-cell network interactions. We will discuss the metabolic network and the use of flux balance analysis (FBA) to establish a cell's metabolic state. The traditional linear models have been successful at identifying such things as lethal gene knockouts. The linear model assumes that every cell's metabolism is similar to the average metabolism, which is a questionable assumption. We will re-model the problem stochastically and show that the new model can be solved efficiently as a second order cone problem.

• Topic:  Wheel of Misfortune
• Speaker:  Nicole Burton, Grange Insurance Company
• Date:  06 Feb 2012
• Abstract:  The "Wheel of Misfortune" is a metaphor for the risks that individuals experience through the course of their daily lives. Just by driving a car or owning a home, people are exposed to "pure risk" and may take action such as purchasing insurance to eliminate this risk. We will simulate the act of buying and selling insurance to avoid spinning this metaphorical "wheel of misfortune" and learn why it is so important for insurance companies to price their insurance products accurately. I will also describe typical projects for a property and casualty actuary and why students might be interested in a career as an actuary

• Topic:  Singular Perturbations and Reductive Asymptotics via the Renormalization Group - Part I
• Speaker:  Vin Isaia
• Date:  01 Feb 2012
• Abstract:  Forced oscillators (like those describing a musical instrument: simple versions are the Van Der Pol oscillator and Duffing's equation) and fluid boundary interactions (e.g. Stokes-Oseen) give rise to ODEs (and PDEs) which exhibit behavior that may require care in establishing. Such problems share the distinction of having a regularization parameter epsilon which tends to be small in practice, and interesting behavior can arise in the limit as epsilon approaches zero. This talk will show how RG ideas sidestep the inherent difficulties of approximating solutions to these problems with a systematic approach rather than ad hoc methods. Two examples with boundary layers and a WKB problem will all be handled by the same RG approach, and it will capture a transcendentally small term ("beyond all orders"), correctly generate an asymptotic expansion without a priori information about the form of the solution and derive a WKB approximation. The viewpoint will then shift in an attempt to completely bypass the solution, so that the correct asymptotic expansion can be generated directly from the ODE itself, which tends to be a much simpler process. The result is a proof of asymptotic validity of the RG expansion for sufficiently general ODEs on infinite domains, which will make use of the notion of almost periodic functions.

• Topic:
• Speaker:  Mark Ward, Purdue University, Statistics
• Date:  25 Jan 2012
• Abstract:  We survey some of the ways that symbolic methods can be used for counting the number of large objects of various types, or for finding the average, variance, distribution, etc., of large randomly generated objects. Examples include integer compositions and partitions, set partitions, permutations, sequences, trees, words, etc. We discuss the basics of probabilistic, combinatorial, and analytic techniques for the analysis of algorithms and data structures. All of the discussion will be given from a very elementary level. Anybody who understands Taylor series will be able to comprehend the discussion. The talk can be viewed as an invitation to learn about the methods of Analytic Combinatorics, as in Philippe Flajolet and Robert Sedgewick's 2009 book.

• Topic:  Introduction to the Mathematics of Optical Tomography
• Speaker:  Joe Eichholz
• Date:  18 Jan 2012
• Abstract:  Optical tomography (OT) is an emerging class of biomedical imaging techniques in which volumetric information about the target object is computed from measurements of light emitted and scattered from the object. This class of modalities offers several benefits over other techniques; they are regarded as safe for the patient, have the potential for very high resolution, have potential for functional imaging, and are (relatively) portable and inexpensive. This talk provides a high level overview of OT and a description of the mathematics and computational science that arise when recovering volumetric information from measurements of emitted light. Topics include inverse problems, high performance computing, partial differential equations, and optimization. Details of some recent work with undergraduates may be presented as time permits.

• Topic:  Holditch's Theorem, Ambiguous Bicycle Tracks and Floating Bodies
• Speaker:  David Finn
• Date:  11 Jan 2012
• Abstract:  A curious result of Rev. Hamnet Holditch, the President of Caius College, which was published in 1858 in the Quarterly Journal of Pure and Applied Mathematics (a one page paper), states that if a chord of a closed curve is divided into two parts of length a and b respectively, the difference of the area of the closed curve and of the locus of the dividing point (as the chord is moved along the given closed curve) will be .

• Topic:  From Chaos to Ellipse: Using Eigenvalues and Eigenvectors to Explain Phenomenon
• Speaker:  Charles Van Loan, Cornell University - SIAM Distinguished Lecturer
• Date:  14 Dec 2011
• Abstract:  Let P(0) be a given random polygon in the plane. Let P(k+1) be obtained from P(k) by connecting P(k)'s side midpoints in order and then normalizing the vertex vectors x and y so that they each have unit 2-norm length . Why is it that the vertices of P(k) converge to an ellipse having a 45-degree tilt? A simple eigenvalue/singular value analysis explains it all and carries with it a story about computational science and engineering and its connection to mathematics. RHIT Math Department Presents DISTINGUISHED LECTURER Charles Van Loan, Cornell University Wednesday, November 9, 2011 10th period – 4:20 p.m. – 5:10 p.m., Rm. E-104 TITLE: "From Chaos to Ellipse: Using Eigenvalues and Eigenvectors to Explain Phenomenon" Abstract: Let P(0) be a given random polygon in the plane. Let P(k+1) be obtained from P(k) by connecting P(k)'s side midpoints in order and then normalizing the vertex vectors x and y so that they each have unit 2-norm length . Why is it that the vertices of P(k) converge to an ellipse having a 45-degree tilt? A simple eigenvalue/singular value analysis explains it all and carries with it a story about computational science and engineering and its connection to mathematics. Bio: Charles Van Loan, Professor of Computer Science and John C. Ford Professor of Engineering at Cornell University and a leading authority on computational matrix algebra, is a Society for Industrial and Applied Mathematics Visiting Lecturer this year. He is co-author of the well-known reference "Matrix Computations" (with Gene Golub). Dr. Van Loan will be visiting Rose-Hulman on Wednesday November 9th. He'll be on campus much of the day to speak with those who have interests in computing science and engineering and to consult on a soon-to-be-proposed major in Computational Science at Rose-Hulman. He will be delivering a lecture during 10th hour in E-104. The subject of his talk became a paper in SIAM Review but is accessible to those with a basic matrix algebra background. In addition, he states that "the topic is also a metaphor for computational science and engineering plus it speaks to the teaching/research connection." Please contact Jeff Leader, Professor of Mathematics, at leader@rose-hulman.edu if you have any questions.

• Topic:  Mathematical Modeling of Ocular Blood Flow and Its Relation to Glaucoma
• Speaker:  Giovanna Guidoboni, IUPU Mathematics
• Date:  07 Dec 2011
• Abstract:  Glaucoma is a disease in which the optic nerve is damaged, leading to progressive, irreversible loss of vision. Glaucoma is the second leading cause of blindness worldwide, and yet the mechanisms underlying its occurrence remain elusive. Elevated intraocular pressure (IOP) remains the current focus of therapy, but unfortunately many glaucoma patients continue to experience disease progression despite lowered IOP, even to target levels. Clinical observations show that alterations in ocular blood flow play a very important role in the progression of glaucoma. Significant correlations have been found between impaired vascular function and optic nerve damage, but the mechanisms giving rise to these correlations are still unknown. This talk will present some recent results of a mathematical study aimed at investigating the bio-mechanical connections between vascular function and optic nerve damage, in order to gain a better understanding of the risk factors that may be responsible for glaucoma onset and progression. In particular, two mathematical models will be discussed. The first entails partial differential equations for solid and fluid mechanics and aims at modeling the interaction between lamina cribrosa and central retinal artery, the main vessel nourishing the retina. The second consists of a system of nonlinear ordinary differential equations which represents the whole retinal circulation.

• Topic:  From Chaos to Ellipse: Using Eigenvalues and Eigenvectors to Explain Phenomenon
• Speaker:  Charles Van Loan, Cornell University - SIAM Distinguished Lecturer
• Date:  9 Nov 2011
• Abstract:  Let P(0) be a given random polygon in the plane. Let P(k+1) be obtained from P(k) by connecting P(k)'s side midpoints in order and then normalizing the vertex vectors x and y so that they each have unit 2-norm length . Why is it that the vertices of P(k) converge to an ellipse having a 45-degree tilt? A simple eigenvalue/singular value analysis explains it all and carries with it a story about computational science and engineering and its connection to mathematics. Bio: Charles Van Loan, Professor of Computer Science and John C. Ford Professor of Engineering at Cornell University and a leading authority on computational matrix algebra, is a Society for Industrial and Applied Mathematics Visiting Lecturer this year. He is co-author of the well-known reference "Matrix Computations" (with Gene Golub). Dr. Van Loan will be visiting Rose-Hulman on Wednesday November 9th. He'll be on campus much of the day to speak with those who have interests in computing science and engineering and to consult on a soon-to-be-proposed major in Computational Science at Rose-Hulman. He will be delivering a lecture during 10th hour in E-104. The subject of his talk became a paper in SIAM Review but is accessible to those with a basic matrix algebra background. In addition, he states that "the topic is also a metaphor for computational science and engineering plus it speaks to the teaching/research connection." Please contact Jeff Leader, Professor of Mathematics, at leader@rose-hulman.edu if you have any questions.

• Topic:  "Programming in a world of GPUs and pervasive parallelism
• Speaker:  Eric Holk, Ph.D. student - Indiana University
• Date:  09 Nov 2011
• Abstract:  Today's mid-range gaming PC easily outperforms the world's fastest supercomputer from 15 years ago, and yet high performance parallel programming is as much a black art as ever. Over the past decade, CPU clock speeds have not increased at the rate they once did. Increases in computing power now come from increasing parallelism. This trend is seen in the proliferation of multicore CPUs and the use of graphics processors for a broader variety of tasks. The majority of programmers remain ill-equipped to effectively use multiple processor cores. As parallelism becomes pervasive, parallel thinking must become the norm rather than the exception. Programming languages and algorithms must evolve to incorporate parallelism as a fundamental feature. I'll be discussing the trends in programming language research that support ubiquitous parallelism, as well as some of the changes that will be needed to design effective parallel algorithms. Bio: Eric Holk is a Ph.D. student studying programming languages for parallel computing at Indiana University. He graduated from Rose-Hulman in 2006 with a B.S. in computer science and mathematics.

• Topic:  Perturbation Methods: When Do Small Parameters Have Large Consequences? - Part II
• Speaker:  David Goulet
• Date:  02 Nov 2011
• Abstract:  This is the second part of my talk on applied asymptotic methods, including discussions of the following concepts. Boundary Layers How shock waves propagate and why peeing in your wetsuit is more than just fun. Multi-Scaling When adding more time dimensions to your problem actually makes it easier to solve. Homogenization How fine structure determines bulk behavior, e.g., what the Riemann-Lebesgue lemma says about waves traveling through chaotic environments. Overview: Mathematical models sometimes include one parameter whose size is dramatically different than the others. Small parameters often arise in models governing systems with dramatically different spacial or time scales. In these cases, the ratio of scales can be very small. Typical situations include chemical reactions with rate limiting steps, composite materials with a fine microstructure, and the rapid flow of low-viscosity fluids. In models with a small parameter, it is tempting to set the parameter to zero, with the hope of obtaining a rough approximation to reality. This talk will include examples where setting the small parameter to zero gives disastrous results. Several methods of correcting this mistake will be presented.

• Topic:  Perturbation Methods: When Do Small Parameters Have Large Consequences? - Part I
• Speaker:  David Goulet
• Date:  26 Oct 2011
• Abstract:  This two part talk will be a smorgasbord of applied asymptotic methods, including discussions of the following concepts. Boundary Layers How shock waves propagate and why peeing in your wetsuit is more than just fun. Multi-Scaling When adding more time dimensions to your problem actually makes it easier to solve. Homogenization How fine structure determines bulk behavior, e.g., what the Riemann-Lebesgue lemma says about waves traveling through chaotic environments. Overview: Mathematical models sometimes include one parameter whose size is dramatically different than the others. Small parameters often arise in models governing systems with dramatically different spacial or time scales. In these cases, the ratio of scales can be very small. Typical situations include chemical reactions with rate limiting steps, composite materials with a fine microstructure, and the rapid flow of low-viscosity fluids. In models with a small parameter, it is tempting to set the parameter to zero, with the hope of obtaining a rough approximation to reality. This talk will include examples where setting the small parameter to zero gives disastrous results. Several methods of correcting this mistake will be presented.

• Topic:  Balanced Matching: A Generalized Approach for Causal Inference
• Speaker:  Jason Sauppe, University of Illinois
• Date:  19 Oct 2011
• Abstract:  Causal inference is applied in a wide range of scientific disciplines in order to determine the effect of a treatment or procedure. In observational studies one does not have the luxury of controlling who receives treatment and who does not. To address this problem, the technique of matching similar treated and control individuals are widely used to estimate a treatment effect. A matching paradigm may be too restrictive in many cases because exact matches often do not exist in the available data. One mechanism for overcoming this issue is to relax the requirement of exact matching on some or all of the covariates (attributes that may affect the outcome) to one of balance on the covariate distributions of individuals in the treatment and control groups. Such a relaxation is considered here, and several complexity results are presented for the resulting problem.

• Topic:  Automatic Image Registration Techniques
• Speaker:  Tony Kellems, METRON
• Date:  05 Oct 2011
• Abstract:  Image registration is the process of aligning an acquired image with a reference image for the purposes of making some comparisons in a common frame of reference, a problem which arises in many fields including medical imaging, environment mapping, and photography. The acquired image may be geometrically deformed and may have contrast differences that arise from being imaged under different sensing modalities; it is the parameters of these transformations which we require in order to accurately register an image. While it is possible to perform registration by hand in some cases, this is very expensive when the number of images to register is large and it also sacrifices quantitative accuracy for qualitative matching. This talk will examine a few automated techniques for image registration that handle different classes of transformations, each requiring the solution of a multivariable optimization problem, and will show the strengths and weaknesses of each technique with some illustrative examples.

• Topic:  Exponential Equations and p-adic Numbers
• Speaker:  Josh Holden
• Date:  28 Sep 2011
• Abstract:  The discrete logarithm is a problem that surfaces frequently in the field of cryptography as a result of using the transformation x goes to gx reduced modulo n. Analysis of the security of many cryptographic algorithms depends on the assumption that it is statistically impossible to distinguish the use of this map from the use of a randomly chosen map with similar characteristics. For instance, we can ask when gx is equal to x modulo n (a "fixed point"), or when repeating the map twice gets you back to x (a "two-cycle"), or when gx and gy are the same modulo n for different x and y (a "collision"). When n is a prime p, much is known about the answers to these questions, although much is left to be learned. When n is a prime power pe, however, these questions have not been studied as much. It turns out that they can be answered using a type of number called a "p-adic" number, which is neither real nor complex. I will introduce these numbers, which are interesting in their own right, and then show how they can be used to find solutions to equations involving exponentials modulo pe. If students are interested, I may teach a course in p-adic numbers in the Winter. So come see if you are interested!

• Topic:  Nyquist Beat Down: Applications of Compressed Sensing
• Speaker:  Kurt Bryan
• Date:  14 Sep 2011
• Abstract:  Last week, I showed how incorporating randomness into sampling techniques and seeking sparse solutions allows one to solve drastically under determined systems of linear equations (but if you missed the first talk, I'll start with a warp speed review, so you'll be able to follow.) This week I'll flesh out a little more of the theory and show how these ideas allow one to do an end run around the cherished Nyquist sampling rule, which in its simplest form states that if you want resolve frequencies up to "f" hertz in a signal, you need to sample the signal twice as fast, at "2f" hertz. Then I'll show how researchers at Rice University have used these ideas to construct a one-pixel camera!

• Topic:  Making Do With Less: The Mathematics of Compressed Sensing
• Speaker:  Kurt Bryan
• Date:  07 Sep 2011
• Abstract:  Abstract: Suppose a bag contains 100 marbles, each with mass 10 grams, except for one defective off-mass marble. Given an accurate electronic balance that can accommodate anywhere from one to 100 marbles at a time, how would you find the defective marble with the fewest number of weighings? (You've probably thought about this kind of problem and know the answer.) But what if there are two bad marbles, each of unknown mass? Or three or more? An efficient scheme isn't so easy to figure out now, is it? Is there a strategy that's both efficient and generalizable? The answer is "yes," at least if the number of defective marbles is sufficiently small. Surprisingly, the procedure involves a strong dose of randomness. It's a nice example of a new and very active topic called compressed sensing (CS) that spans mathematics, signal processing, statistics, and computer science. In this first talk I'll explain the central ideas, which require nothing more than simple matrix algebra and elementary probability. Next week I'll show some applications, including how one can use this to beat the Nyquist sampling rule in signal processing, and build a high-resolution one-pixel camera.

### 2010-11 (latest first)

• Topic:  Non-trivial Composite Sequences through Digit Appendage
• Speaker:  John Rickert
• Date:  18 May 2011
• Abstract:  Begin with a number whose base ten representation is s0 = anan-1 ...a0 and append the digit dn times to obtain the terms sn = anan-1 ...a0dd . . . d. For what seed values s0 is the sequence {sn} always composite? L. Jones showed that when d = 1 the seed s0 = 37 produces a composite sequence and provided upper bounds for the smallest seeds that produce composite sequences when for other values of d. Grantham, Jarnicki, Rickert, and Wagon have conjectured minimals seeds for d =3, 7, 9 and have proven the value for d = 7, up to certification of some probable primes. Grantham, Jarnicki, Rickert, and Wagon also have conjectures for seed values for the sequences produced when bases other than ten are used, and have found seed values base ten to which any digit can be appended an arbitrary number of times and produce only composite values.

