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Mathematics Modules
  • Mathematics Modules

     

    Discrete vs. Continuous Population Models 
    Discrete vs. Continuous Population Models: Continuous Logistic  
    Discrete vs. Continuous Population Models: Discrete Logistic  
    Discrete vs. Continuous Population Models: Geometric 
    Discrete vs. Continuous Population Models: Exponential vs. Geometric  

    Module Author: Lewis D. Ludwig
    Author Contact: ludwigl@denison.edu
    Funded By: National Science Foundation

    This module is designed for students in an introductory calculus course. Completion of the module requires knowledge of dierentiation and integration of exponential functions, basic knowledge of logarithms, and basic algebra. If students have the pre-requisite math skills, this module could also be used in any course studying introductory ecology. To make the module as accessible as possible, the module includes Microsoft Excel R spreadsheets to explore various aspects of discrete and continuous growth models: Geometric.xls, Discrete Logistic.xls, Continuous Logisitc.xls, and Exponential vs. Geometric.xls.

     

     

    Evolutionary Game Theory: The Hawk-Dove Game

    Module Author: Craig Jackson
    Author Contact: chjackso@owu.edu
    Funded By: National Science Foundation
    Evolutionary game theory successfully applies the mathematical techniques of classical game theory and simple dynamical systems to study the evolution of animal behavior. In particular, by placing a given animal contest in a game theoretic context, theoretical results concerning the "best strategies" for playing the game are shown to be directly related to the long term stable states of the animal population.

    This module is intended for use in introductory courses on mathematical modeling, biomathematics, and computational science, though it could successfully be used in introductory biology and zoology courses if the students have a strong background in mathematics. In any case, the highest level of mathematics assumed is a first course in calculus, though this prerequisite can be relaxed by the instructor at the expense of amending the discussion slightly in one or two places. However, of particular importance is the concept of recursively defined quantities. This is because population frequencies in one generation will naturally depend on the frequencies of the previous generation.

    The strength of this module is its use of systems dynamics software (specifically Vensim) to model the time-evolution of animal populations. Vensim allows the user to change the parameters of a model in real time while in the midst of a simulation. This lets students see the effect of such a change immediately and intuitively.

    Diagrams for building the models in Vensim, as well as descriptions of their defining equations, are included below. Moreover, model files are attached for the Hawk-Dove game and several of its extensions. However, detailed instructions on using Vensim are not included in this module. 

     

    Public Key Cryptography and the RSA Cryptosystem

    Module Author: Nuh Aydin
    Author Contact: aydinn@kenyon.edu  

    Funded By: National Science Foundation
    Cryptography, or cryptology, is a subject that is concerned with privacy or confidentiality of communication over insecure channels, in the presence of adversaries. It seeks to find ways to encrypt messages so that even if an unauthorized party gets a hold of a message, they cannot make sense out of it. The ways to break encryption systems, called cryptanalysis, is also part of the subject. Cryptography is sometimes confused with the related but distinct field of coding theory that deals with reliability of communication over noisy channels. See the author’s earlier module titled “An introduction to coding theory via Hamming codes” for an introduction to coding theory.

    There are two basic methods in cryptography: classical cryptography and public key cryptography. The latter is a more recent idea and this module will focus on that method through one of its best known and widely used examples: RSA cryptosystem. Proposed in 1977, the RSA cryptosystem has survived many attacks and is still commonly used. 

     

     

    Mathematical Models for a Chemical Reactor

    Module Author: Ignatios Vakalis
    Author Contact: ivakalis@calpoly.edu  

    Funded By: National Science Foundation
    The module presents the development and solution methodology of mathematical models for the concentration of a chemical in a reactor. The steady state case is first examined by providing the development and the solution (both analytical and numerical) process of the underlying models. The modeling process is extended to include time dependency by presenting the formation of the mathematical model and its numeric solution process.



