Mathematics Research for the Beginning Student, Volume 1: Accessible Projects for Students Before Calculus

Description

Mathematics research opportunities for undergraduate students have grown significantly in recent years, but accessible research topics for first- and second-year students with minimal experience beyond high school mathematics are still hard to find. To address this need, this volume provides beginning students with specific research projects and the tools required to tackle them. Most of these projects are accessible to students who have not yet taken Calculus, but students who know some Calculus will find plenty to do here as well. Chapters are self-contained, presenting projects students can pursue, along with essential background material and suggestions for further reading. Suggested prerequisites are noted at the beginning of each chapter. Some topics covered include:games on graphsmodeling of biological systemsmosaics and virtual knotsmathematics for sustainable humanitymathematical epidemiologyMathematics Research for the Beginning Student, Volume 1 will appeal to undergraduate students at two- and four-year colleges who are interested in pursuing mathematics research projects. Faculty members interested in serving as advisors to these students will find ideas and guidance as well. This volume will also be of interest to advanced high school students interested in exploring mathematics research for the first time. A separate volume with research projects for students who have already studied calculus is also available. Preface Contents Games on Graphs: Cop and Robber, Hungry Spiders, and Broadcast Domination 1 Introduction 2 The Game of Cops and Robbers 3 Hungry Spiders 4 (t,r) Broadcast Domination 5 Further Investigation 6 A Short Primer on Graph Theory References Mathematics for Sustainable Humanity: Population, Climate, Energy, Economy, Policy, and Social Justice 1 Introduction 2 Quantifying Change 2.1 Absolute and Relative Change and Rate of Change 2.2 Linear and Exponential Change 2.3 Measuring and Estimating 3 Population Growth and Ecological Footprint 4 Climate Change 5 Energy Production, Consumption, and Efficiency 6 Economic Growth (and Collapse) 7 Policy and Social Justice References Mosaics and Virtual Knots 1 Math and Knots 2 Gauss Codes 3 Virtual Knots 4 Mosaics 5 Virtual Mosaics 6 Further Reading References Graph Labelings: A Prime Area to Explore 1 Introduction 1.1 Families and Classes of Graphs 1.2 Graph Operations 1.3 Introduction to Graph Labeling 2 Coprime and Prime Labelings 2.1 Minimal Coprime Labeling 3 Consecutive Cyclic Prime Labelings 4 Neighborhood-Prime Labelings 4.1 Building on Cycles 4.2 Building on Trees 4.2.1 Further Projects on Neighborhood-Prime Labelings 5 Conclusion References Acrobatics in a Parametric Arena 1 Analogies to Motivate Parametric Thinking 2 Overview 3 Parametric Basics 3.1 Parametric Function 1 3.2 Parametric Function 2 3.3 Parametric Function 3 4 Function Concepts 4.1 Functions and Nonfunctions, Based on the Context 4.2 Connecting Component Functions and the Parametric Function 5 Try Some Parametric Acrobatics Yourself 5.1 General Desmos Graphing Instructions 5.2 Exercises with some Desmos Instructions 5.3 Experimentation 6 True Acrobatics: Parametric Modeling of a Flower Stick 6.1 Flower Sticks and Digitized Motion 6.2 Modeling the Left End of the Flower Stick 6.3 More on the Flower Stick 6.4 Your Work 7 Financial Acrobatics: Modeling US Wireless Subscribers 7.1 About the Data 7.2 Views of the Data 7.3 Data Projections 7.4 Linear Fits