• Topic:  How many slopes are produced
• Speaker:  Leanne Holder
• Date:  11 May 2011
• Abstract:  We modify the typical Euclidean geometry to remove the concept of parallel lines, which supports the development of a projective plane. We continue by discussing the relationships between Euclidean and projective planes, emphasizing the existence and construction of blocking sets in the projective plane. A blocking set is a collection of points in the plane that do not contain a line, but have the property that every line meets that set. We use blocking sets to obtain bounds on the number of distinct slopes produced by q points distributed on two lines in a projective plane. Moreover, we offer a construction that classifies the sizes of all possible slope sets obtained in this configuration.

• Topic:  Really, how hard is it to schedule final exams?
• Date:  04 May 2011
• Abstract:  In this talk we explore the scheduling of final exams here at Rose-Hulman. We will discuss how the exams are currently scheduled and some ideas for reducing student conflicts. Finally, we provide some preliminary results which indicate better schedules can be found in a relatively short amount of extra time and effort. This talk explores some of the work done during the speaker's recent sabbatical.

• Topic:  The Joy of Computational Complexity
• Speaker:  Jeff Kinne, Indiana State University, Computer Science
• Date:  27 Apr 2011
• Abstract:  The overriding theme of computational complexity theory is the question - what problems can be solved efficiently? Some specific questions of importance are - Is cryptography and provably secure communication possible (current best answer - maybe/probably, see P versus NP...)? Can optimization problems always be solved optimally, or are some optimization problems "hard" (current answer - maybe/probably, see P versus NP...)? Can use of randomness significantly speed up computation (sometimes yes, and sometimes no)? In this talk, I will look at how computational complexity phrases these types of questions and what is currently known (for many questions, not nearly as much as we would like). I will focus mostly on high level ideas and the big picture, but will also plan to give a few "nice ideas/proofs". I will also be happy to diverge from "the plan" at times based on the interests of the audience.

• Topic:  The Heisenberg Uncertainty Principle is Necessary for Life on Earth
• Speaker:  Rick Ditteon, Physics and Optical Engineering
• Date:  13 Apr 2011
• Abstract:  This talk examines the physics of stars, specifically how the Sun generates the energy which makes life, as we know it, possible on Earth. I will show that the Heisenberg Uncertainty Principle plays an essential role in energy production in the Sun. Even though numerical calculations will be presented which explicitly illustrate the important concepts, the talk is at a level which should be understood by the general public.

• Topic:  Terre Haute Warming
• Speaker:  Tim Ekl and Dr. Allen Holder
• Date:  23 Mar 2011
• Abstract:  The question of whether or not human behavior is altering the earth's global weather pattern is among the most intense, popular, and scientifically charged of our day. Whereas global atmospheric models are complex and debatable, we postulate a simple, linear regression model that calculates the historic trend locally, i.e. right here in Terre Haute, IN. Numerical evidence suggests that the model is accurate enough to identify the solar cycle. Our Terre Haute model is an adaptation of R. Vanderbei's at Princeton. However, unlike Vanderbei's result for the McGuire Air Force Base, which is near Princeton, we find a predicted temperature change of approximately 0.6 degrees Fahrenheit per century, as compared to his predicted 3.63 degrees Fahrenheit per century at the McGuire Air Force Base.

• Topic:  Yeast and Mathematics
• Speaker:  Kyla Lutz, Rose Student, Biomedical Engeineering
• Date:  16 Mar 2011
• Abstract:  Mathematics underlies many biological problems, including the metabolism of Saccharomyces cerevisiae, or baker's yeast. The metabolic network of this organism is modeled using flux balance analysis (FBA), which incorporates linear algebra, computer science, and the chemical reactions within a cell to determine what a single cell is doing while it is in steady-state. More basic mathematics is also used for this metabolic model. Specifically, Boolean algebra is used to represent the reactions so that the experimental data can be used accurately in the model. The metabolites given to the cell initially are collectively called the environment, or medium, which can be controlled by the user in the model to mimic experimental conditions so that accurate predictions can be made. Using this tool, some questions that can be asked are "What are the minimum number of metabolites that the cell needs and what are they?" and "What are the compositions of all of the possible 'minimal media' in which the cell can survive?" These questions were addressed in Dr. Jason Papin's Biomedical Engineering laboratory over the course of an REU at the University of Virginia. Another problem that was addressed is the connectedness among all of the reactions in a cell and each metabolite with which the cell interacts. These connections can be found using a modification of the Floyd- Warshall algorithm from computer science. Mathematics played a major part in this research project and others very similar to it.

• Topic:  Algebra in Geometric Combinatorics
• Speaker:  Chris McDaniel
• Date:  19 Jan 2011
• Abstract:  Geometric combinatorics studies shapes or figures made up out of a finite number of pieces. Convex polytopes play a prominent role in this field of mathematics and, although their study dates back to antiquity, polytopes continue to serve as a rich source of problems for us even today. One basic problem is that of counting faces of convex polytopes. It turns out that numerical constraints on the number of faces a polytope can have can be gleaned (in the "simplest" cases) from algebraic properties of a certain ring associated with the polytope. In this talk, I will introduce convex polytopes and simple polytopes, showing several examples along the way. Then I will describe how to associate a ring to a simple polytope, and I will describe some important properties of this ring. Finally I will show you how these properties "classify" the number of faces that a simple polytope can have. More concretely, I will use this result to answer the following question: Does there exist a four dimensional convex polytope with 7 vertices, 14 edges, 13 two-dimensional faces, and 6 three-dimensional faces? Come and find out. The answer will shock and amaze you!

• Topic:  Using inverse functions to solve equations
• Speaker:  Cabral Balreira, Trinity University Mathematics
• Date:  03 Nov 2010
• Abstract:  We will discuss the problem of solving an equation as a functional problem in Mathematics. We shall observe that solving an equation entails finding the inverse of a map, a task that is generally difficult. Based on several applications from Economics, Geometry, and Differential Equations, we will show that a natural setting to answer such problems lies in Topology.

• Topic:  Combinatorial Structures with Prescribed Automorphism Groups
• Speaker:  Tanya Jajcay
• Date:  27 Oct 2010
• Abstract:  The concept of an automorphism group of a combinatorial structure is a fundamental concept in the cross-section of Combinatorics and Group Theory. Finding the automorphism group of a specific structure is a notoriously hard problem whose general complexity has not been resolved but it is believed to be exponential. In the talk, I will address the opposite problem of constructing a combinatorial structure for a given automorphism group. I will survey the known results for the classes of oriented and non-oriented graphs, outline the solution to this problem for the class of general combinatorial structures, and present a strategy for solving this problem for the class of hypergraphs. I will start from the basic concepts and present the theory through a series of examples so that the talk will be accessible to all mathematically minded students.

### 2009-10 (latest first)

• Topic:  All Things Infinite
• Speaker:  Bill Butske
• Date:  12 May 2010
• Abstract:  Why can’t you cancel $\frac{\infty}{\infty}=1$? (Being a trained mathematician, I am licensed by the state of Indiana to safely do so.) How much is $\infty$? In this talk I’m going to answer these questions (“because” and “lots” respectively, there, talk is done) and use the math you know against you, in the process rendering your finals frazzled minds inert with shock and awe. This will make grading the math finals much easier and settle a bet I have going with the other math profs about whether or not I can use $\infty$ to hypnotize a room of students.

• Topic:  Just what does the number of slopes of a collection of points have to do with a flock?
• Speaker:  Leanne Holder
• Date:  05 May 2010
• Abstract:  We being this talk with an introduction to Projective Geometry and some uses for Projective Geometry. Afterwards, we will use the axioms for a projective plane to collectively construct several examples of finite projective planes. Then, we discuss the history and evolution of the geometric structure known as a flock. Finally, we examine the problem of determining the number of slopes produced by a collection of points in a projective plane. (Considering taking MA423, Topics in Geometry, next fall? Then consider coming to this talk and get a small taste of what the future could have in store for you.)

• Topic:  Why be an Actuary?
• Speaker:  Richard Lenar (Chief Actuary) and Brad Jones (Rose grad, Associate Actuary), McCready and Keene, Inc. (based in Indianapolis)
• Date:  27 Apr 2010 (special time 10th hour)
• Abstract:  hese two actuaries from McCready and Keene, Inc. (based in Indianapolis) are coming to Rose-Hulman to discuss their careers as Actuarial Scientists.

• Topic:  Cryptography, Finite Groups, and the Discrete Log Problem
• Speaker:  Jonathan Webster, 2001 Rose alum, Ph.D graduate University of Calgary
• Date:  21 Apr 2010 (special time 10th hour)
• Abstract:  Modern day cryptosystems typically rely on the computational difficulty of one of two problems: integer factorization (RSA) or the discrete log problem (ECC). It is usually easy to convince people of the computational difficulty of integer factorization; therefore, we will focus on the discrete log problem. In order to understand this problem, we will examine the ln(x) function and its discrete analogue, define what a group is, and give many examples.

• Topic:  Compositions and the Jacobsthal Numbers
• Speaker:  Ralph Grimaldi
• Date:  14 Apr 2010
• Abstract:  What is a composition? For a positive integer n, there are 2^(n-1) ways to write n as an ordered sum of positive integers. These 2^(n-1) summations constitute the compositions of n. Since order is relevant here, the compositions of a positive integer are different from the partitions of the integer. When the summands for the compositions are restricted, certain notable sequences, such as the Fibonacci numbers, arise. When one requires that the last summand in the composition be odd, the Jacobsthal numbers come into play. These compositions constitute the major part of this presentation, where characteristics of these compositions will be examined and enumerated.

• Topic:  A Computational Study of the Dynamics of Human Tear Film
• Speaker:  Kara L. Maki, Institute for Mathematics and its Applications (IMA) - University of Minnesota
• Date:  24 Mar 2010
• Abstract:  Each time someone blinks, a thin multilayered ﬁlm of ﬂuid must reestablish itself, within a second or so, on the front of the eye. This thin ﬁlm is essential for both the health and optical quality of the human eye. An important ﬁrst step towards eﬀectively managing eye syndromes, like dry eye, is understanding the ﬂuid dynamics of the tear ﬁlm. In this talk, I will describe, through demonstrations and experiments, the physical phenomena that play an important role in the tear ﬁlm mechanics. These will be used to create a mathematical model that is realistic enough to reﬂect the essential aspects of the tear ﬁlm dynamics, but simple enough so that it can be analyzed and solved through the construction of a numerical algorithm. Simulations of the tear ﬁlm equations using an overset grid based computation method can then be compared to experimental observations to show both the model’s strengths and shortcomings.

• Topic:  Modeling Complex Fluids - A Primer on Continuum Mechanics
• Speaker:  David Finn
• Date:  10 Feb 2010
• Abstract:  To mathematically describe a complex fluid, a fluid that exhibits properties of both a solid and a liquid, we need mathematics capable of modeling deformation, continuous changes of the shape of an object, and the interaction of the fluid with its boundary. We also need to be able to model the forces acting on the material through the surface of the material, and how the material's deformation influences the forces and possibly exert forces on the material, and even how material properties can be incorporated into the description of the forces. This is done through the subject of continuum mechanics, which is part mathematics and part physics, and heavily used in certain areas of engineering. This talk will be an introduction to continuum mechanics and its application in describing complex fluids. Only knowledge of vectors and partial derivatives are required to understand the mathematical methods of continuum mechanics in this talk, plus enough physics to understand F=ma.

• Topic:  Oobleck, Silly Putty, Shampoo, Syrups and Other Complex Fluids
• Speaker:  David Finn
• Date:  03 Feb 2010
• Abstract:  In this first of a series of two talks on complex fluids from my sabbatical at the Institute of Mathematics and its Application, I will give examples and motivation of complex fluids or non-Newtonian fluids, and how they differ from usual fluids or Newtonian fluids. This talk will be more demonstrations and experiments to give some of the phenomenological differences between non-Newtonian fluids and Newtonian fluids. The mathematical descriptions of the phenomenon given in this talk will be given in the next talk.

• Topic:  THE WORLD’S HARDEST EASY GEOMETRY PROBLEM
• Speaker:  Herb Bailey
• Date:  18 Jan 2010
• Abstract:  If you Google the title you will find the problem and many solutions -‘some short, some long, some right, some wrong’. It is easy to solve with trig, but hard if you use only high school geometry. There is also a second hardest problem of the same type that has been published. The common theme is that all angles must be an integer multiple of 10 degrees. Using trig, I have shown that there are only four more problems of this type. I can solve three of them using geometry and have tried for many hours, without success, to find a geometric solution of the fourth. As a prize, if you can help me solve the fourth, we can coauthor a paper describing our results and propose a problem that is harder than 'The World’s Hardest'.

• Topic:  Deciding Complex Feasibility by Reducing to Finite Fields
• Speaker:  Arnold Yim, Rose Student
• Date:  16 Dec 2009
• Abstract:  Last summer, I participated in an REU program at Texas A&M University where I worked with Dr. Rojas and a couple of other students on deciding complex feasibility. This problem was to determine whether a system of polynomials has a complex root or not. Our approach was to reduce the polynomial to different finite fields and determine whether the system had roots in those finite fields. If enough of those fields had a root, then we can conclude with some certainty that the system had complex roots. We coded up an algorithm outlined by Koiran, then looked at how different families of polynomials behaved in different finite fields in attempt to improve the algorithm. In particular, we looked at polynomials whose Galois groups are dihedral groups and symmetric groups. After running some tests, we were able to find certain patterns in the density of prime numbers for which the polynomials had a solution, which we later used to generalize specific formulas for the prime density based on the Galois group of the polynomial. Although a little background would be in algebra would be helpful, this talk should be accessible to all audiences.

• Topic:  Using Mathematical Models and Operations Research to Tackle the Risky Business of Aviation Security
• Speaker:  Sheldon H. Jacobson, Professor and Director, Simulation and Optimization Laboratory, Department of Computer Science University of Illinois
• Date:  09 Dec 2009
• Abstract:  Aviation security has become a topic of intense national interest, as the risk of terrorism and of other hazardous threats to the nation's air system increase. Recent events have hastened changes to improve the security of the air traffic industry. This includes multi-million dollar investments in new security technologies and equipment. Passenger screening is a critical component of such aviation security systems. This paper introduces the sequential stochastic security design problem (SSSDP), which models passenger and carry-on baggage-screening operations in an aviation security system. SSSDP is formulated as a two-stage model, where in the first stage security devices are purchased subject to budget and space constraints, and in the second stage a policy determines how passengers that arrive at a security station are screened. Passengers are assumed to check in sequentially, with passenger risk levels determined by a prescreening system. The objective of SSSDP is to maximize the total security of all passenger-screening decisions over a fixed time period, given passenger risk levels and security device parameters. SSSDP is transformed into a deterministic integer program, and an optimal policy for screening passengers is obtained. Examples are provided to illustrate these results, using data extracted from the Official Airline Guide.

• Topic:  Actuary: The Best Job in the World
• Speaker:  Phil Banet, Allstate - Rose math alumnus
• Date:  04 Nov 2009
• Abstract:  Ever wanted to know more about being an actuary? Not sure what it is? It’s only one of the highest rated jobs in the country. Please join us Wednesday, November 4, at the Mathematics Colloquium to hear more from a practicing actuary with Allstate Insurance Company who just happens to also be a graduate of Rose-Hulman. Phil Banet has been an actuary since he graduated from Rose in 1991. He’ll be discussing his experiences as well as how Rose helped prepare him for this fascinating career.

• Topic:  Classical Markov Logic and Network Analysis
• Speaker:  Ralph Wojtowicz and Geoff Ulman, Metron Inc
• Date:  07 Oct 2009
• Abstract:  Markov logic is a set of techniques for estimating the probabilities of truth values of formulae written in first-order languages. In network analysis applications, the formulae describe properties of and relationships or links among entities. The truth values tell if an entity has a property or whether or not a link exists. The networks may involve many different sorts of entities and types of links. Estimates are based on the values specified in training and test data. We refer to the special case involving two truth values as classical Markov logic. Data in this case must assign either ‘false’ or ‘true’ to all (closed) formulae. In practical applications, however, we may have limited confidence in some information sources or data values. To model such uncertainties, we generalize Markov logic in order to allow non-classical sets of truth values. The concepts and methods of category theory give precise guidelines for selecting sets of truth values based on the form of a network model. We plan to give an overview of Markov logic; discuss applications to alias detection, cargo shipping, insurgency analysis, and social network analysis; and describe open problems.

• Topic:  Moment Convergence Rates and Method of Moments Central Limit Theorems via Induction
• Speaker:  Mark Inlow
• Date:  30 Sep 2009
• Abstract:  In 1965 von Bahr proved that the difference between each finite moment of the sample mean and the corresponding normal moment is O(n^{-1/2}) by appealing to results by Esseen, Loeve, and Cramer. We present two proofs of his result using only elementary properties of expectation plus mathematical induction. Since the normal distribution is determined by its moments, if all moments of the sample mean exist then, by a converse of the second limit theorem, it is asymptotically normal. Thus our results also provide simple versions of Markov's 1898 method of moments central limit theorem.