    Relaxation Methods for Partial Differential Equations: Student
    Relaxation Methods for Partial Differential Equations: Instructor
    Relaxation Methods for Partial Differential Equations: Samples

    Module Author: David G. Robertson
    Author Contact: drobertson@otterbein.edu   

    Funded By: The National Science Foudation
    Approaches for numerically solving elliptic partial differential equations such as that of Poisson or Laplace are discussed. The basic technique is that of \relaxation," an iterative scheme based on a discretization of the domain of interest. Applications to problems in electrostatics in two and three dimensions are studied. Extensions including overrelaxation and the multigrid method are described.

     

    Models for Birth and Death Processes of Organisms

    Module Author: Ignatios Vakalis
    Author Contact: ivakalis@calpoly.edu  

    Funded By: The National Science Foundation
    The module presents the development and solution methodology of deterministic and stochastic models for a cadre of scenarios pertaining to the birth and death processes of organism populations. The module starts with the construction of simple models for pure birth or pure death processes. Gradually it presents the process of developing and solving more complex scenarios leading to general coupled birth-death models (deterministic and stochastic) under various assumptions.

     

     

    Solving the Time-Independent Schrodinger Equation: Student
    Solving the Time-Independent Schrodinger Equation: Instructor

    Module Author: David G. Robertson
    Author Contact: drobertson@otterbein.edu
    Funded By: The National Science Foundation
    Approaches for numerically solving the time-independent Schrodinger equation in one dimension are discussed. Possible simulation projects include solving the TISE for various potentials, testing the validity of stationary state perturbation theory, and developing a simple model of covalent bonding. Some background on numerical techniques for integrating differential equations is provided.

     

    First Order Difference Equations

    Module Author: Ignatios Vakalis
    Author Contact: ivakalis@csc.calpoly.edu
    Funded By: National Science Foundation (9952806)
    First order difference equations are explored.

     

    First Order Differential Equations

    Module Author: Ignatios Vakalis
    Author Contact: ivakalis@csc.calpoly.edu  

    Funded By: National Science Foundation (9952806)
    In this chapter we will examine the solution methodology of specific classes of ODEs. Thus whenever we refer to a differential equation, we mean an ordinary differential equation, unless we explicitly say otherwise.

     

    Higher Order Difference and Differential Equations

    Module Author: Ignatios Vakalis
    Author Contact: ivakalis@csc.calpoly.edu  

    Funded By: National Science Foundation (9952806)
    We now turn to the solution of differential equations of order two or higher. We will now examine various techniques for solving linear homogeneous and non-homogeneous differential equations. First we will concentrate on second-order equations for two main reasons. First, their theory is typical of that of linear differential equations on any order n and second they have important applications in many fields of science and engineering. Numerical methods for higher-order differential equations will also be presented.

     

    Higher Order Linear Difference Equations

    Module Author: Ignatios Vakalis
    Author Contact: ivakalis@csc.calpoly.edu
    Funded By: National Science Foundation (9952806)
    The main part of this section covers the solution methodology of linear second order difference equations (discrete dynamical systems). We examine two cases: i) homogeneous, 2nd-order, linear discrete dynamical systems; and ii) non-homogeneous, 2nd – order, linear discrete dynamical systems. Dynamical systems are powerful tools for modeling and analyzing a variety of situations.

     

    Laplace Transformations

    Module Author: Ignatios Vakalis
    Author Contact: ivakalis@csc.calpoly.edu
    Funded By: National Science Foundation (9952806)
    We investigate a technique that transforms (1) into an algebraic equation that can often be easily solved, so that the solution to the differential equation can be obtained. It works for smooth and non-smooth functions g(x). Furthermore, the technique is also valid, even if the coefficients of the left-hand side of (1) are functions of the independent variable t.

     

    Systems of Linear Differential Equations

    Module Author: Ignatios Vakalis
    Author Contact: ivakalis@csc.calpoly.edu  

    Funded By: National Science Foundation (9952806)
    In this chapter we present the solution methodology for a system of linear first-order differential equations as well as for linear discrete dynamical systems. As you will see, any high order linear differential equation can be transformed to a system of linear first order differential equations. The same holds true for the case of discrete dynamical systems. Up to this point we have focused our attention in solving differential equations and discrete dynamical systems that involve one dependent variable. However, many physical situations are modeled with more than one equation and involve more than one dependent variable. For example to determine the population of two interacting species such as foxes and rabbits, we would have two dependent variables that represent the two populations and one independent variable that represent time. 