to 2004–2014 Data 7.5 Proportional Growth 8 GeoGebra to Practice 3D Parametric Equation 9 Your Project References Further Reading Software But Who Should Have Won? Simulating Outcomes of Judging Protocols and Ranking Systems 1 Introduction 2 Fundamentals of Probability 2.1 A Little Set Theory 2.2 Computing Probability 2.3 Conditional Probability 2.4 Random Variables and Probability Functions 2.4.1 Discrete Random Variables 2.4.2 Continuous Random Variables 3 Introduction to Simulation 3.1 Random Number Generation 3.1.1 Sampling from Discrete Distributions 3.1.2 Sampling from Continuous Distributions 3.2 ``If/Else'' Statements 3.3 ``While'' and ``For'' Loops 3.3.1 ``While'' Loops 3.3.2 ``For Loops'' 3.4 Writing More Complex Simulation Code 4 Suggested Research Projects 4.1 Scenario 1: Objective Ranking 4.2 Scenario 2: Subjective Ranking 4.3 Scenario 3: Comparing Voting Methods References Modeling of Biological Systems: From Algebra to Calculus and Computer Simulations 1 Introduction 1.1 A Description of Mathematical Modeling 1.2 Building a Model with Bias 1.3 A Note About Computer-Based Simulations 1.4 Active Learning 2 Grey Squirrels in Six Fronts Park: Modeling a Changing Population 2.1 The Problem 2.1.1 Step 1: Goals, Questions, and Assumptions 2.1.2 Step 2: Build a Model 2.1.3 Step 3. Apply the Model 2.1.4 Step 4. Assess and Revise Your Model 2.2 Conclusion and Exercises 3 Non-contact Cardiovascular Measurements 3.1 Context 3.2 The Challenge 3.3 The Initial Experiment 3.4 Weigh a Bed? 3.5 Let Us Get Our Hands Dirty! 3.6 Interesting Observations 3.7 Conclusions 4 Difference Equations in Population Ecology 4.1 Introduction 4.2 Population Growth with Difference Equations 4.3 Coding Difference Equations 4.4 Difference Equations for Predator-Prey Problems 5 Modeling the Spread of Infectious Diseases with Differential Equations 5.1 Modeling the Demise of Candy 5.2 Population Growth Models in Continuous Time 5.2.1 A Basic Model of Infectious Disease Spread 5.3 Discussion 6 Conclusions References Population Dynamics of Infectious Diseases 1 Mathematical Models in Epidemiology 2 An Individual-Based Epidemic Model 2.1 Model Description and Physical Simulation 2.2 Computer Simulation 2.3 Section 2 Exercises 2.4 Section 2 Challenge Problem 2.5 Section 2 Projects 3 Continuous-Time Dynamical Systems 3.1 The Derivative 3.2 Dynamical Systems 3.3 Section 3 Exercises 4 Dynamical System Models 4.1 Classification of Dynamical System Models 5 Building the SEIR Epidemic Model 5.1 Quantifying the Processes 5.1.1 Transition Processes 5.1.2 The Transmission Process 5.2 The Final Model 5.3 Section 5 Exercises 6 Modeling 6.1 Identifying Parameter Values 6.2 The Basic Reproduction Number 6.3 Section 6 Exercises 6.4 Section 6 Challenge Problem 7 Model Analysis 7.1 Early Phase Exponential Growth 7.2 The End State 7.3 Section 7 Exercises 7.4 Section 7 Challenge Problems 7.5 Section 7 Project 8 Simulations 8.1 Numerical Simulation of Continuous Dynamical Systems 8.2 Implementation of Numerical Simulations 8.3 Section 8 Exercises 8.4 Section 8 Projects Appendix: Programs hpsr.m HPSR_onesim.m HPSR_avg.m seir.m SEIR_onesim.m SEIR_comparison.m SEIR_paramstudy.m References Playing with Knots 1 Knots: Knotted and Unknotted 1.1 What Is a Knot? (The Basics) 1.2 Knot Equivalence and Reidemeister Moves 1.3 Some Useful Knots and Links 1.4 Unknotting Operations and Numbers 1.5 Knot Invariants 2 The Knotting-Unknotting Game 3 The Region Unknotting Game 4 The Linking-Unlinking Game 5 The KnotLink Game 6 The Link Smoothing Game 7 Conclusion References