• Topic:  Roll-ups and Differential Geometry
• Speaker:  S. Allen Broughton
• Date:  16 Sep 2009
• Abstract:  We all know that cylinders and (frustums of) right cones can be formed by rolling up a flat strip of paper, metal, plastic, or other flexible material. In fact there are pictures of such "roll-ups" in Calculus books. However, what happens when we do not have such a standard cone shape? What region do we cut out of the paper or metal to achieve a desired cone shape? The problem started as a phone call from a local manufacturing design company who had to solve this problem. They wanted to build a specific shape but did not know what the flattened out shape would be. Since their plan was to build the part from a flattened sheet of metal, the answer to the roll-up problem was crucial. In this talk we discuss the geometry problem and show a solution using the techniques of differential geometry. The techniques are not advanced, in fact everything can be done with multi-variable Calculus and the simple separation of variables in Differential Equations. The time for the talk does not allow for a complete discussion of the "ghastly derivations" but we will discuss the formulas that allow us to solve the practical problem. The formulas can be evaluated using numerical integration (Calculus II) and we show the flattened out shape form the given problem. For those interested in full details see the technical report at http://www.rose-hulman.edu/math/MSTR/MSTRpubs/2009/RHIT-MSTR-2009-01.pdf

### 2008-09 (latest first)

• Topic:  Modeling Pattern Formation in the Auditory and Visual System
• Speaker:  Kim Montgomery
• Date:  22 Apr 2009
• Abstract:  In the course of biological development interesting hexagonal patterns are formed in both the non-mamalian auditory system and the visual system of the fly's eye. I'll discuss how mathematical models for intercellular signaling and cell motility can be useful in explaining the formation of these patterns.

• Topic:  The Mathematics of Cloaking
• Speaker:  Kurt Bryan
• Date:  18 Mar 2009
• Abstract:  Cloaking and invisibility are old staples of popular fiction, especially science fiction. The pseudo-explanation usually given is that "the selective bending of light rays" (to quote Mr. Spock) around the object to be cloaked can render the object invisible. But with the laws of physics in the real world, is this actually possible, even in theory? Scientists and mathematicians have recently found that the answer to this question is a qualified "yes." In this talk I'll give a quantitative, but accessible account of an essential mathematical idea behind cloaking, in the context of an electromagnetic imaging technique called "impedance imaging."

• Topic:  Braids, Cables, and Cells: An Interesting Intersection of Mathematics, Computer Science, and Art
• Speaker:  Joshua Holden
• Date:  11 Mar 2009
• Abstract:  The mathematical study of braids combines aspects of topology and group theory to study mathematical representations of one-dimensional strands in three-dimensional space. These strands are also sometimes viewed as representing the movement through a time dimension of points in two-dimensional space. On the other hand, the study of cellular automata usually involves a one- or two-dimensional grid of cells which evolve through a time dimension according to specified rules. This time dimension is often represented as an extra spacial dimension. Therefore, it seems reasonable to ask whether rules for cellular automata can be written in order to produce depictions of braids. The ideas of representing both strands in space and cellular automata have also been explored in many artistic media, including drawing, sculpture, knitting, crochet, and macrame, and we will touch on some of these.

• Topic:  Linear Volterra Inverse Problems - Formulation and Regularization
• Speaker:  Cara Brooks
• Date:  18 Feb 2009
• Abstract:  When solving practical problems, one often tries to gain intuition by first making many assumptions to obtain a simplified model. As the problem becomes better understood, the assumptions can be relaxed and a more complex model can be considered. In this spirit, we will start by examining the problem of differentiating data, then move on to the problem of computerized tomography, demonstrating how a few simplifying assumptions and a lot of calc II lead to solving a linear Volterra integral equation of the first kind. Depending on the function spaces involved, this means solving an ill-posed (inverse) problem. We will then examine regularization techniques for handing some linear Volterra problems and discuss some of the work involved in obtaining good'' approximations to the exact solution when using measurement data corrupted with noise.

• Topic:  Nonlinear Design Problems, Model Discrimination, and Impossible Solutions
• Speaker:  Mike DeVasher
• Date:  11 Feb 2009
• Abstract:  This talk will cover three distinct topics. First, an introduction to the fundamental conundrum of optimal design for nonlinear experiments will be discussed. A novel approach in applying Bayesian ideas to the need for prior information will be compared to the historical standard of local optimality. Next, a related nonlinear design problem, that of model discrimination for exponential regression models will be introduced. A brief review of model discrimination techniques for linear models will be offered as well as a discussion of model discrimination techniques particular to nonlinear models. Finally, time permitting, a solution to a variant of Freudenthal's "Impossible Problem" attributable to Lee Sallows will be discussed. Solutions to the so-called "Superimpossible Problem" will have to wait until a later date.

• Topic:  Inverse Problems on Resistor Networks
• Speaker:  Kurt Bryan
• Date:  04 Feb 2009
• Abstract:  Suppose we have a rectangular grid (of finite extent) of resistors in the 2D plane. The "interior" resistors are not accessible to us and have unknown resistance. However, we have access to the resistors on the boundary of the grid, to which we can apply voltages and measure the resulting currents. Can we use this kind of information to determine the resistance of the inaccessible interior resistors? What if we have access to only part of the boundary? This kind of problem is a natural discrete analogue to certain problems in nondestructive testing and "impedance imaging", but easier to analyze---all we need is linear algebra. I'll show some results obtained by students in the mathematics REU last summer at Rose-Hulman.

• Topic:  Solutions to the Pure Parsimony Problem
• Speaker:  Joshua Burbrink, Nicole Fehribach, Tony Ferrell, Fred Freers, Casimir Ksiazek, Jason Sauppe, Jeremy Schendel and Al Holder
• Date:  28 Jan 2009
• Abstract:  The students in MATH 444 addressed the problem of finding the least amount of genetic diversity needed to describe a population, which is known as the Pure Parsimony Problem. This talk will start with a succinct introduction to the problem and then will proceed to a discussion of the proposed solution methods. In particular, local search methods and their efficiency will be discussed. The talk will end with a mathematical result that identifies a sub-class of these APX-Hard problems that can be solved in polynomial time.

• Topic:  The role of beauty in the search for world-record cages
• Speaker:  Robert Jajcay, Indiana State University
• Date:  21 Jan 2009
• Abstract:  A (k,g)-cage is a very neat and efficient mathematical creature; a k-regular graph of girth g that has the smallest number of vertices possible. As finding the (absolutely) smallest cage is extremely hard, researchers often settle for finding a graph that is smaller than anyone else's in the world -- the world record cage. This gives rise to a curious area of mathematics, an area where tables of current record holders are constantly being updated and closely watched, and every new entry gives rise to frantic attempts at beating it. It is also an area where everybody has an equal chance (well, not really, being smart helps quite a bit), and even newcomers may get their chance for their 15 minutes of fame (or how long it takes until someone else beats their record and erases their name from the tables). In our talk we intend to introduce some order into the competition by looking into the relation between beauty and efficiency. We make a very non-mathematical claim that beautiful (i.e., highly symmetric) structures have the best chance for being the world records, and we support this claim with a little bit of evidence and a lot of speculation. We will take great care not to start fights with proof-demanding mathematicians, but cannot make any promises.

• Topic:  Differentiating the QZ Algorithm with Application to Gradient Based Output Feedback Optimization
• Date:  17 Dec 2008
• Abstract:  Special time 10'th Hour PARTA: The QZ algorithm gives a robust way of computing solutions to the generalized eigenvalue problem. The generalized eigenvalue problem is used in linear control theory to find solutions to Ricatti equations, as well as to determine system transmission zeros. In state space linear system analysis, the system poles and transmission zeros are particularly important for determining system time and frequency response. Here we embed calculation of the eigenvalue derivatives in the QZ algorithm such that the derivatives of system poles and transmission zeros are computed simultaneously with the poles and zeros themselves. The resulting method is further exercised in finding generalized eigenvalues and their sensitivities required for finding the derivatives of system residues. This technique should openthe door to solutions of problems of interest by unconstrained gradient based methods. Typical numerical results are presented. PART B: A new method for gradient based determination of H2 optimal output feedback gains is presented. Constraints representing the dynamics of a linear time invariant system are substituted into the quadratic cost function. Sylvester's expansion is used to write the matrix exponential in a form which can then be integrated closed-form. The cost function and its derivatives can then be written as algebraic expressions in terms of the system eigenvalues. PART C (time permitting): A numerical model of the Ares I upper stage main propulsion system is ormulated based on first principles. Equations are written as non-linear ordinary differential equations. The GASP Fortran code is used to compute thermophysical properties of the working fluids. Complicated algebraic constraints are numerically solved. The model is implemented in Simulink and provides a rudimentary simulation of the time history of important pressures and temperatures during re-pressurization, boost and upper stage firing. The model is validated against an existing reliable code, and typical results are shown.

• Topic:
• Speaker:  S. Allen Broughton, Voronoi Tesselations, Delaunay Tesselations and Flat Surfaces
• Date:  10 Dec 2008
• Abstract:  Voronoi tessellations are all about us. In crystallography, the can be used to define a unit cell. In coding theory the can be used measure effectiveness of detection and correction of errors in transmission. The sizes of the cells can give us information about uniform placement of points on a sphere such as satellites in the sky. Delaunay tessellations are dual to Voronoi tessellations and have their own uses. In the first part of this talk we will give some examples of the tessellations and discuss algorithms for determining them. In the second part of the talk we will look at how these tessellations can be used to understand the geometry of flat surfaces, such as a cube or icosahedron. This talk is the second of two sabbatical report talks from Professor Broughton's sabbatical at Indiana University last spring. The first talk "Billiards and Flat Surfaces" was a motivational introduction to flat surfaces intended for a general audience. This second talk, will discusses additional geometrical concepts and problems about flat surfaces suitable for undergraduate research.

• Topic:  (Almost) The Poincare Conjecture or "What's the difference between a ball and a doughnut?"
• Speaker:  Bill Butske
• Date:  29 Oct 2008
• Abstract:  In this talk I'll discuss how to (mathematically) answer one of the burning questions of our time and indeed of this election cycle. Namely, what is the difference between a doughnut and a ball (there's a giant hole in one of them) and how it relates to the recently proven Poincare Conjecture (which has to do with holes in things). This talk is intended for a general audience (this means you M. Fouts) though you'll learn lots of fancy words that you can use to impress the public at large.

• Topic:  Algebraic Tori and Their Applications
• Speaker:  Arnold Yim, Rose Student
• Date:  22 Oct 2008
• Abstract:  In this talk, we will discuss the structure of algebraic tori. In particular, we will go over what it takes for an algebraic variety to be rational. We will then look at an example of a small torus. Finally, I will describe the applications of algebraic tori in public key cryptosystems. Most of the mathematics involved in this talk should be accessible to anyone, however, familiarity with finite fields and basic algebraic ideas will help.

• Topic:  Billiards and Flat Surfaces
• Speaker:  Allen Broughton
• Date:  01 Oct 2008
• Abstract:  What do flat surfaces like a cube or icosahedron have to do with billiards? The billiard question is simply: If you hit a billiard on a polygonally shaped billiard table and it continues indefinitely, will it eventually get near to every point on the table? The answer is fairly easy for rectangular shaped tables but more complicated for other shapes. In this talk we will discuss how flat surfaces arise from the discussion of billiards and look at some of the properties of flat surfaces, including a suitable interpretation of Euler's formula. This talk is the first of two sabbatical report talks from Professor Broughton's sabbatical at Indiana University last spring. The first talk is a motivational introduction to flat surfaces and is intended for a general audience of Rose faculty and students. The second talk, to be given later in the year, will discuss additional concepts and problems about flat surfaces suitable for undergraduate research topics.

• Topic:  Mathematical Programming, Systems Biology, and Undergraduate Research
• Speaker:  Allen Holder
• Date:  24 Sep 2008
• Abstract:  Several recent advances in biology, medicine and health care are due to computational efforts that rely on new mathematical models. The underlying mathematics largely lies within discrete mathematics, statistics & probability, and optimization, which are combined with savvy computational tools and an understanding of cellular biology to advance our biological insights. One of the most significant areas of growth is in the field of systems biology, where we are using information from high-throughput computing to construct models that describe larger entities. We will introduce the overriding goal of systems biology and will highlight the role of mathematical programming. In particular, places for undergraduate research will be discussed.

• Topic:  My Summer as an Actuarial Intern
• Speaker:  Casimir G. Ksiazek III, Rose Student
• Date:  17 Sep 2008
• Abstract:  In this talk, I will describe the work I did as an actuarial intern this summer at Allstate Insurance in Northbrook, IL. The talk will be informal, with discussion and questions encouraged. I STRONGLY recommend that anyone even remotely interested in actuarial science attend. Most of the mathematics involved should be accessible to anyone, though a familiarity with regression will help.

### 2007-08 (latest first)

• Topic:  Generalizations of Niven Numbers
• Speaker:  Robert Lemke Oliver, Rose Student
• Date:  14 May 2008
• Abstract:  A Niven number is an integer that is divisible by the sum of its base q digits. For example, 2008 is Niven both in base 3 and in base 5 (see abstract). Several people have derived asymptotic formulae for the function N(x) that counts the number of Niven numbers less than x. We proceed in a more general case, studying functions that act only on the base q digits of an integer. An asymptotic formula for the counting function of these generalized Niven numbers is known, but the question of divisibility by multiple functions is still open. We present partial work toward acquiring an asymptotic formula in this case, as well as conjectures based off of numerical evidence.

• Topic:  A Generalization of the Fibonacci and Jacobsthal Sequences
• Speaker:  Ian Rogers, Rose Student
• Date:  07 May 2008
• Abstract:  Among the sequences of discrete mathematics, the Fibonacci sequence is probably the most well-known. Turning up in myriad areas from geometry to graph theory, seashells to the stock market, the Fibonacci numbers display an amazing number of interesting properties. The Jacobsthal numbers, another well-known sequence, are defined by a different, yet closely related, recurrence relation to that of the Fibonacci numbers. While slightly less popular, the Jacobsthal numbers too display many desirable properties. In this talk, we will describe a new class of generalizations of the Fibonacci and Jacobsthal numbers. We then look at a few examples in which the Fibonacci and Jacobsthal numbers are known to occur, and expand them to produce the new sequences. Finally, we show that many of the desirable properties of the Fibonacci numbers still hold in the general case, and provide suggestions for further research into this new family of sequences.

• Topic:  Total Variation Image Restoration
• Speaker:  Ely Spears, MIT Lincoln Labs - Rose Alum
• Date:  26 Mar 2008
• Abstract:  One of the most widely studied areas of applied mathematics is image processing. Image restoration, also called image inpainting, is one of the most prominent uses for these mathematical techniques. In this talk, a particular procedure for restoring damaged or corrupted images, called total variation, is discussed. Most of the material will be accessible to students familiar with linear algebra. A brief description of numerical methods, in particular the Fast Level Set Transform, is included.

• Topic:  Modeling a Slice of French Bread
• Speaker:  David Finn
• Date:  23 Jan 2008
• Abstract:  Why does a slice of French or Italian bread have a somewhat elliptical shape? In this talk, I will provide a heuristic model to describe the shape based on treating dough as a liquid. Then from data from slices of bread, I will show that this model provides a good description of a slice of bread.

• Topic:  Introduction to the Life Table
• Speaker:  Casimir G.Ksiazek III, Rose Student, Mathematics
• Date:  19 Dec 2007
• Abstract:  Buying life insurance is a quite a common occurrence. But how do people determine how much life insurance premiums should cost? Historically, actuaries have used life tables to assist in pricing insurance and annuities. In this talk, the concept of a life table will be introduced. In addition, examples will be given to show how from seemingly simple data, quantities such as life expectancy and insurance premiums can be calculated. A knowledge of probability is recommended, but not required. Anyone interested in actuarial science is strongly encouraged to attend.

• Topic:  Introduction to Infinity or Why Johnny Can't Add
• Speaker:  Bill Butske
• Date:  07 Nov 2007
• Abstract:  First I want emphasize that this talk is for anyone who has wondered what mathematics has to say about the concept of infinity. In particular non-math majors are encouraged to attend and the talk is aimed primarily at them. I'm going to talk about infinity in two ways, first in counting, where we will see that there are two different kinds of infinity (at least) and second in geometry where we know that parallel lines DO intersect, namely at infinity. Of course this is a math talk and the underlying intent is to warp your mind.

• Topic:  FETCHING WATER WITH MINIMUM RESIDUES: Generalization of a problem from Die Hard 3
• Speaker:  Herb Bailey, Emeritus Rose Math professor
• Date:  31 Oct 2007
• Abstract:  Bruce Willis can disarm a bomb if he is able to get exactly 4 gallons of water from a well using only a 3 gallon jug and a 5 gallon jug. This problem dates back to the 13th century. A generalization of this problem is to determine all possible integer gallons that can be obtained using an M gallon jug and an N gallon jug, with M < N. We solve the generalized problem using some congruence results. It turns out that there are only two distinct pouring sequences to get a given number of gallons. The shorter of the two can be determined by solving a linear congruence equation. Short is good since Bruce has but 5 minutes prior to detonation. Not to worry, no previous knowledge of number theory will be needed to enjoy this talk.