     

    Systems of Linear Difference Equations

    Module Author: Ignatios Vakalis
    Author Contact: ivakalis@csc.calpoly.edu  

    Funded By: National Science Foundation
    We will examine the solution methodology of 1st –order linear systems of difference equations / discrete dynamical systems. Note that any high order linear difference equation can be transformed (by introducing new variables) to a system of 1st-order linear difference equation. The same methodology was used in transforming a high order linear differential equation to a system of 1st-order linear differential equations. First we present analytic solutions for linear discrete dynamical systems of several variables. Matrix notation is used to represent each system and results from linear algebra are utilized in the solution methodology. Numerical solutions for systems of difference equations are also outlined.

     

    Fourier Transforms, Fourier Series and the FFT
    Fourier Transforms, Fourier Series and the FFT: Zipped Files

    Module Authors: Lisette de Pillis, Ami Radunskaya
    Author Contact: depillis@hmc.edu  

    Funded By: W. M. Keck Foundation
    Given a set of data, one is often faced with the question: Do the data have hidden patterns not immediately visually apparent? Alternately, one might wish to take a particular time series, such as one coming from sound signals, and ask the question: How does one represent this time series as a sum of periodic components? Fourier theory can be used to address both types of questions: How does one look for periodic patterns in data and how does one re-represent sets of periodic data as a sum of simpler periodic functions. The goal of this module is to give students an understanding of the one-dimensional Fourier Transform, both mathematically and computationally, with a focus on finding periodicity in data. To motivate the study of the Fourier Transform, the students will be presented with certain application areas, such as searching for periodic patterns in CO2 data and differentiating between two sound signals using their power spectra. Students will explore some computational issues and challenges of the Discrete and Fast Fourier Transforms.

     

    Did he do it? Evaluating Evidence Using Bayesian Networks 
    Did he do it? Evaluating Evidence Using Bayesian Networks: Zipped Files

    Module Authors: Lisette de Pillis, Ami Radunskaya, Dani Mata, Tony Sevold
    Author Contact: depillis@hmc.edu  

    Funded By: W. M. Keck Foundation
    The accuracy of DNA testing and the assumption that the blood could only come from the one offender became very important. When identifying the origin of DNA in paternity tests, the following fundamental theory is used. An individuals DNA profile consists of measurements on several markers, each yielding a genotype which consists of a pair of alleles, one inherited from the father and the other from the mother. It is impossible, however, to tell which allele came from the mother, and which came from the father. This theory can be used at crime scenes when there is mixed trace blood. The alleles can help to distinguish the people who have contributed DNA to the mixture. This module will explore the following problem: Given a DNA mixture at a crime scene and given the DNA of a suspect, how likely is it that the suspect was in fact the offender? You will have to consider possible contamination and whether the fact that the suspect might be a close relative of the offender might affect the analysis. Bayesian networks will be used to model the different scenarios, and to produce likelihoods for the guilt (or innocence) of the suspect. In order to use Bayesian networks, you will also learn the basics of conditional probability and Bayes Rule.

     

    Pharmacokinetics: Mathematical Analysis of Drug Distribution in Living Organisms 

    Module Author: Igantios Vakalis
    Author Contact: ivakalis@capital.edu
    Funded By: W. M. Keck Foundation
    Pharmacokinetics is the study of time course of a drug or a metabolite in different fluids and tissues and of the mathematical relationships required to develop models to interpret such data. The mathematical theory of drug phenomena is a branch of the mathematical theory of metabolism. Even though drugs are not normal metabolites they do affect different metabolic processes. While the drug is acting, it does take part in some phases of metabolism. The theory of drug phenomena can be subdivided into two categories: (1) The modeling of distribution of drugs in the organism, and (2) The biochemical kinetics of the interaction of the drug with different components of the organism. The first category will be used as the background info, while the module focuses on the mathematical treatment of the second category.

