Mathematics research opportunities for undergraduate students have grown significantly in recent years, but accessible research topics for first- and second-year students are still hard to find. To address this need, this volume provides beginning students who have already had some exposure to calculus with specific research projects and the tools required to tackle them. Chapters are self-contained, presenting projects students can pursue, along with essential background material and suggestions for further reading. In addition to calculus, some of the later chapters require prerequisites such as linear algebra and statistics. Suggested prerequisites are noted at the beginning of each chapter. Some topics covered include:- lattice walks in the plane
- statistical modeling of survival data
- building blocks and geometry
- modeling of weather and climate change
- mathematics of risk and insurance
Mathematics Research for the Beginning Student, Volume 2 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 not yet studied calculus is also available.
Author(s): Eli E. Goldwyn, Sandy Ganzell, Aaron Wootton
Series: Foundations for Undergraduate Research in Mathematics
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 313
City: Cham
Preface
Contents
Constructible Pi and Other Block-Based Adventures in Geometry
1 Introduction
1.1 In the Beginning There Were Circles
1.2 The Isoperimetric Problem and the Story of Queen Dido
2 Block Headed Thinking
2.1 Block Curves and Block Isoperimetry
2.2 The Block Isoperimetric Problem
2.3 The Dual Block Isoperimetric Problem
2.4 Isoperimetric Pairs
3 Additional Research Projects
3.1 New Neighbors on the Block
3.2 Block Knots
3.3 What Is So Special About Rectangles?
3.4 Isoparametry and Blocks
References
Numerical Simulation of Arterial Blood Flow
1 Introduction
2 Planar Model of Poiseuille Flow in 2D Channel
3 Exact Solution of the 2D Planar Model
4 Numerical Solution of the 1D Model
5 Exercises
6 Challenging Problems
7 Suggested Projects
7.1 Cylindrical Model
8 Conclusions and Future Work
Appendix
Bibliography
Statistical Tools and Techniques in Modeling Survival Data
1 Introduction
1.1 Probability Density Function
1.2 Survival Function
2 Common Distributions in Survival Analysis
2.1 Exponential Distribution
2.2 Weibull Distribution
2.3 Gamma Distribution
2.4 Generalized Gamma Distribution
3 Parameter Estimation with Likelihood Function
3.1 Likelihood Function
3.2 Maximizing the Likelihood Function
3.2.1 Profiling Approach for Maximizing Likelihood Functions
3.3 Censored Data
3.4 Parameter Estimation with Right-Censored Data
4 Data Simulation Methods
4.1 Simulation Methods for Fully Observed Data
4.2 Simulation Methods for Right-Censored Data
5 Model Selection
5.1 Log-Likelihood Value
5.2 AIC and BIC
5.3 Likelihood Ratio Test
6 Research Projects
7 Concluding Remarks
References
So You Want to Price and Invest in Options?