• Topic:  Blow-up Solutions to Differential Equations
• Speaker:  Kurt Bryan
• Date:  24 Oct 2007
• Abstract:  Nonlinear differential equations of the form u' = f(u) where u=u(t) are common in applied mathematics. Usually t is time, u(t) is the amount of some "stuff" in a system, and f(u) models the rate stuff is produced or destroyed, as a function of the amount present. If the function f is positive and increasing (the stuff catalyzes its own production) then solutions may grow to infinity in a finite time, a phenomena called "blow-up". In this talk I'll start with the simple ODE above, then describe some recent progress in analyzing blow-up phenomena for similar partial differential equations in which diffusion is present.

• Topic:  Models for Emergent Behavior
• Speaker:  Ely Spears, Rose Student, Mathematics
• Date:  17 Oct 2007
• Abstract:  Emergent behavior is a division of biology that seeks to understand and better explain phenomena that appear in group situations but not on an individual basis. Fish schooling, bacterial growth properties, and bird flocking are just a few prominent examples of this sort of behavior. The latter of these examples motivated summer research at the City University of Hong Kong, in China. This introductory presentation will give the details behind some popular mathematical models for bird flocking behavior. Additionally, numerical simulations of these models will be discussed at length and the various model parameters will be explored. The talk is such that students of any background are encouraged to attend.

• Topic:  Generalized Niven Numbers
• Speaker:  Robert Lemke-Oliver, Rose Student, Mathematics
• Date:  27 Sep 2007
• Abstract:  A base-q Niven number is one which is divisible by the sum of its digits. For example, 18 is a base 10 Niven number, since 9 divides 18. We will be interested in simultaneous Niven numbers, numbers that are Niven in more than one base. Returning to the example, 18 is also base 9 Niven, since 18 is 20 in base 9. Thus, 18 is a simultaneous base 9 and base 10 Niven number. We are interested in counting the number of simultaneous Niven numbers up to a point, x. One approach to this is to look at completely q-additive functions. These functions essentially act on the digits of a number, so that f(124)=f(1)+f(2)+f(4). Note that the sum of digits function is completely q-additive. If we can understand these generalized Niven numbers, we can hopefully gain some information about the standard Niven numbers. In this talk, we will prove an asymptotic formula for the number of generalized Niven numbers, and we will present the work that has been done to relate this to Niven numbers.

### 2006-07 (latest first)

• Topic:  Modeling Hysteresis (PART II): A load dependent hysteresis model for a simple shape memory wire actuator.
• Speaker:  Steve Galinaitis
• Date:  31 Jan 2007

• Topic:  Modeling Hysteresis: A load dependent hysteresis model for a simple shape memory wire actuator.
• Speaker:  Steve Galinaitis
• Date:  24 Jan 2007

• Topic:  Alignment of Protein Structures
• Speaker:  Yosi Shibberu
• Date:  17 Jan 2007
• Abstract:  Proteins play a key role in nearly all of the biochemical processes of living organisms. A protein is a long molecular chain constructed from twenty types of molecules called amino acids. Proteins produced by living organisms fold up into unique, tightly packed, structures called folds. The particular sequence of amino acids in a protein's chain determines its unique fold. The geometry of a protein’s fold largely determines the protein's specific biological function.

Identifying the biological function of individual proteins is an important and challenging problem. A better understanding of the evolution of protein folds will help us decipher the function of individual proteins and will lead to major advances in biology and new treatments for many human diseases.

The evolution of proteins is studied by making comparisons. Proteins are typically compared by comparing their sequence of amino acids, by comparing the geometry of their folds, and more recently, by comparing their expression profiles.

Fold-based comparisons of proteins is believed to be much more informative and robust than sequence based comparisons. However, the problem of aligning protein folds is not as well understood as the problem of aligning protein sequences. In this talk, we describe a new mathematical framework for describing the geometry of protein folds. This mathematical framework may lead to a better understanding of the fold alignment problem.

• Topics:  Cookius Maximus by Robert Lemke Oliver and On the Minimum Vector Rank of a Graph by Ian Rogers
• Speakers:  Robert Lemke Oliver and Ian Rogers , Rose Students
• Date:  20 Dec 2006
• Abstracts:
Shape of a Cookie How can the shape of a sugar cookie be modeled mathematically? It turns out that it’s a solution of a non-linear partial differential equation. In this talk, we examine a simplified version of this “cookie equation” to find the highest point on the cookie. Our eyes seem to be very good at locating it, but whatever process we’re using turns out to be hard to explain mathematically. We will look in particular at convex regions, which are known to have only one maximum.

Graphs Given a graph or multigraph G on n vertices, we associate a set of nonzero complex vectors to the vertices of G in the following manner: If vertices i and j are not joined then the corresponding vectors are orthogonal, and if i and j are connected by a single edge, the associated vectors are not orthogonal. The rank of a vector representation is the maximum number of linearly independent vectors in the representation. The minimum vector rank of G, mvr(G), is the minimum rank among all vector representations of G. We present methods for determining mvr(G) if G is among certain classes of graphs, including perfect graphs, complete graphs, and cycles. Further, we present upper and lower bounds on mvr(G) for all multigraphs that contain only multiedges, and provide two conjectures on the exact value of mvr(G) for a graph.

• Topic:  Optimizing 4th-Order and 5th-Order Explicit Runge-Kutta Formulas
• Speaker:  Stephen Dupal, Rose Student
• Date:  13 Dec 2006
• Abstract:  Differential equations have been solved numerically with explicit Runge-Kutta methods for over a century. Runge-Kutta methods are used in the sciences as well as mathematical software such as Matlab’s ode45 solver. Utilizing techniques in polynomial theory based on Gröbner bases, it becomes more manageable to find Runge-Kutta formulas that minimize higher-order truncation error. In this talk, we will discuss the connection between the Runge-Kutta method and Gröbner bases, and we will present some of the results of exploring the optimization of fourth- and fifth-order Runge-Kutta formulas. This presentation is based on work done by Iowa State University’s summer 2006 Numerical Analysis REU group consisting of Stephen Dupal (Rose-Hulman) and Michael Yoshizawa (Pomona College).

• Topics:  Characterizing Holes in Wires and Plates Inverting the Heat Equation: Tom Werne and
Characterizing Refinable Rational Functions: Ely Spears
• Speakers:  Thomas Werne and Ely Spears, Rose Students
• Date:  06 Dec 2006
• Abstracts:
Heat Equation: The heat equation is a classical partial differential equation that can predict the temperature distribution on some domain subject to certain boundary conditions. Motivated by the field of nondestructive testing, the equation turns out to be a useful tool for characterizing defects in metallic plates. In this talk we will discuss solution methods and results that show how to characterize certain defects in two dimensional regions using only boundary data. The presentation is based on work done during the summer of 2006 by the Inverse Problems REU group of Thomas Werne (Rose-Hulman) and Jay Preciado (The College of New Jersey) at Rose-Hulman under Dr. Bryan (Rose-Hulman Mathematics Department).

Refinable Functions: A k-refinable function is a function f(x) that can be re-written in terms of the function f(x). In recent decades, refinable functions have become increasingly popular due to their desirable properties in many applications, such as wavelet analysis. While the refinability properties of many popular classes of functions, such as compactly supported splines, have been known for a while, rational functions had seemed to escaped notice in terms of refinability. This talk is based on research investigating the refinability of rational functions that took place at Texas A&M University during the summer. Preliminary simplifications to the general problem are presented in a chronological collection of lemmas. A complete characterization of refinable rational functions follows with an interesting connection to an open problem in number theory.

• Topic:  Knots, Braids, and an Application followed by Probability, Electrical Circuits, and Rectangles
• Speaker:  Jennifer Franko and Michael Bateman, Indiana University
• Date:  29 Nov 2006
• Abstract:  This week we have two mathematics seminars on Wednesday that may be of special interest to students. The seminars are 9th and 10th period in G221 on Wednesday by two Graduate Students from the Mathematics Department of Indiana University. The first (during all of 9th period) is by Jennifer Franko entitled “Knots, Braids, and an Application” which concerns the application of topology to quantum computing, and the second (during the first half of 10th period) is by Michael Bateman entitled “Probability, Electrical Circuits, and Rectangles”. Following Michael Bateman’s talk, both graduate students will answer questions about graduate school, the application process, what life is like as a graduate students, etc, so if you are considering Graduate School in your future it might be worthwhile to attend.

Knots, Braids, and an Application: One method proposed to build quantum computers is based on braid representations. In this talk, we will define the braid group and discuss the connection between braids and knots. Any invertible matrix which satisfies the Yang Baxter Equation can be used to obtain representations of the braid group, and we will study these types of representations and as well as link invariants they might yield. Finally, we will mention how these representations might be used in a topological model of quantum computation.

• Topic:  Actuarial Mathematics
• Speaker:  Nate Dorr, Rose Student
• Date:  08 Nov 2006
• Abstract:  Actuarial Mathematics refers to the mathematics of the insurance industry. Actuaries use probability and statistics in calculating premiums, determining reserves, and modeling insurance products. In this talk, actuarial components of a whole life insurance product will be covered. Life insurance, life annuities will be discussed and will lead to how premiums are calculated. In addition, information about the actuarial profession will be presented with time for questions at the end. Probability should be sufficient background for this talk.

• Topic:  Geometry from Chemistry II - The Geometry of Nanotubes
• Speaker:  Allen Broughton
• Date:  01 Nov 2006
• Abstract:  Carbon nanotubes are an interesting but as of yet incompletely understood part of nanotechnology, an area of science that has really grown up in just that last 15 years. From the mathematical perspective nanotubes have an interesting molecular structure based on the hexagonal honeycomb structure of graphite. In this talk I will describe the geometry and symmetries of nanotubes. There is an infinite family of such nanotubes, so describing the structure takes some care. Multivariable calculus should provide plenty of background to make this talk accessible. There will be a brief recap from lecture I of this series to motivate the atom labelling problem - a graph theory problem - for nanotubes.

• Topic:  Geometry from Chemistry I - Understanding Molecular Dynamics of Bucky Balls
• Speaker:  Allen Broughton
• Date:  25 Oct 2006
• Abstract:  Buckminsterfullerene is a complex molecule consisting of sixty carbon atoms is an arrangement like a soccer ball, and so the molecules are often called bucky balls. Trying to understands the molecular dynamics of bucky balls leads to some interesting problems in geometry, algebra and differential equations. In the talk, the theory will be described in some detail for very simple objects such as triangular molecules such as water. We then will examine the geometrical issues that come about in modeling the much more complex bucky balls. We are only go to talk about classical dynamics as quantum mechanics add a level of complexity well beyond an hour's talk This work is a collaboration with Dan Jelski of the chemistry department and Guo-Ping Zhang of the ISU physics department. We do not have complete results at this stage, in fact I'd like to describe some problems that could be tackled by undergraduates. I don't plan to use much more beyond multi variable calculus, though understanding of differential equations helps.

• Topic:  How to Paint Your Way out of a Maze
• Speaker:  Joshua Holden
• Date:  18 Oct 2006
• Abstract:  Many people don't realize that what we now call "algorithm design" actually dates back to the ancient Greeks! Of course, if you think about it, there's always the "Euclidean Algorithm". A more dubious example might be Theseus's use of a ball of string to solve the "Labyrinth Problem". (Google "Theseus, Labyrinth, string".) Solutions to this problem got a lot less dubious after graph theory was invited, since a graph turns out to be a good way of representing a maze mathematically. We will examine the classical solutions to this problem, and then throw in a twist --- a Twisted Painting Machine that puts restrictions on which paths we can take to explore the maze. Applications to sewing may also appear, depending on the presence of audience interest and string.

• Topic:  More Talks on the Numerical Range
• Speaker:  Thomas Werne, Ted Lyman and Robert Lauer, Rose students EE/MA, ME/MA , EE/MA
• Date:  27 Sep 2006
• Abstract:
First Talk
Title: Finding the Centroid of W(A)
Student: Thomas Werne

Abstract: The numerical range of a matrix A is a subset of the complex plane. One method of generating this subset is to choose random vectors on the unit ball in complex hyperspace. The method of generating these random vectors induces a probability density function on the numeric range. In this talk, we examine these density functions and a possible connection with the centroid of the numerical range and the spectrum of the matrix.

Second Talk
Title: Pre-Images of Points in the Numerical Range
Students: Ted Lyman (speaking) and Robert Lauer

Abstract: If A is an n x n matrix, the numerical range of A is the set of complex numbers W(A) = (Ax,x), where x is a unit vector in Cn and (Ax,x) denotes the dot product between Ax and x. Although W(A) appears simple, it has many intriguing properties. We give a brief overview of some of these properties and take a look specifically at the connectedness of the pre-image of points in W(A).

• Topic:  Home on the (Numerical) Range
• Speaker:  Dr. Roger Lautzenheiser
• Date:  20 Sep 2006
• Abstract:  Like the eigenvalues, the numerical range of a matrix is a subset of the complex plane. However, unlike the eigenvalues, the numerical range will not be a finite set except when the matrix is a multiple of the identity. Indeed, the numerical range of A is the singleton set {a} if and only if A = a I. In addition to containing the eigenvalues, the numerical range has many interesting properties. In this talk we survey the history of the numerical range, the relationships between the geometric properties of the numerical range and the algebraic properties of the matrix, and perhaps most importantly, how the numerical range is used as a research project in Linear Algebra 2.

### 2005-06 (latest first)

• Topic:  Fuzzy Topological Spaces Part II (of II) - "Correct" Fuzzification of Topological Spaces: Functors and the General Tychonoff Theorem
• Speaker:  Stephan Carlson*
• Date:  17 May 2006
• Abstract:  In this second part of his presentation, the speaker will discuss Lowen's modified definition of a fuzzy topology on a set and its ramifications for the investigation of fuzzy topological spaces. Emphasis will be placed on the use of category theory as a test for a correct generalization of set-based topology and the success in proving a general theorem on products of compact fuzzy topological spaces. *Research on the results presented was completed during the presenter's 2004-2005 sabbatical leave.

• Topic:  Fuzzy Sets and Fuzzy Topologies: Early Ideas and Obstacles
• Speaker:  Stephan Carlson
• Date:  10 May 2006
• Abstract:  Fuzzy set theory and fuzzy logic were introduced in the 1960s by electrical engineers as tools for understanding and developing efficient control methods. Since fuzzy sets in a fixed set generalize subsets of the set, mathematicians - especially topologists - took on the challenge of generalizing existing set-based theories to the fuzzy set context. In this first part of his presentation, the speaker will survey the initial development of the field of fuzzy topology, which yielded some elegant results but also left some challenging gaps. The presentation will be intended for a general audience, in the sense that no previous background in either fuzzy set theory or topology will be necessary in order to comprehend basic ideas.

• Topic:  Bicycle Tracks on the Plane and the Sphere
• Speaker:  David Finn
• Date:  29 Mar 2006
• Abstract:  The title problem of the MAA book "Which way did the bicycle go? … and other intriguing mathematical mysteries" by Konhauser, Velleman and Wagon considers the following situation: Imagine a 20-foot wide mud patch through which a bicycle has just passed, with its front and rear tires leaving tracks as illustrated below. In which direction was the bicyclist traveling? This problem is motivated by the Sherlock Holmes mystery, The Priory School, in which the great detective encounters a pair of tire tracks in the mud and immediately deduces the direction the bicycle was going. This evidence then leads to the finding of a duke's son and the arrest of a murderer. In this talk, we will describe solutions to two variations of this problem on both the plane and the sphere in which a criminal could potentially fool the great detective as it is possible for an incredible bicyclist to create tracks for which it is impossible to determine which direction the bicycle went by only the geometry of the tracks. Moreover, an incredible bicyclist can also defeat the great detective by riding in such a way to leave only one track, possibly causing the detective into believing he is pursuing a unicyclist instead.

• Topic:  Breaking the MD5
• Speaker:  Brandon Borkholder, Rose Student, Computer Science
• Date:  22 Mar 2006
• Abstract:  The MD5 hash function and its family are security algorithms that have been used world-wide for nearly a decade. Just a few years after creation there were hints of weakness and now there are algorithms to crack it efficiently. How do these algorithms work? Is the MD5 completely broken? How can a potential hacker exploit this weakness to undermine the trust of those who use it?

• Topic:  Investigating the Shape of a Cookie
• Speaker:  Hari A. Ravindran, Rose Student, Mathematics
• Date:  15 Feb 2006
• Abstract:  This is a continuation of the previous two talks on the Shape of a Cookie. The goal of the investigation is the establishment of an asymptotic expansion for the shape of a cookie with an elliptical base domain. The talk summarizes Hari's work towards this goal over the past summer and during this academic year. This research was funded in part by a Joseph B. and Reba A. Weaver Undergraduate Research Award.

• Topic:  Existence for a Heuristic Model for the Shape of a Cookie (Part II Cookie Series)
• Speaker:  David Finn
• Date:  08 Feb 2006
• Abstract:  Have you ever wondered why cookies are generically round? Well, I have. And, the reason involves some interesting mathematics: Calculus of Variations, Nonlinear Partial Differential Equations, and Differential Geometry (Sorry, cookie picture is too large to e-mail!) In this second of two talks on the shape of a cookie, I will first give an overview of the first talk developing a heuristic model for determining the shape of a cookie. Then, we will prove that this model can be solved mathematically and outline the method used to generate the numerical solutions presented. Finally, I will present some interesting questions that will be examined during the REU this summer. Homemade Cookies will be provided during the talk.