     

    Vector Spaces and Linear Transformations  

    Module Author: Greg Baker
    Author Contact: baker.27@osu.edu  

    Funded By: W. M. Keck Foundation
    What is a vector? The answer may surprize you. But let's start with the simplest view of a vector. It is an arrow that records distance and direction. By stringing together a sequence of arrows we can provide detailed directions for a journey, or outline an object. It is the way we add arrows to produce a new arrow that really identi es what a vector is. We can incorporate this addition property to other quantities such as velocities, forces, and even functions. What quickly emerges is that it is the linear combination of vectors that allows great diversity in applications and provides deep understanding to the nature of solutions to linear problems. This module starts with the basic description of vectors and then proceeds to elucidate their role in the formation of systems of linear equations. 

     

    Systems of Linear Equations

    Module Author: Greg Baker
    Author Contact: baker.27@osu.edu  

    Funded By: W. M. Keck Foundation
    To further our knowledge in the sciences and engineering we try to formulate mathematical models that quantify behavior. We trust these models only when they can predict observed behavior. Normally, this step requires the solution of the unknown quantities in our model when written as a system of equations. So the first major question is: Do our equations have solutions? The theme in this module is to face squarely the question of existence of solutions to systems of linear equations. Specifically, we will learn how to determine when no solutions exist, many solutions exist, or there is just one solution.

     

    Street Lighting

    Module Author: Ignatios Vakalis
    Author Contact: ivakalis@csc.calpoly.edu
    Funded By: National Science Foundation (9952806)
    Have you ever wondered what the optimal height of a street lamp-post should be; what should the intensity of the light bulb be; how far apart the lamp-posts should be placed; if there is any spot (between two consecutive lamp-posts) that is minimally illuminated. This module examines three variations of the illumination problem: (1) Calculating a point, between two light sources that is minimally illuminated; (2) Calculating the height of one lamp-post in order to maximize the illumination at the above point; and (3) Calculating the height of two lamp-posts in order to maximize the illumination at the above point; (this is given as a project assignment). The module presents the derivation of the mathematical models of each situation and uses a Computer Algebra System (Maple) to examine symbolic, numeric and graphical solutions to the models.

     

    An Introduction to Coding Theory via Hamming Codes

    Module Author: Nuh Aydin
    Author Contact: aydinn@kenyon.edu
    Funded By: National Science Foundation (0618252)
    The theory of error-correcting codes is a relatively recent application of mathematics toinformation and communication systems. The theory has found a wide range of applicationsfrom deep space communication to quality of sound in compact disks. It turns out that a richset of mathematical ideas and tools can be used to design good codes. The set of mathematicaltools used in the field generally comes from algebra (linear and abstract algebra). The purposeof this module is to introduce the basics of the subject to students with an elementaryknowledge of linear algebra, via a well-known class of codes known as Hamming codes.
    Interesting properties and projects related to Hamming codes are introduced.

     

    The FFT and Functions

    Module Author: Greg Baker
    Author Contact: baker.27@osu.edu  

    Funded By: National Science Foundation (0618252)
    The incredible importance of the Fast Fourier Transform (FFT) is its use in a vast number of applications appearing in many disciplines. Of particular interest is its use in approximateing functions and their derivatives with applications to solving differential equations. In this regard, the FFT based on powers of 2 provides the most convenient tool because of its simplicity and speed.

     

    Modeling Seashell Form
    Seashell Modeling Supporting Document 1
    Seashell Modeling Supporting Document 2

    Module Author: Judy Holdener
    Author Contact: holdenerj@kenyon.edu
    Funded By: National Science Foundation (0618252)
    In this module a mathematical model of seashell form is presented. The model is based on the observation that most shells grow isometrically, meaning that they remain the same shape as they enlarge. Using parameterizations of space curves and matrices, we create spiraling surfaces by rotating an aperture curve about the z-axis while simultaneously shrinking all of its spatial dimensions by a constant scaling factor. We then use the computer algebra system MAPLE to produce visualizations of the results.