1 Introduction
1.1 Binomial Model
1.2 Brief History of Quantitative Finance: From Bachelier to Black, Scholes, and Merton
1.3 Building Tree Models
1.3.1 Example: Forward Contract
1.3.2 Example: Call Option
1.4 Calibration of p and q
2 Utility Theory
2.1 What Is Your Level of Risk Aversion?
2.2 Expected Utility Hypothesis
2.2.1 Example: A Fair Game
2.2.2 Example: Optimizing a Portfolio
2.3 Iso-Elastic Utility Functions
3 Maximal Utility and Indifference Pricing
3.1 Setting Up the Optimization Problem
3.1.1 An Initial Example of Optimal Investing
3.1.2 Our Initial Example of Optimal Investing as Constrained Optimization
3.2 An Example of Indifference Pricing
4 Conclusion and Further Reading
References
The Spiking Neuron Model
1 Introduction
1.1 Table of Symbols and Notation
1.2 Exercises and Programming Challenge Problems
2 Modeling the Neuron
2.1 The Neuronal Spike
2.2 Ionic Concentrations, Conductance, and Reversal Potentials
3 Integrate-and-Fire Spiking Neuron Model
3.1 Numerical Implementation of the Model
3.2 Coding Challenge Problems
4 Integrate-and-Fire with Spike-Rate Adaptation
4.1 Coding Challenge Problems
5 Hodgkin–Huxley Spiking Neuron Model
5.1 Coding Challenge Problems
6 Integrate-and-Fire with Synaptic Conductance
6.1 Coding Challenge Problems
7 Conclusion
8 Suggested Research Projects
Exercises: Answers
References
Counting Lattice Walks in the Plane
1 Introduction
2 Combinatorial Proof Techniques
2.1 Walks of Minimal Length
3 The Problem
4 Summary of Known Results
4.1 Enumeration When S Is Small
4.2 Enumerating Walks with Steps of Fixed Length
5 An Interlude on Computational Techniques
6 Open Problems for Lattice Walk Enumeration
6.1 Generating Functions
6.2 Counting According to a Statistic
7 Counting Walks with Restrictions
7.1 Restricted Walks with a Range of Steps
8 Conclusion
References
The Mathematics of Host-Parasitoid Population Dynamics
1 Introduction
1.1 What Are Parasitoids?
1.2 Host and Parasitoid Behavior
1.3 Prerequisites
2 Discrete Modeling
2.1 Host-Only System
2.2 Nicholson–Bailey Model
2.3 Host Refuge
3 Continuous Modeling of the Vulnerable Period
3.1 General Parasitism
3.1.1 Constant Rate of Attack
3.1.2 Functional Response
3.2 Host Mortality
3.3 Host-Feeding
3.4 Other Mechanisms
4 Semi-discrete Modeling
4.1 General Framework
4.1.1 Constant Attack
4.1.2 Functional Response
4.2 Host Mortality
4.3 Host-Feeding
5 Suggested Research Projects
Appendix: Stability Theorem for 3-D Systems
References
Mathematical Modeling of Weather and Climate
1 Introduction
2 Modeling with Data
2.1 Linear Models
2.2 Transformation to Linear Models
3 The Keeling Curve
3.1 Linear Regression
3.1.1 The Method of Least Squares
3.1.2 The Coefficient of Determination
3.2 Modeling with Differential Equations
4 Energy Balance Models
4.1 Observation
4.2 Units
4.3 Variable
4.4 Physical Parameters
4.5 Assumptions
4.6 Phase Line Analysis
4.7 Using Excel's Solver
6 Suggested Projects
References
Beyond Trends and Patterns: Importance of the Reproduction Number from Narratives to the Dynamics of Mathematical Models
1 Introduction
5 Empirical Models for Tropical Storm Windspeeds After Landfall
5.1 Hurricane Forecasting Models
2 Concept of Reproduction Number for an Epidemic
2.1 Characteristics and Dynamics of an Epidemic Model
2.2 Epidemiological Interpretation of R0
2.3 Computation of Reproduction Number Using Next Generation Method
3 Application of Reproduction Number for Understanding Dynamics
3.1 The Reproduction Number for the Single-Host-Vector Model
3.2 The Reproduction Number in Multi-host Setting
3.3 Invasion Reproduction Number in Multi-strain Model
3.4 Threshold Host Size to Sustain Macroparasite Infection
3.5 The Reproduction Number in Social Issues
3.6 Time-Varying Reproduction Number
4 Application of Reproduction Number for Design of Interventions
4.1 Critical Vaccination Rate
4.2 Critical Virus Replication Rate in Diagnostic Testing
4.3 Virus Generation Rate in a Host's Immune Cell
4.4 Basic Reproduction Number in Healthcare Settings
5 Distribution of Reproduction Number (Capturing Variance in R0) or K-number
6 Concluding Remarks
References
Application of Mathematics to Risk and Insurance
1 Introduction
2 Definitions and Notation
3 The Big Three Claim Frequency Models
3.1 Poisson Distribution
3.2 Binomial Distribution
3.3 Negative Binomial Distribution
4 Examples of BMS
4.1 The Switzerland BMS
4.2 The Belgian BMS
4.3 The Brazilian BMS
4.4 The Japanese BMS
4.5 The Hong Kong BMS
5 Stationary Distribution of a BMS
5.1 Numerical Examples
Appendix
References