• Topic:  Modeling the Shape of a Cookie
• Speaker:  David Finn
• Date:  01 Feb 2006
• Abstract:  Have you ever wondered why cookies are generically round? Well, I have. And, the reason involves some interesting mathematics: Calculus of Variations, Nonlinear Partial Differential Equations, and Differential Geometry. In this first of two talks on the shape of a cookie, I will overview a heuristic model for determining the shape of a cookie, and show under some physically reasonable assumptions that a cookie is generally round. Some interesting questions suggest themselves, when the generically round is stated in a mathematically precise language, and the physically reasonable assumptions are allowed not to hold. An investigation of some of aspects of this mathematical model for the shape of a cookie will examined during the REU this summer. Homemade Cookies will be provided during the talk.

• Topic:  A Combinatoric Proof of the Chan-Robbins-Yuen Theorem
• Speaker:  Daniel Litt, High School Student from Ohio
• Date:  25 Jan 2006
• Abstract:

• Topic:  Solving the Rubik's cube: An Introduction to Group/Graph Theory
• Speaker:  William Butske
• Date(s):  02 Nov 2005, 09 Nov 2005
• Abstract:  The Rubik's cube is one of the most concrete examples of a finite non-abelian group that one is likely to come across. If these terms don't mean anything to you, don't worry, they will by the end of the talk(s). We will see how group theory and graph theory can be used to solve fundamental problems about the Rubik's cube. For example, how many different positions are possible is the same as asking what is the order of the Rubik's group. How many moves are necessary to solve the worst possible scrambling (God's Algorithm) is a question about the diameter of the associated Cayley graph.

• Topic:  Do Dogs Really Know Calculus?
• Speaker:  Eric Reyes, Rose Student, Math/Econ Major
• Date:  26 Oct 2005
• Abstract:  Least squares is a regression technique frequently used by engineers and scientists to gain insight into data generating processes. In 2003, Timothy Pennings of Hope College asked the question: "Do Dogs Know Calculus?" In an effort to see if his dog Elvis minimized the retrieval time when playing fetch, Professor Pennings collected data during a game of fetch on the beach. We take a second look at his data and statistical analyses. We show how a simple-looking problem can require intricate analysis. We use advanced methods, including weighted least squares, to detect and compensate for violations in the standard least squares assumptions. And, we seek to answer the question: Do Dogs Really Know Calculus?

• Topic:  Nonparametric estimation of volatility models with serially dependent Innovations
• Speaker:  Michael Levine, Purdue University
• Date:  20 Oct 2005
• Abstract:  We are interested in modeling the time series process yt = ¾(xt)"(xt)) where "t = Á0"t¡1 + vt. This model is of interest as it provides a plausible linkage between risk and expected return of financial assets. Further, the model can serve as a vehicle for testing the martingale difference sequence hypothesis, which is typically uncritically adapted in financial time series models. When xt has a fixed design, we provide a novel nonparametric estimator of the variance function based on the difference approach and establish its limiting properties. When xt is strictly stationary on a strongly mixing base (hereby allowing for ARCH effects) the nonparametric variance function estimator by Fan and Yao (1998) can be applied and seems very promising. We propose a semiparametric estimator of Á0 that is pT-consistent, adaptive, and asymptotic normally distributed under very general conditions on xt.

• Topic:  Finite Groups of Matrices with Integer Entries
• Speaker:  James Kuzmanovich (joint work with Andrey Pavlichenk), Wake Forest University
• Date:  28 Sep 2005
• Abstract:  Finite groups of nonsingular matrices with integer entries are some of the first groups seen in an undergraduate algebra course, since they only require knowledge of matrix multiplication and inversion. They nevertheless have many interesting properties and associated problems (some unsolved), and they have been the object of study by many famous mathematicians. Not much of this theory (or history) appears in undergraduate texts (and was not known by at least one algebraist - me), even though it is a good source of problems and projects. Indeed, this talk is a report on what Andrey and I learned as he wrote a term paper for my undergraduate algebra course and we followed it up with independent study. Most of this talk should be accessible to students who have had a linear algebra course. It will introduce ideas and concepts from many areas of mathematics, but prior knowledge will not be assumed.

### 2004-05 (latest first)

• Topic:  An Introduction to Constructible Numbers
• Speaker:  Kurtis Katinas, Rose Student, Mathematics
• Date:  18 May 2005
• Abstract:  Around 2500 years ago, the ancient Greeks proposed a set of three geometry problems about constructing certain lengths with an unmarked straightedge and compass. These are trisecting an arbitrary angle, doubling the cube, and "squaring" the circle. It turns out that all three of these feats are impossible. Trisecting the angle was the first to be disproved, but the ancient Greeks were not the ones who did it. It wasn't until the 1800's, when all three were disproved. What is most surprising about these solutions was that they did not use any heavy geometry. Instead, they relied on field theory and number theory. This talk is aimed primarily at undergraduate students as a walkthrough of two of these proofs and some details on the proof of the third. No knowledge of field theory is required or assumed. Basic number theory and little geometry will be used, but not required either.

• Topic:  New Goodness-of-Fit Tests
• Speaker:  Dr. Mark Inlow, inlow@rose-hulman.edu
• Date:  11 May 2005
• Abstract:  Goodness-of-fit tests are formal procedures for assessing the fit between a given model and the distribution of some quantity of interest. Using the moment-matching property of the exponential family of distributions, we derive new generalizations of the smooth goodness-of-fit test. (The exponential family of distributions encompasses many distributions including the normal, t, chi, exponential, gamma, beta, and Poisson families.) We compare the performance of our new tests with standard goodness-of-fit tests for the normal distribution.

• Topic:  Computational Modeling with Partial Differential Equations
• Speaker:  Chad Westfall, Wabash College
• Date:  13 Apr 2005
• Abstract:  Partial differential equations (PDEs) are used in many areas of science to model the behavior of quantities that depend on several independent variables. In this talk we will look at the process of modeling physical phenomena with partial differential equations. Working through a simple example we will highlight the issues and challenges in the discretization and solver stages of the process.

• Topic:  Batch Calculation of the Residues and Their Sensitivities, Or: How to compute almost any derivative using sum(prod([combnk(factors), dfactors.]))
• Date:  30 Mar 2005
• Abstract:  In determining the time-domain response of linear time invariant systems, the inverse LaPlace transform technique using partial fraction expansions has both practical and historical significance. The values which constitute the numerators of the partial fraction expansion are commonly known as the residues. A recent application of interest is brute force computation of the quadratic cost function for optimal output feedback which can be facilitated by the Sylvester expansion. Sylvester's expansion requires computation of the system residues. Computation of the residues is typically accomplished by deconvolving the system transfer function and evaluating ratios of polynomials at a system pole. In this work, the first order form of the partial fraction expansion is investigated. A general matrix equation is derived for computation of the residues. This equation is generalized for cases involving repeated as well as distinct system poles. The sensitivities of the residues to changes in system parameters can then be computed by differentiating this matrix equation. Typical numerical results are presented.

• Topic:  The $20,000,000,000 Eigenvector - Part II • Speaker: Kurt Bryan, bryan@rose-hulman.edu • Date: 26 Jan 2005 • Abstract: In the last talk I showed a key idea that lies behind how Google ranks the importance of each page in a web of interconnected pages. The problem boils down to computing an eigenvector of a certain n by n matrix, where n is the number of pages in the web. But Google currently indexes over 8 billion pages---how does one do linear algebra on matrices of that size? Gaussian elimination? If you believe that, I have got a bridge for sale. In part II we will look at how one can reasonably compute an eigenvector for these very large matrices, and I will address a few questions that were raised in the first talk. • Topic: The$20,000,000,000 Eigenvector - Part I
• Speaker:  Kurt Bryan, bryan@rose-hulman.edu
• Date:  19 Jan 2005
• Abstract:  When Google went online in the last decade, one thing that set it apart from other search engines was that its search result listings always seemed to deliver the good stuff up front. With other search engines you often had to wade through screen after screen of links to unimportant web pages that just happened to match the search text. Part of the magic behind Google is its ability to quantitatively rate the importance of each page on the web and so rank the pages, then present to the user the more important pages first. In these two talks I will explain one popular approach to rating web page importance. It turns out to be a delightful application of standard linear algebra.

• Topic:  Why Number Theorists Care About Elliptic Curves - Part II
• Speaker:  Ken McMurdy, mcmurdy@rose-hulman.edu
• Date:  08 Dec 2004
• Abstract:  Let E be an elliptic curve whose Weierstrass equation has rational coefficients. In the first installment of this talk, we defined an abelian group structure on E. We then showed how to compute the p-torsion subgroup, denoted E[p], which must always be isomorphic to two copies of the integers mod p. In Part II, we will show how a certain Galois group acts on E[p], resulting in a Galois representation into the group of invertible two-by-two matrices over the field Fp. This will all be done in great detail for the specific curve whose 3-torsion was worked out explicitly in Part I. Time permitting, I will then discuss an analogous construction of l-adic Galois representations, and connections with modular forms such as the Shimura-Taniyama-Weil Conjecture.

• Topic:  Why Number Theorists Care About Elliptic Curves - Part I
• Speaker:  Dr. Kenneth McMurdy, mcmurdy@rose-hulman.edu
• Date:  10 Nov 2004
• Abstract:  In two earlier talks, Allen Broughton discussed elliptic curves from the point of view of classical algebraic geometry over the real or complex numbers. Much of the recent interest in elliptic curves, however, has come from modern algebraic number theorists. Elliptic curves even played a prominent role in the proof by Andrew Wiles of Fermat's Last Theorem. This raises the question of why number theorists would be so interested in such a classically geometric object. In a series of two talks, I will attempt to give one answer to this question, by showing how certain Galois representations can be attached to elliptic curves with rational coefficients via their torsion subgroups. In the first installment, I will define the group structure of an elliptic curve E, and state the main results regarding the torsion subgroups E[N]. We will then use MAGMA to explore the structure of E[N] in some meaningful examples. In the second installment, I will focus more on the attached Galois representations, again embellishing the main results with a few meaningful examples.

Note: this will be during period 10 not the usual period 9.

• Topic:  Algebraic Cycles on Abelian Varieties
• Speaker:  Reza Akhtar, Miami University of Ohio
• Date:  27 Oct 2004
• Abstract:  The theory of algebraic cycles was initially developed with an view towards studying intersections on algebraic varieties. Since then, it has found many applications to K-theory, number theory, and most recently to the theory of motives. This talk will provide an introduction to algebraic cycles and abelian varieties, and will describe the interaction between the product structure on cycles and the group law on an abelian variety. Some recent results of the speaker in this area will also be discussed.

• Topic:  Equivalence of Real Elliptic Curves - Part II - Birational Equivalence
• Speaker:  Allen Broughton, brought@rose-hulman.edu
• Date:  13 Oct 2004
• Abstract:  This second talk on real elliptic curves will complete the picture of birational equivalence of real elliptic curves by looking at the complex elliptic curve defined by the original curve. The complex curve is called a complexification of the real curve and the real curve is called a real form of the complex curve. The complex curve is a torus and it interesting to visualize the real forms as curves on the torus. We will spend most of the talk exploring the very interesting relationship among the real forms, mirror reflections on the torus, and the automorphisms of the complex curve. Non-isomorphic real curves can have can have isomorphic complexifications. The main result we will show is that each complex elliptic curve defined by real equations has exactly two real forms which are birationally inequivalent. The most interesting part is that there is exactly one complex elliptic curve that has a real form with one component and another real form with two components. We will not use any calculations more complex than high school algebra and nor any geometric concepts beyond what we cover in our multi-variable calculus course. The calculations are made quite easy by using the Weierstrass form discussed in the first talk. The first part of the talk will be a recap of the first talk in the context of complex elliptic curves. There will be lots of pictures.

• Topic:  Equivalence of Real Elliptic Curves - Part I - Linear Equivalence
• Speaker:  Allen Broughton, brought@rose-hulman.edu
• Date:  06 Oct 2004
• Abstract:  This is the first of several talks on elliptic curves given by Allen Broughton and Ken McMurdy. In the two talks by Allen Broughton a complete answer will be given to a question posed by Ken McMurdy during his job talk last spring.
What is the moduli space of real elliptic curves like?
Since then a complete answer has been worked out and it is surprisingly simple.

In the first talk a basic introduction to real elliptic curves will be given -- starting from definitions, smoothness, projective completion, the geometry of the group law, the geometry of tangents and inflection points and ending up with the notions of embedded linear equivalence, normal Weierstrass form, and linear classification. The main result is that there are two families of curves each depending on a single real parameter. Each curve in one family has one component and each curve in the other family has two components*. The talk does not use calculations more complex than high school algebra and the geometric concepts that we cover in our multi-variable calculus course (except a smidgen of topology at one point). There will be lots of pictures. *Well that statement is almost true. The explanation of almost true will be given in the second talk, which will cover the complexifications of real elliptic curves, real forms of complex elliptic curves, the moduli space complex elliptic curves, and the automorphism groups of curves.

• Topic:  Fast Reconstruction of Cracks using Impedance Imaging
• Speaker:  Dr. Kurt Bryan, bryan @rose-hulman.edu
• Date:  22 Sep 2004
• Abstract:  This talk is based on the work done in our mathematics REU in the summers of 2002-2004, concerning some mathematical problems that arise in the non-destructive testing of materials. I will present an absurdly simple and fast algorithm to reconstruct linear cracks inside an object, by using electrical currents applied to the outer boundary of the object and then measuring the induced voltages on the outer boundary (or if you prefer to think in terms of heat, one applies a known heat source to the outer boundary and measures the resulting steady-state boundary temperatures). An insightful result by the 2003 group (extended by the 2004 students, using results from the 2002 group) turns this apparently hard problem into a DE I exercise!

### 2003-04 (latest first)

• Topic: The Celestial Sphere: Geometry and Astrolabes
• Speaker: Tanya Leise   leise@rose-hulman.edu
• Date: April 14, 2004
• Abstract: In the first and second centuries BC, Greek thinkers took the Babylonian beginnings of astronomy, which included the zodiac, and incorporated their brilliant geometrical ideas to create a mathematical model of the heavens that was both useful and accurate. Ptolemy's Almagest (ca. 100-150 AD) marks the peak of the development of the Greek mathematical astronomy. This early astronomy viewed the heavens as a great rotating celestial sphere with a stationary Earth at its center. The stars were fixed to the celestial sphere, while the sun moved along the zodiac, making one full circle each year. We will survey some of the geometry used in developing coordinate systems on the celestial sphere and in projecting the sphere onto a plane to result in a working two-dimensional model of the heavens -- the astrolabe. In order to visualize this sphere-to-plane stereographic projection, we will work some basic computations with astrolabes that I will provide to the audience, and compare the 2D astrolabe to a 3D celestial globe.

• Topic: The Joy of Zero Divisors (and possibly the horror if time permits )
• Speaker: Mike Axtell, Wabash College,    axtellm@wabash.edu
• Date: March 31, 2004
• Abstract: The talk will focus on a beautiful and surprising result linking Abstract Algebra to Graph Theory.  You need not know anything about Graph Theory (the speaker doesn't either).  You need not know anything about Abstract Algebra - relevant ideas are basic and will be introduced.  Warning: The speaker may use this opportunity to trash talk Rose prior to the ICMC (Indiana Collegiate Mathematics Competition) on Friday.

• Topic: Numerical ODE Solving for a Chaotic System
• Date: March 17, 2004
• Abstract: A simple non-linear dynamical system with chaotic properties is used to illustrate the advantages and limitations of Runge-Kutta (RK) based ODE solving.  Herein we describe the course "Computer Applications in Engineering 2" (ME 323): how it fits in the ME curriculum, and course objectives.  We quickly review the techniques of fixed and adaptive step fourth order RK (RK4).  The definition of stability for non-linear autonomous systems is reviewed.  We then present the physical system and its ODE representation.  Results are shown for adaptive and fixed-step RK4 where the system stability boundary estimate visibly changes due to numerical inaccuracies.

• Topic: Probability Models in Genetics
• Speaker: Amanda Lynn Stephens, Rose student, stephanal@rose-rulman.edu
• Date: February 18, 2004
• Abstract: A discussion of probability models in genetics. Genetics models such as the Wright-Fisher and the Moran Model will be analyzed with Probability Modeling.  The talk is based on an undergraduate research project by the speaker.

• Topic: Theme and Variations from Geometric Function Theory (3 talks)
• Speaker: Jerry Muir,  muir@rose-rulman.edu
• Dates:  January 28, 2004, February 4, 2004, February 11, 2004
• Abstracts of the talks:
• I. Convex Mappings of the Unit Disk: The theory of univalent (one-to-one and analytic) functions of the unit disk in the complex plane has been an area of active research for almost a century. Bieberbach's conjecture, proposed in 1916 and proved by de Branges in 1985, that a univalent function defined on the unit disk of the form

must satisfy for all n motivated a great deal of this research. In particular, many elegant results were proved for families of univalent functions that are defined by some geometric condition on the image of the function. Usually, there is no nice extension of results from one complex variable into higher dimensions, and this topic is no exception. Because of this, the geometric classes of functions are of special importance in that setting. In this, the first of three talks, we will consider univalent mappings of the unit disk whose image is a convex set in the plane. A sequence of appealing results will be given that draw upon some of the classical principles from Complex Analysis.

II: Some Examples and Obstacles in Higher Dimensional Geometric Function Theory: Having been introduced to some of the basic and elegant results of one variable geometric function theory, we turn our attention to the higher dimensional setting. Although natural to consider, this setting yields problems of much greater difficulty. Many of the simplest one variable results either have no reasonable extension or the extensions require difficult unintuitive arguments. We will introduce the basic ideas of function theory in higher dimensions, including all of the necessary definitions, and examine some situations where difficulties arise. This will include some counterexamples to natural generalizations of the one variable theory. We will conclude by considering different norms on the space C2 of two-dimensional complex vectors. The impact that changing norms has on the function theory is substantial. Recently developed constructions of convex mappings of the unit ball of C2 with certain non-Euclidean norms will be given.

III: Analysis of Convex Mappings of the Ball in Cn Onto Sets Containing a Line: In the last talk, we saw some instances in which elementary properties of convex mappings of the unit disk do not easily extend to a higher dimensional setting. Few examples of higher dimensional mappings are known, and those that are known fail to extend the familiar properties that some one-dimensional mappings posses. In this talk, we will focus on mappings F of the Euclidean ball B in Cn such that F(B) is a convex subset of Cn containing a line. These provide an interesting generalization of mappings of the unit disk onto strips and half-planes and may eventually be useful in the determination of the extreme points of the family of convex mappings.

• Topic: The Banach Fixed Point Theorem and Solvability of Integral Equations
• Speaker: Dan Abretske, Rose student, Daniel.A.Abretske@rose-rulman.edu
• Date:  January 21, 2004
• Abstract: As part of my independent study last quarter I studied various solvability conditions that can be placed on both linear and non linear operators. As an extension of that course I will be discussing the Banach Fixed Point Theorem and the Geometric Series Theorem. I will then show how they can be applied to integral equations of the second kind.

• Topic: Black Box Linear Algebra
• Speaker: William Turner, Wabash College,  turnerw@wabash.edu
• Date:  November 12, 2003
• Abstract: In symbolic computation and its subfield of computer algebra, we desire algebraic methods to compute an exact solution to a problem, as opposed to the numerical approximations supplied in numerical analysis.  In this talk, we introduce the black box model for symbolic linear algebra.  We investigate Wiedemann's approach to solve a system of linear equations and compute the determinant and rank of a black box matrix.

• Topic: Inverse Electrocardiography
• Speaker: Lorraine Olson, Mech Eng, Lorraine.Olson@rose-rulman.edu
(Joint work with Robert Throne, Rose-Hulman Institute of Technology and John R. Windle, University of Nebraska Medical Center )
• Date:  November 5, 2003
• Abstract: The heart is an electromechanical device. In its resting state, the heart is electrically polarized. For each heartbeat, a wave of "depolarization" travels through the heart muscle, causing the tissues to contract. If the electrical pathways in the heart malfunction, this leads to arrhythmias and poor blood flow. Hence, knowledge of the electrical patterns on the heart is extremely useful in diagnosing and correcting certain types of heart-conduction related defects. In recent years there have been a growing number of attempts at reconstructing surface potentials on the heart from minimally invasive remotely measured signals. Two basic approaches have been taken. In the oldest approach, body surface potentials are measured and used to estimate the potential patterns on the endocardium (outside surface of the heart). More recently, a probe which can be inflated within a heart chamber has been developed and is used to estimate potential patterns on the interior surface of the heart. Both of these estimation problems are "inverse problems", and they are very sensitive to small errors in the measurements. We therefore need to use some form of "regularization", or smoothing, to ensure that the answers we obtain are reasonable. The key question is how much smoothing to use, so that we obtain accurate answers. This talk will focus on the mathematical details behind the inverse electrocardiography problem for the inflated probe case: the governing equations, finite element methodology, regularization techniques, and methods for selecting the regularization parameters. We will also show preliminary results for the probe data.

• Topic: Small Cycles of the Discrete Logarithm (2 talks)
• Speaker: Joshua B. Holden, holden@rose-hulman.edu
• Dates: October 22, 2003 and October 29, 2003
• Abstract of the talks: Brizolis asked the question: does every prime p have a pair (g,h) such that h is a fixed point for the discrete logarithm with base g? In other words, is g^h congruent to h modulo p? We will extend this question to ask about not only fixed points but also two-cycles, and examine methods for estimating the number of such pairs given certain conditions on g and h. This problem has applications to cryptography, since one well-known cryptographically secure random number generator uses the idea of iterating the discrete logarithm and we hope that it does not fall into cycles too often!

• Topic: Fast Reconstruction of Internal Cracks with Thermal Imaging
• Speaker: Nic Trainor, Rose student, Nic.A.Trainor@rose-rulman.edu
• Dates:  October 1, 2003 and October 8, 2003
• Abstract: The ability to characterize the interior of an object without damaging the object is an invaluable tool in industry.  One useful technique of recent interest is "impedance imaging", or equivalently, "steady-state thermal imaging".  The idea, in thermal terms, is to use temperature measurements on the boundary of an object---specifically, imposed thermal energy fluxes and measured boundary temperatures---to determine interior structure, for example, to find internal cracks or voids. In these two talks we'll discuss some new mathematical results on thermal imaging for cracks, obtained in Rose-Hulman's summer REU mathematics program.  In the first talk we'll examine a new and very rapid approach to finding a single crack in the interior of an object, under the assumption that the crack blocks the flow of heat.  In the second talk we'll discuss how to extend the procedure to the problem of finding several interior cracks, and look at the issue of what types of input fluxes provide optimal resolution and stability.

• Topic: The Best Way to Knock ‘em Down
• Speaker: Art Benjamin, Harvey Mudd College,  benjamin@math.hmc.edu
• Date: October 9,2003
• Abstract: `Knock 'em Down'' is a game of dice that is so easy to learn that it is being played in classrooms around the world as a way to develop students' intuition about probability.  However, as our analysis will show, lurking underneath this deceptively simple game are many surprising and highly unintuitive results.

Disclaimer:  Professor Benjamin takes no responsibility for any scams or "get rich quick" schemes that students may learn by applying the ideas of this talk!

• Topic: Imaging the Inner Wall Profile of a Blast Furnace
• Speaker: Kurt Bryan, bryan@rose-rulman.edu
• Date: September 249,2003
• Abstract: A blast furnace is essentially a large vessel filled with molten material.  It turns out that the inner wall of the furnace, which is in contact with the molten interior, can change shape over time, becoming either thinner due to the corrosive nature of the furnace interior, or the wall can become thicker due to the build up of deposits.  Walls that become too thin are obviously dangerous, and it's also undesirable to have the walls become too thick.

It's obviously difficult to directly measure the profile of the inner wall when the furnace is in operation, so one would like a means of determining the profile indirectly, from the outside. One approach is to use thermal methods, by measuring the temperature and heat flux at positions on the outer wall and from this information infer the inner wall profile.

In this talk we'll consider a simple one-dimensional model of the situation, in which the furnace wall (or a cross section of it) is modeled as a thermally conductive bar, whose length changes slowly over time. We'll look at how one can use temperature and heat flow measurements at one end of the bar to determine the length of the bar at any time.  This is work done during our summer REU program.

### 2002-03

• Topic: Approximate solutions to the Boussinesq equation
• Speaker: Aleksey Telyakovskiy
• Date: October 2,   2002
• Abstract : The Boussinesq equation is a nonlinear diffusion equation that models the behavior of groundwater in unconfined aquifers. Solutions of the Boussinesq equation are considered in many areas of hydrology. In case of zero initial conditions, solutions of the Boussinesq equation exhibit wetting fronts that propagate with finite speed. For certain types of initial-boundary value problems the Boussinesq equation can be reduced to boundary-value problems for an ordinary differential equation for a scaling function. In this talk we construct approximate closed-form solutions to the one-dimensional Boussinesq equation.

• Topic: An Inverse Problem Arising In Non-destructive Testing for Cracks
• Speaker: Kurt Bryan,  bryan@rose-hulman.edu
• Date: October 9,2002
• Abstract : Consider some material object which may or may not have an internal "crack ". You want to find out if there is indeed such a crack, and if so, determine the location of the crack. The catch is that you must do it non-destructively---there's no point to cutting the thing in half only to find out it was good. Recently, two methods for imaging the interior of an object to find defects have been much investigated. The techniques use either heat or electrical energy to "see " inside objects, non-destructively. In this seminar I'll talk about mathematical research done with undergraduates in our REU program last summer, in which we extended some known theoretical and computational techniques for finding cracks in objects using thermal and electrical methods.

• Topic: The Distribution of the Kolmogorov-Smirnov Statistic for Exponential Populations with Estimated Parameters
• Speaker: Diane Evans, evans@rose-hulman.edu
• Date: October 23,2002
• Abstract: I will present the derivation of the distribution of the Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling test statistics in the case of exponential sampling when the parameters are unknown and estimated from sample data for n = 1 and n = 2 via maximum likelihood.

• Topic: Factoring Integers via Lenstra's Elliptic Curve Method
• Speaker: Noor Martin,  noor.martin@rose-hulman.edu
• Date: October 30,2002
• Abstract: This talk examines a method for factoring integers based on the use of Elliptic Curves modulo some composite number n. Published by H. W. Lenstra in 1987, this method is a modification of Pollard's p-1 method for factoring integers. Background information on both Elliptic Curves and Pollard's p-1 method will be covered as well.

• Topic: Elliptic Curve Cryptography
• Speaker: Matthew Ford,  matthew.ford@rose-hulman.edu
• Date: November 6,2002
• Abstract: Elliptic Curve Cryptography (ECC) provides an alternative method of public key cryptography. While RSA is based on the factorization of a composite number, ECC is based on the Elliptic Curve Discrete Log Problem. The difference in these problems makes ECC not vulnerable to some of the attacks against RSA. The current best known attack against ECC is an exponential time algorithm.

• Topic: The Combinatorics of Symmetric Functions
• Speaker: Thomas Langley, thomas.langely@rose-hulman.edu
• Date: November 13,2002
• Abstract: There is a remarkable connection between representations of the symmetric group and symmetric multivariable polynomials polynomials that are unchanged when the variables are permuted). This correspondence, in which characters of irreducible representations are mapped to Schur functions, allows the combinatorics of symmetric functions to be used to solve representation theoretic problems. This talk will provide an introduction to this complex and beautiful combinatorial world, introducing symmetric functions, tableaux, Schur functions, plethysm, and the Robinson-Schensted correspondence.

• Topic:Vanishing Cycles and Kaleidoscopic Quadrilateral Tilings
• Speaker: Allen Broughton,  brought@rose-hulman.edu
• Date: December 11,2002
• Abstract: For the last 5 years the focus of the Rose-Hulman REU Tilings group has been hyperbolic, kaleidoscopic tilings of Riemann surfaces by triangles. A lot has been discovered about these objects including a complete classification up to genus 13. Last summer we pushed beyond triangles to consider quadrilateral tilings. On the plus side the group theory did get a bit simpler; on the minus side we lost rigidity. A surface constructed from triangles is rigid in the sense that there are no transformations that preserve both angles and area. This is not true in the quadrilateral case. The euclidean analog is that all triangles with congruent corresponding angles and the same area are congruent. However, there is a one-parameter family of mutually non-congruent rectangles with the same area. On hyperbolic surfaces the same holds true, but there is an interesting twist. As we vary the quadrilaterals through an infinite family of equiangular, equal area quadrilaterals some curves on the surface take on arbitrarily small lengths, and shrink to a point as we go to infinity. These are the so-called "vanishing cycles" studied in algebraic geometry. We will show how to identify the vanishing cycles in simple geometric terms. Much of the talk will be explaining the basic concepts in terms of small visual examples. Students Isabel Averill, Michael Burr, John Gregoire and Kathryn Zuhr all contributed to this project.

• Topic: Equations, Scramblings, and Random Walks in Finite Groups
• Speaker: Gary Sherman, gary.sherman@rose-hulman.edu
• Date: December 18, 2002
• Abstract: We prove (casually) that the probability of solving an equation in a (finite) group is just about the reciprocal of the cardinality of the groups derived subgroup. Our approach is to:
• view your favorite group equation, xy = yx, in terms of a permutation action,
• introduce a new class of permutations, so-called scramblings, which are combinatorially related to derangements
• spawn a natural random walk on the derived subgroup. Natural research questions suitable for Natural research questions suitable for undergraduates ensue

• Topic:de Casteljau's Algorithm in Hyperbolic Space
• Speaker: Alla Genkina, Rose CS major
• Date: January 22, 2003
• Abstract: Geometric Modeling can be defined as the application of mathematics to describe the shape and properties of physical or virtual objects. This application of mathematics extends to various industrial and graphical fields. Since most of the fields are computerized, the algorithms developed to describe objects in mathematical terms can be programmed and analyzed by computers. This presentation will describe de Casteljau's algorithm which is used to generate Bezier curves. The curves that are created can then be utilized to model various objects. The presentation will demonstrate the use of the algorithm in both Euclidean and Hyperbolic Space, but the main concentration will be on its application and use in Hyperbolic Space.

• Topic:Prescribing the curvature of level curves
• Speaker: Dave Finn, finn@rose-hulman.edu
• Date: January 29, 2003
• Abstract: Given a function u(x,y), it is a straight forward calculation in vector calculus to determine the curvature K[u] of the level curves of u. This curvature can be computed using the Hessian of u. In this talk, we consider the problem of prescribing the curvature of level curves of a function ,

Given a function k(x,y), is it possible
to find a function u(x,y) with K[u]=k

As a problem in nonlinear partial differential equations, this problem poses some interesting questions, starting with the nature of the equation, the correct boundary values to consider, the effect of the domain on solvability, the effect of the boundary values on solvability, and finally the existence of a solution.

• Topic: Why (and how!) we should all use group projects in all introductory statistics courses
• Speaker: Douglas Andrews, Wittenberg University
• Date:  March 12, 2003
• Abstract: The overwhelming consensus emerging from the statistics education community over the past twenty years is for greater emphasis on exploratory data analysis, design, interpretation, and concepts, at the expense of probability, theory, recipes, and techniques. Moreover, education reform efforts in many fields highlight the benefits of active and collaborative learning pedagogies, as well as more authentic forms of assessment. Group data analysis projects -- in which students analyze data from simple observational studies and experiments of their own design -- can be an ideal way to implement these stat ed recommendations and realize these broader educational benefits in introductory statistics courses for all audiences. In this talk, I'll lay out some of the rationale for using such projects and give plenty of concrete advice for how to structure the experience.

• Topic: Automatic Differentiation of Algorithms (2 talks)
• Dates: March 26, 2003 and April 2, 2003
• Abstract: Many algorithms used in scientific computation require derivatives. Typically the function is provided--often in the form of a computer program--and the user must find or approximate its derivative. Automatic differentiation is a technique for automatically generating a program that produces those derivatives, by reading in the code of the program defining the function, considering its computational graph, and then finding its exact derivative. In Part I of this talk I will define the problem and outline its solution using this technique; in Part II of this talk I will discuss the two principal modes of automatic differentiation, forward and reverse, and how they use the computational graph to produce code for the required derivative(s).

• Topic: Homogenization: It's Not Just for Dairy Products
• Speaker: Kurt Bryan bryan@rose-hulman.edu
• Date: April 9, 2003
• Abstract: A material is homogeneous if its physical properties don't vary with position, at least at the physical scale of interest.  But many (one could argue all) materials are not homogeneous at the microscopic level, but possess a structure with small-scale periodic or random variation. Indeed, composite materials are intentionally designed with such small-scale variations, in order to have certain desirable physical properties. In many cases one would like to predict the bulk or macroscopic physical properties of a composite material from the microscopic structure.  Homogenization is a set of mathematical techniques for modeling a material with microscopic inhomogeneous structure as a macroscopically homogeneous material. In this talk I'll show one mathematical framework in which this is done, and illustrate with simple examples .

• Topic:Kaleidoscopic tilings on surfaces, this time with the groups (1st of 2 talk series)
• Speaker: Allen Broughton,    brought@rose-hulman.edu
• Date: April 30, 2003 and May 7, 2003
• Abstract: In the past I have given several lectures on kaleidoscopic tilings by triangles and quadrilaterals on surfaces, and asserted in these talks that the tiling group completely determined the combinatorial and topological structure of a tiling. However, I have never really talked about the influence of the group theory!  In this series of two talks I will give two examples of determining combinatorial and topological structure, by group computations. Each talk will focus on a problem I intend to give to REU students this summer. Thus, there will be no general theorems just problems statements with suggestions of attack, the talks will focus on developing the background to get to the problem statements. The first talk will include the necessary review of tilings and hyperbolic geometry.  You don't need to know much about group theory or hyperbolic geometry.

First talk:  Constructing a fundamental domain for kaleidoscopically tiled surfaces. We are all familiar with the process of creating a torus by identifying opposite sides of a euclidean rectangle. For higher genus surfaces of genus s > 1, a surface may be constructed by identifying sides of a hyperbolic 4s-gon. For a kaleidoscopically tiled surface can this be done so that the polygon is a "nice" collection of tiles?  The group theory computation will be focus on relating the infinite tiling group on the hyperbolic plane to the finite tiling group on the surface.

Second talk: When are kaleidoscopic tilings separating? Every edge of a kaleidoscopic tiling generates a reflection of the surface to itself fixing the edge. In the case of a sphere the fixed point set (or mirror) of the reflection is a great circle which separates the sphere into two pieces. This is very misleading example, since for higher genus the mirror very rarely separates the surface.  The question is: is there a fast way to determine this splitting property from the properties of the tiling group? The talk will present a method of attack using the group algebra of the talk.  Again, no previous knowledge of group theory is assumed.

• Topic: Guessing Secrets
• Speaker: Jon Mastin - Rose CS major
• Date:  May 21 , 2003
• Abstract: This talk will present a variation on the game "20 questions" which has arisen in the last few years in relation to internet security.  In the two player game, one player holds two or more secrets (IP addresses) while the second player asks yes or no questions.  If the first player must answer truthfully using one of his secrets, how much can the second player discover?  We will discuss the answer to this question and discuss strategy from the point of view of both players.

### 2001-02

• Topic: Mathematical Phylogeny
• Date: September 19 and 26,   2001
• Abstract (September 19): I will discuss how search engines use the singular value decomposition (SVD) to improve, and to score, the relevance of the results they return. This material will be needed for the second talk.
• Abstract (September 26): I will define the problem of mathematical phylogeny and the reconstruction of phylogenetic trees, then discuss current research being performed by Gary Stuart of ISU and myself that uses the ideas of the first talk to create such trees.

• Topic: Species Phylogenies from Whole Genomes using SVD
• Speaker: Gary Stuart (ISU),  Gary.Stuart@isugw.indstate.edu
• Date: October 3,   2001
• Abstract  : Following Jeff Leader's fine series of seminars intoducing SVD (Singular Value Decomposition) as a tool for generating biomolecular phylogenies, I will describe some of our very recent attempts to solve some very large problems using the same method.  In particular, I will describe the generation of a tree summarizing the evolutionary relationships of 19 bacterial species.  Unlike most trees, which result from the analysis of only one or a few genes or proteins, this tree is based on an exhaustive comparison of over 35,000 proteins predicted from whole genome sequence. Along the way, I plan to explore the "meaning" of the SVD relative to our application, and to present some of the (questionable?) assumptions upon which our method is based.

• Topic: The Mathematics of Financial Derivatives and Option Pricing
• Speaker: Kurt Bryan, bryan@rose-hulman.edu
• Date: October 17, 24, 31,   2001
• Abstract  : Most people are familiar with traditional investments like stocks, bonds, and commodities.  However, in the past few decades a huge market has arisen in the trading of "options" and other "financial derivatives", contracts in which payment is based on the value of some benchmark, e.g., the price of a given stock on a certain date.  In short, the value of the contract is derived from the price of some underlying asset (hence the term "derivative').

As an example, suppose a contract is written in which I give you the option (but not the obligation) to buy from me one share of Microsoft stock for a guaranteed price of $50 on January 1, 2002 (today, October 15, it's selling for$57).  This is an example of a European Call Option, in which you have the right to buy some asset at a guaranteed price sometime in the future.  How much should you pay to enter into such an agreement?  Surprisingly, there is a very quantitative strategy for determining the price of this option contract.

In these talks (3 or 4) we'll examine the problem of option pricing.  We'll start by looking at some common options, then at basic probabilistic models for asset prices.  Finally, we'll derive the celebrated Black-Scholes partial differential equation which shows how one can rationally determine option prices.  This is work for which Robert Merton and Myron Scholes won the 1997 Nobel Prize in economics.

• Topic: Linear Algebra for Cryptography
• Speaker: Jonathan Webster (Rose alumnus, U of I graduate student),    jewebste@students.uiuc.edu
• Date: December 5,2001
• Abstract  : Two of the most challenging cryptographic problems are the Discrete Log Problem (DLP) and the number factorization problem.  The best known solutions of these problems involve finding many relations among elements and then using linear algebra to solve a large system of equations.

The unique features of these matrices and what methods are used to solve them will be discussed.  The two specific methods will be discussed: Wiedemann and Lanczos.  Currently my work is using the Lanczos method to solve the DLP in a class group setting.

• Topic: Ideal Error-Correcting Codes: Reed-Solomon Decoding without Fourier Analysis
• Speaker: Matt Lepinski  (Rose alumnus, MIT graduate student),   lepinski@theory.lcs.mit.edu
• Date: December 19,   2001
• Abstract  : An error-correcting code is a set of strings (called codewords) such that any two strings in the set differ in a large number of positions. Error-correcting codes are very useful in data transmission (where a noisy channel may corrupt some positions in the string). This is because many positions must be corrupted by the channel in order for the receiver to mistake one codeword for another.

This talk deals with the decoding problem for error-correcting codes. That is, given a string, how do we find the codeword that differs from the given string in the fewest number of positions. The codes considered in this talk will be the commonly used Reed-Solomon codes and the number theoretic Redundant Residue Number System codes. The talk will present a new algebraic framework for thinking about error-correcting codes and show how this framework allows us to use the same ideas to decode both Reed-Solomon codes and RRNS codes. These ideas are of particular interest because they can also be applied to decoding Algebraic Geometry codes which are asymptotically the best known codes. (Although a discussion of Algebraic Geometry codes is beyond the scope of this talk).

This talk assumes no prior knowledge of error-correcting codes. However, familiarity with polynomial algebra and finite fields is helpful in understanding some of the ideas in this talk. Most of the material in this talk comes from the work of Madhu Sudan, Venkatesan Guruswami and Amit Sahai.

Probably the most common Error-correcting codes in practice are the Reed-Solomon codes which are based on polynomials over finite fields. These codes are used by everyone from NASA to CD manufacturers.

• Topic: Optimizing College Enrollments Under Uncertainty
• Speaker: Concetta DePaolo, Indiana State University,  sdcetta@befac.indstate.edu
• Date: January 16,2002
• Abstract  : Each year an institution must decide which students to admit in order to accomplish its goals (e.g. quality, enrollment, etc.) while satisfying various capacity constraints.  This presentation details a mathematical optimization model for this problem, which assumes that students are of different types that exhibit different (random) behavior.  The presentation will describe the properties of the optimal solution, as well as an implementation and a heuristic algorithm that are both Excel-based.  How the model is being used by Indiana State University to compare alternative admissions strategies and forecast the long-term effects of those strategies will also be touched upon.

• Topic: Automorphisms of Riemann Surfaces,  Galois  Groups, and Hecke Algebras
• Speaker: Allen Broughton, Rose-Hulman,   brought@rose-hulman.edu
• Date: March 20 and 27,   2002
• Abstract  : There is a classical and very well-understood connection between automorphism groups of compact Riemann surfaces and Galois groups of branched coverings of surfaces.  In the first of this series of two talks we will introduce and explore this idea. In the second talk we will consider non-Galois coverings, and  see how this situation can be partially captured by Hecke Algebras. These talks will highlight past and continuing work by students in the "tilings group" of the Rose-Hulman REU.

• Topic: The Theodorus Equations
• Date: May 1,   2002
• Abstract  : The square-root spiral, or spiral of Theodorus, will be introduced, then generalized to a map on R^n with many strange attractors.

• Topic: Calculation of Bernoulli Numbers and Values of Zeta Functions
• Speaker: Josh Holden, Rose-Hulman, josh.holden@rose-hulman.edu
• Date: May 15,   2002
• Abstract  : This talk will discuss some of the methods known to calculate Bernoulli numbers,
• with an emphasis on asymptotic analysis of their running times. Definitions (and some motivation) will be provided.  We will also discuss some more modern extensions of the Bernoulli number concept, and explore how and whether the methods for calculating Bernoulli numbers extend.

### 2000-01

• Topic: Statistics, Earwax, and the Bering Strait
• Speaker: Dr. Doug Wolf, Department of Statistics, Ohio State University
• Topic: Teaching statistics the EESEE way
• Speaker:Dr. Elizabeth Stasny, Department of Statistics, Ohio State University
• Date: September 12, 2000
• Abstract: This was a visit to recruit students into graduate statistics programs

• Topic: Singular Solutions to a Partial Differential Equation Arising in Corrosion Modeling
• Speaker: Kurt Bryan, bryan@rose-hulman.edu
• Date: September 20 and 27, October. 4, 2000
• Abstract for Sept. 20 and 27: I'll talk about some joint work with Michael Vogelius at Rutgers University, specifically a partial differential equation (PDE) that arises in the modeling of electrochemical systems. Although the PDE is linear, the boundary conditions contain an exponential type of nonlinearity. Under certain conditions the problem has a unique solution, but in other cases the boundary value problem has an infinite family of solutions with logarithmic singularities on the boundary of the domain. I'll show some numerical simulations, what we've been able to deduce about the nature of the solutions, and talk about what remains to be proved.

Abstract for Oct. 4: I'll discuss joint work with Lester Caudill at the University of Richmond, specifically a partial differential equation that arises in the modeling of heat flow through an object with an interior "crack" or flaw. The flaw is modeled as a discontinuity or jump in the temperature over the flaw, with a nonlinear relationship between the heat flux over the flaw and the temperature jump. I'll look at conditions under which the PDE has a unique solution, and discuss the inverse problem that motivates this: how to determine the location and nature of the interior flaw from boundary measurements.

• Topic: Cwatsets
• Speaker: Gary Sherman Gary.Sherman@rose-hulman.edu, and Dennis Lin, Rose student
• Date:  October 18, 25, November 1, November 8, 2000
• Abstract: A cwatset is a subset of binary n-space that is closed (c) with (w) a (a) twist (t) For example, C = {000,110,101} is a cwatset because;

C + 000 = C,
C + 110 = {110,000,011} is C with the first two components of each element transposed,
C + 101 = {101,011,000} is C with the first and last components of each element transposed

That is, for each element c of C there exists a permutation, pi, of three symbols such that the coset C + c is just C with pi applied to the components of each element of C.

The theory of cwatsets has roots in statistics (a cwatset determines a confidence interval for the mean or median of a symmetric random variable) and blossoms in graph theory (each isomorphism class of simple graphs has a unique cwatset associated with it) and algebra (constructions, morphisms, representations).  In this sequence of four talks we trace the development of the theory from the first cwatset sighting at Rose-Hulman in 1987 to the latest results on isomorphism classes of cwatsets while highlighting the contributions undergraduates have made to the theory

Talk 1: The statistical motivation for cwatsets, examples of cwatsets, and constructions of cwatsets.
Talk 2:  The group theoretic ideas which bare the soul of the theory of cwatsets.
Talk 3: The connection between representation and isomorphism of cwatsets.
Talk 4:  The determination of all cwatsets of order at most 23.

Talks 1,2 and 3 will be given by Gary Sherman and talk 4 will be given by Dennis Lin (a Rose student).

• Topic: Pi in the Mandelbrot set
• Speaker: Aaron Klebanoff
• Date:  Dec. 6, 2000
• Abstract: The Mandelbrot set is arguably one of the most beautiful sets in mathematics.  In 1991, Dave Boll discovered a surprising occurrence of the number pi while exploring a seemingly unrelated property of the Mandelbrot set.  Boll's finding is easy to describe and understand, and yet it is not widely known -- possibly because the result has never before been shown rigorously.  In this presentation, I will provide the necessary background material to understand what the Mandelbrot set is and what Boll's discovery was.  I will then outline a proof of the result.

• Topic: Why Chaos Toys are Chaotic.
• Speaker: Aaron Klebanoff
• Date:  Dec. 13 and 20, 2000
• Abstract for Dec. 13: The Horseshoe Map.  The horseshoe map is a simple map of the unit square into itself that is the prototypical example for a chaotic map.  I will define the map, explore its dynamics, and subsequently define what is meant by a chaotic dynamical system. Although this talk stands alone, it is preliminary material for next week's talk.

Abstract for Dec. 20: A Simple Chaotic Toy.  I will describe a simple (chaotic) toy that my colleague and I developed, built, and analyzed.  I'll outline a rigorous argument for showing that the toy (along with many executive-type "chaotic" desk toys) is chaotic by showing that it is well modeled by a system that is conjugate to the horseshoe map.  I'll also show a picture of the real toy as well as some computer generated animations.

• Topic: A New Formula for Computing Frobenius Numbers in Three Variables.
• Speaker: Janet Trimm, Rose student
• Date:  Jan 24, 2001
• Abstract :  It is well known that if a and b are relatively prime positive integers, then the Frobenius number of a and b is equal to ab-a-b.  Many authors have developed "explicit" formulas and algorithms for computing Frobenius numbers of relatively prime integers a1,a2, ... an when n>2.  But these formulas and algorithms are clumsy and complicated even for n=3.  In this paper, we will prove that there is surprisingly a nice formula that computes the Frobenius number of three positive integres a, b, and c where a and b are relatively prime.

• Topic: On the Probability that a Monic Integral Polynomial Is Irreducible
• Speaker: Timothy Kilbourn, Rose student
• Date:  Jan 31, 2001
• Abstract :  It is proved that if m is any positive integer, then the limiting value, as the prime-power q goes to infinity, of the probability that an m-th degree polynomial in F_q[X] is irreducible is 1/m. As a corollary, one obtains an identity which is indexed by the partitions of m and whose terms are unit fractions. Analogous probabilistic studies are carried out for various classes of integral polynomials, where the underlying notion of "probability" is defined in the spirit of "natural density," namely, as the limiting value, as n goes to infinity, of the usual combinatorial probability of irreduciblity in Q[X] (equivalently, Z[X])  for integral polynomials whose coefficients are bounded in absolute value by n. With this notion of "probability", it is shown that if m is between 2 and 6, then with probability 1, the random integral polynomial X^m+aX+b is irreducible; and if m is between 1 and 5, the same conclusion holds for the random monic integral m-th degree polynomial. Numerical evidence is presented in support of related conjectures.

• Topic: Mathematical Modeling with Categories
• Speaker: Ralph Wojtowicz
• Date:  Feb 7 2001
• Abstract :  Every concept arises from the equation of unequal things.  Just as it is certain that one leaf is never totally the same as another, so it is certain that the concept "leaf" is formed by arbitrarily discarding these individual differences and by forgetting the distinguishing aspects. ...What then is truth?  A movable host of metaphors, metonymies, and; anthropomorphisms:  in short, a sum of human relations which have been poetically and rhetorically intensified, transferred, and embellished, and which, after long usage, seem to a people to be fixed, canonical, and binding.  Truths are illusions which we have forgotten are illusions...  it is originally "language" which works on the construction of concepts, a labor taken over in later ages by "science".
--Friederich Nietzsche
"On Truth and Lies in a Nonmoral Sense" (1873)

A theory is a mathematical model for an aspect of nature.  One good theory extracts and exaggerates some facets of truth.  Another good theory may idealize other facets.  A theory cannot duplicate nature,  for if it did so in all respects, it would be isomorphic to nature itself and hence useless, a mere repetition of all the complexity which nature presents to us, that very complexity we frame theories to penetrate and set aside.    With this sober and critical understanding of what a theory is, we need not see any philosophical conflict between two theories, one of which represents a  gas as a plenum, the other as a numerous assembly of punctual masses.   Models of either kind represent aspects of real gases;  if they represent those properly, they should
entail many of the same conclusions, though of course not all.
---Clifford A.  Truesdell and Robert G. Muncaster
"Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas" (1980)

...in mathematical practice we must, more than in any other science, hold a given object quite precisely in order to construct, calculate, and deduce; yet we must also constantly transform it into other objects.
---F. William Lawvere
"Some thoughts on the Future of Category Theory"  (1990)

Categories are abstract mathematical structures which may be viewed as the places where mathematical models live.  A category consists of two sorts of things:  objects and morphisms.  Every morphism has source and target objects:       Source ----" Target Each object has an identity morphism and there is an associative composition operation on adjacent pairs of morphisms. An example is the category having sets as objects and functions  as morphisms.

The language of category theory is rich enough to describe diverse structures which arise in mathematical modeling and to express precise comparisons between models of different types. After discussing basic definitions and examples and giving a brief history of the theory, I will describe categories of sets and of stochastic matrices and a category having transition probabilities as morphisms.  I will give examples of  deterministic and stochastic, discrete-time dynamical systems and show how the former may be viewed as special cases of the  latter. Certain constructions that can be made with sets (points, cartesian products, disjoint unions) have useful interpretations in other categories.  I will also present an implementation of the category of stochastic matrices using Maple.

• Topic: Improving Solar Car Strategy
• Speakers: Brad Berron, Todd Goldfinger, Mike Ritter, Tom Schneider, Bill Stephen, Jerod Weinman, Rose students
• Date:  Feb 14 2001
• Abstract :  The MA331 Mathematical Modeling class has been working for the past four weeks on modeling a few important aspects of the Rose-Hulman Solar Phantom VI solar car project, with the goal of improving race strategy  for the upcoming 2001 American Solar Challenge.  We have focused on two basic issues: calculating available power from sunlight and computing torque-power curves for various speeds and hill grades.  The intensity of solar radiation changes over the course of a day, and depends on the current latitude, time of year, cloud conditions, and angle of the solar cell array on the car.  The resulting power available versus time of day curve can then be used to help determine race strategy for that day (e.g., the maximum speed allowed by the available power for current road and weather conditions).  To complement these calculations, we combined the efficiency curves provided by the engine manufacturer with the vast amounts of data compiled by past solar car runs to find torque-power curves for different constant speeds and hill grades.  These models can use the GPS data supplied by the race coordinators, giving information like latitude and altitude along the racecourse, to help determine optimal race strategies.

• Topic: Elliptic Curve Cryptography
• Speakers: John Rickert, rickert@rose-hulman.edu
• Date:  March 21, 28  and April 4,    2001
• Abstract : Public key cryptography has been applied to many systems in which encoding and decoding a secret message needs to be simple, while cracking to code must be difficult. Computer security, Internet sales and smart cards are three of the places in which public key cryptosystems are being used. In 1978, Rivest, Shamir, and Adleman proposed the RSA cryptosystem, which is currently used in many secure applications. As attacks on RSA have grown more sophisticated, other cryptosystems have been proposed. The Elliptic Curve cryptosystem is in currently use, and is growing in popularity, especially in applications, such as smart cards, in which memory is limited.

March 21 -  Number Theory and Public Key Cryptography
An introduction to public key cryptography and how some basic ideas from number theory are used to generate relatively secure cryptographic systems. This talk will discuss elementary modular arithmetic and SA-cryptography.

March 28 - Introduction to Elliptic Curves
A look at some simple examples of elliptic curves and the emergence of algebraic structure through some simple geometry and basic polynomial algebra.  We will also look at how the correspondence between the algebra and the geometry is used to work with elliptic curves over finite fields.

• Topic: Mathematical and Computer Models of Specifications for Complex Systems
• Speakers: Bill Schindel, ICTT,   Bill.Schindel@ictt.com
• Date:  April 18, 2001
• Abstract : ICTT, in conjunction with System Sciences, LLC, carries out systems engineering projects for industrial clients with high complexity systems composed of many technologies--mechanical, electronic, hydraulic, computers and communication, and business processes. (Refer to www.ictt.com.)  The
company has evolved over many years a methodology called Systematica Methodology(tm). This methodology is specialized for modeling not single systems but (economically more important) families (product lines) of systems, in which patterns can be detected and their common content propagated and managed. Among the patterns we are interested in are patterns of intelligent behavior. The resulting approach establishes class hierarchies of system models that show the degree to which families of systems share common content (behaviors in particular) and the extent to which these are variant to satisfy local (e.g., market) specialization needs. A broad group of rules, called Gestalt Rules, are used to express commercial engineering guidelines for individuals trying to keep their divisional designs consistent with the general patterns of corporate architectures.

This suggests that metrics could be developed to express a variety of useful commercially and scientifically/mathematically interesting quantities--similarity, re-use, variance, etc., along with a number of other interesting and useful tools. These appear to be potentially good student projects. The general subject of this work is  important (economically and competitively) in almost all large product line oriented product and services enterprises.

• Topic: Mathematical Modeling of Shape Memory Alloys
• Speakers: Tanya Leise,  Tanya.Leise@rose-hulman.edu
• Date:  April 25, 2001
• Abstract : The first shape memory alloy was discovered in 1962 at the Naval Ordnance Lab, when metallurgist
William Buehler passed a NiTi sample around at a meeting and showed how the metal was very flexible and held up well to repeated bending.  You can imagine their surprise when, on a whim, a pipe-smoking scientist heated the bent sample with his lighter and it immediately sprang out straight again.  I'll provide samples of nitinol wire and springs so we can experience this phenomenon firsthand, and then we'll look at some of the mathematical models developed in the past few decades for shape memory alloys.  (In particular, I'll include a model makes a great calculus project that takes students beyond naive implementation of the Second Derivative Rule.)

• Topic: Partial Least Squares Analysis
• Speakers: Yosi Shibberu,  shibberu@rose-hulman.edu
• Date: May 2,   2001
• Abstract : Partial least squares analysis is a relatively new empirical modeling technique. It has been found to be useful in rational drug design where the objective is to relate the chemical structure and properties of a drug molecule to its biological activity. In such problems, the number of variables significantly exceeds the number of equations. We will begin with a review of ordinary least squares and principle component analysis. Partial least squares analysis will then be introduced and compared to the previous two techniques. A simple spring system will be used to illustrate the main ideas.

### 1999-00

• Topic: Hidden Markov Models
• Speaker:Yosi Shibberu shibberu@rose-hulman.edu
• Date: September 22, September 29, and October 6, 1999
• Abstract: Hidden Markov Models were introduced and developed in the late 1960s and early 1970s. They are in wide spread use in speech recognition computer algorithms and more recently are being used as part of computational algorithms in bioinformatics. We will begin with a description of Hidden Markov Models and then proceed to an application of these models to DNA sequence alignment.

• Topic: Algebraic Numbers and Triangle Iterations
• Speaker:Matt Lepinski, Rose-Hulman student
• Date: October 20, 1999
• Abstract: The Hermite problem is to find ways of representing a number which make  algebraic properties of the number apparent. One well studied solution to the Hermite problem is the continued fraction which is a method of
• representing a number as a sequence of integers in such a way that the sequence is repeating if and only if the number is a quadradic irrational.  We present a generalization of the continued fraction based on an iterative mapping of a triangle in the plane. This allows us to represent a point in the plane with a sequence of integers that is periodic only if both coordinates of the point are algebraic numbers of degree at most three. We show how these sequences can be used to construct integer vectors arbitrarily close to a plane in three dimensions.  We then present a link between this linear algebra and the geometry of  the iteration that defines the sequences. Finally, we make use of this  connection to address the question of when an infinite sequence represents  a unique point.

• Topic: Motion of a Hanging Chain after the Free End Is Given an Initial Velocity
• Speaker: Herb Bailey Herb Bailey@rose-hulman.edu
• Date:  October 27, 1999
• Abstract:  One end of a chain is attached to the ceiling and the free end is given a sharp horizontal blow. The resulting pulse travels to the top of the chain, and a few seconds later the reflected pulse causes the free end
• to give a kick. The free end kicks again and again at regular intervals. The time between kicks is constant and has been accurately predicted by the solution of an ordinary differential equation. Close observation of the nature of successive kicks shows that they are not always in the same direction, but they do follow a pattern that repeats every four kicks. We have modeled this experiment by solving of the wave equation with variable tension and summing the resulting series solution. The lateral deflection as a function of time and distance along the chain was calculated. The predicted deflection of the free end is in good agreement with experimental results obtained from a movie of the chain motion.

• Topic: An Introduction to Dynamically Accelerating Cracks
• Speaker: Tanya Leise  Tanya.Leise@rose-hulman.edu
• Date:   November 3, November 10, 1999
• Abstract: Fracture behavior is highly dependent on the microstructure of the material.  Two competing philosophies have arisen in the study of dynamic fracture mechanics.  Lattice models and molecular dynamics treat materials as a lattice of atoms and focus on the microscale to capture the microstructure of the material.  Continuum models treat the material as a bulk with large-scale properties, with special small-scale properties only near the crack-tip, where the microstructure has the strongest effect on the material response.  A continuum model for a dynamically accelerating crack will be presented with a solution method based on the idea of a Dirichlet-to-Neumann map.  The fracture problem is essentially a hyperbolic PDE with boundary conditions that depend on the unknown crack tip path.  To solve for the crack tip motion, an appropriate fracture criterion must be chosen to model the material's microstructure and fracture behavior.  The solution method will be demonstrated for the simplest case of a single crack in the context of antiplane shear in a linearly elastic material.

• Topic: Dynamical Analysis of a Chaotic Electrical Circuit
• Speaker: Evan Graves, Rose student
• Date: December 15, 1999
• Abstract: Rollins and Hunt published a paper in 1982 in which they described the presence of chaotic behavior in a simple electrical circuit consisting of a resistor, inductor, and a diode in series.  They attributed the nonlinear dynamics to the way in which the diode acted as either a capacitor or a voltage source, depending on the flow of current and the reverse recovery time of the diode itself.  They also developed a set of equations that they claimed would model the circuit behavior.  My objective was to experimentally acquire data and use the given model as a guideline to analytically determine the underlying dynamics of the chaotic system.  As voltage increases in the circuit, driven at a frequency around the system's resonance, we are able to see period doubling in both the voltage and the current along with chaotic behavior at high voltages.  Chaotic data of the observed current was analyzed with the use of software developed by Randle Inc. at Applied Nonlinear Sciences, LCC.  A three dimensional representation of the reconstructed chaotic attractor was created, along with an approximation for its true dimension and the corresponding Lyapunov exponents of the system.

• Topic: Which way did that bicycle go?   and other geometric questions about bicycle tracks
• Speaker: David Finn   finn@rose-hulman.edu
• Date: January 12, 2000
• Abstract: You are walking across a snow covered road, and come across a set of  bicycle tracks.  Can you tell which way the bicycle was going?  Which track was produced by the front tire and back tire of the bike? In this talk, we will discuss a mathematical description of bicycle tracks to answer these questions and pose others.

• Topic: Interval Methods for Optimization
• Date: January 26, February 2 and February 9, 2000
• Abstract:  Part I (January 26)  We introduce interval arithmetic, an extension of standard floating
point arithmetic which uses directed rounding to generate intervals guaranteed to contain the result of a given computation.  We indicate some types of problems for which this approach, sometimes called reliable computing, is appropriate, and address the issue of whether or not the width of the intervals can be made sufficiently small. (Interval packages are available for C/C++, Fortran, MATLAB, and other
languages.)

Part II (February 2) We discuss the interval Newton's method as a prelude to introducing Hansen's method, an interval technique for global optimization.  We discuss the advantages and disadvantages of this technique and compare it to other methods for global optimization problems.

Part III (February 9)  We discuss the centered form and its variations.  These are the particular expressions used to obtain quadratic convergence in interval methods.  We also discuss some related applications of interval methods in matrix algebra.

• Topic: Which way did he say that bicycle went?
• Speaker: David Finn   finn@rose-hulman.edu
• Date: April 12, 2000
• Abstract: In a previous talk, I claimed that the only bicycle tracks for which you can not tell which way the bicycle went are either straight lines or concentric circles.  This fact about bicycle tracks is true only in a restricted case.  We need to assume a hypothesis that was not stated during the previous talk.  In this talk, a general procedure for constructing bicycle tracks for which you can not tell which way the bicycle went will be given.

• Topic: Higher Genus Soccer Balls and Kaleidoscopic Tilings in the Hyperbolic Plane.
• Speaker: Allen Broughton   brought@rose-hulman.edu
• Date: April 19 and 26, 2000
• Abstract: Two talks on kaleidoscopic tilings, for a general mathematical audience of students and faculty.  The purpose of the talks is to present an area of intriguing mathematical research, rich with problems suitable for undergraduate research.

A soccer ball has an attractive pattern of pentagons and hexagons on its surface, with a great deal of symmetry. Baseballs and basketballs also have certain patterns of symmetry which are different from the soccer ball pattern. Though the sportsman might never ask, a mathematician would be intrigued by the possibility of  a  higher genus soccer ball (a soccer ball with patterned handles). It turns out that they exist in great abundance though we need to give up on having only hexagons and pentagons.

The key to creating and understanding "soccer balls" are kaleidoscopic tilings of the 2-dimensional geometries: the sphere, the euclidean plane and the hyperbolic plane.  The sphere tilings, of course, yield the patterns of sports balls.  The tilings of the euclidean and  hyperbolic planes form beautiful patterns and have their own artistic interest, as in some of the art of Escher. The higher genus soccer balls, though impractical, are a convenient mental hook for generating questions about patterned surfaces, e.g., constructing simple examples.

In the first talk the relation between (higher genus) soccer balls and tilings will be explored, including an introduction to hyperbolic geometry.

In the second talk I will present some work completed jointly by undergraduates and myself on divisible kaleidoscopic tilings, i.e., simultaneous tilings of the plane by two different kaleidoscopic polygons.  It has a nice interplay between combinatorics (Catalan numbers) and geometry.

• Topic: Linear Fractional Transformations in Complex Euclidean Space
• Speaker: Jerry Muir   Jerry Muir@rose-hulman.edu
• Date: May 3 and 10, 2000
• Abstract:  Linear fractional transformations (LFTs) of a single complex variable are key tools for complex analysts interested in geometric properties of analytic functions in the plane.  For instance, an LFT can be used to conformally map a domain of the complex plane that is either the interior of a circle, the exterior of a circle, or a half-plane onto another prechosen domain of the same type.  Moreover, LFTs are the only functions that will do this.  Using LFTs to switch domains is useful when the geometric nature of one domain is more suitable to a particular problem.  If similar processes existed in higher complex dimensions, they would likely have similar uses.  We will define a higher-dimensional analog to one-variable LFTs and explore what, if any, one-variable properties extend to the higher dimensional case, and we will consider several examples.

### 1998-99

• Topic: The method of lines for solving differential equations.
• Speaker:Dave Voss, Western Illinois University
• Dates: Last half of winter quarter.
• Abstract: The Method of Lines (MOL) provides a flexible and general approach for solving systems of time dependent partial differential equations.  Using MOL, the space variables are discretized on a selected
• mesh yielding an approximating system of ordinary differential equations. The numerical solution of this system can present certain difficulties depending on the method used.  In this talk, the one-dimensional heat equation will be used to introduce MOL with semidiscretization provided by central difference approximations in space.  Some of the numerical difficulties will be exposed by following the time evolution using the Euler methods and the Crank-Nicolson method. Approaches for overcoming these difficulties will be suggested.

• Topic:  Elementary Inversion of the Laplace Transform
• Speaker:  Kurt Bryan  bryan@rose-hulman.edu
• Dates: Last half of winter quarter.
• Abstract:  Last summer while working on a research problem, I discovered a very beautiful and simple formula for inverting the Laplace Transform.  Even the proof that the formula works is very simple and involves only elementary analysis.  After a long search (and a bit of luck---bad luck) I learned that the formula had been discovered by Emil Post in 1930.  I'll show the formula, sketch the proof that it inverts the Laplace Transform, and give a few computational examples.

• Topic:  Root Locus, Feedback, and Block Diagrams
• Speaker: Robert Lopez
• Date:4/21/99
• Abstract:  Our second course in differential equations has traditionally contained the topics of linear systems of ODEs, eigenvalues, and stability.  Of late, there is a growing tendency to solve these linear systems by Laplace transform. When our students enter engineering courses on feedback controls, courses which should build on our presentations of systems of ODEs, they see what is at first glance, a totally different subject.  The ODEs are hidden behind block diagrams, representations of the transfer functions, and "asymptotic stability" never appears in the engineering controls texts.  This talk will make the connection between the linear systems we teach, feedback control as seen in engineering, and the root locus, the locus, in the complex plane, of the roots of a polynomial which depends on a parameter.

• Topic: Maple-Inspired Vignettes in Applied Math
• Speaker: Robert Lopez
• Date: 4/28/99
• Abstract: The following topics from classical applied math will be presented through the medium of Maple: the longitudinal vibrations in an elastic rod, Bezier curves, and the problem of  " two beads on a string through a hole in the table".

• Topic:  Algorithmic RSA Number Factorization
• Speaker: Stephen Young, Rose student
• Date: 5/18/99
• Abstract:  Presentation of an elementary algorithm to factor arbitrarily large RSA encryption numbers.  Also discussion of the practicality of this algorithm and potential for efficient implementation.

### 1997-98

• Topic: THE GAME OF BILLIARDS ON THE PLANE AND ITS CONNECTION WITH ARITHMETIC, GEOMETRY (TOPOLOGY), AND PHYSICS
• Speaker: Dr. Gregory Galperin, Department of Mathematics Eastern Illinois University
• Date:   November 3, November 10, 1999
• Abstract: A billiard system is the simplest dynamical system one can imagine: it's just a region and one point that moves inside that region with unit speed and reflects off the boundary according to the law "angle of incidence equals angle of reflection." It turns out, however, that the billiard system is very rich and can explain many interesting mathematical facts. The speaker will discuss connections between billiards and mathematical notions and results such as hyperbolic geometry, compact surfaces, the "first digit problem" for powers of 2, decimal digits of pi , and geodesic flow on the surface of a polyhedron.

• Topic: Wavelet - based methods in Image Processing
• Speaker:Allen Broughton brought@rose-hulman.edu
• Dates: last half of winter quarter and first half of spring quarter.
• Abstract: Seven  talks on mathematical methods used in image processing, the discussion will give the background on matrix models of images, Fourier and filtering methods and then finally wavelet and filter bank methods.
• Special Notes: Lecture Notes

• Topic:  RSA Cryptography
• Speaker:  Kurt Bryan bryan@rose-hulman.edu
• Dates: last half of spring quarter
• Abstract: 4 to 6 talks explaining  the very elementary number theory behind RSA, how RSA works, and allied topics like primality testing and factoring.
• Resources: outline

### 1996-97

• Topic:  Using Mathematics in Industry
• Speaker: Roy Primus, Rose alumnus
• Date: Friday, December 13, 1996
• Abstract: Roy Primus, a Rose-Hulman graduate (BA in Mathematics in 1975 and a Master's Degree in 1977) and currently the Research and Engineering Director of Combustion Research at Cummings Engine Company, will give a presentation during 4th period on Friday, December 13, in E-104. The topic of his talk will be "Using Mathematics in Industry"

• Topic:  Traveling Salesman Problem and Other Problems in Combinatorial Optimization
• Speakers:  Kurt Bryan bryan@rose-hulman.edu, Lynn Kiaer, David Mutchler  David.Mutchler@rose-hulman.edu
• Dates:   winter and spring quarters
• Abstract: The traveling salesman problem, scheduling problems, computational solution methods, simulated annealing, genetic algorithms

### 1995-96

• Topic:  Wavelets for data analysis
• Speakers:  Dave Bond
• Dates: winter quarter

• Topic:  The USA Mathematical Talent Search: Rose-Hulman's Role in Its Development
• Speaker: George Berzsenyi
• Date: May 2, 1996
• Abstract: The USA Mathematical Talent Search (USAMTS) is a year-round program for talented high school students in creative mathematical problem solving. The present talk will provide an overview of the past seven years of this program, focusing on Rose-Hulman's role in its development, on the last two years of activities, and on the future of the USAMTS. The speaker will also discuss some of the mathematical problems used in the program and some of his other mathematical activities during his recent sabbatical leave.