Mathematical Modeling: Homing Pigeons, Elevator Control, and More

1. Mathematics 191 Research Seminar in Mathematical Modeling Lecture 4February 3rd, 2005

2. Overview • Homing Pigeons, Revisited • Algorithms versus Models • Naïve and Advanced Models • Case Study: Elevator Control Theory • Sources for Mathematical Research • Wide World of Models • Prepare for the MCM and Projects

3. The Modeling Process • Statement of Problem (abstraction) • Define Model Objective / Objective Function • Definitions and Identification of Variables (background research and common sense) • Assumptions (for tractability) • Establish Informal Relationships Based on System • Construct Mathematical Statements • Construct Base Model • Estimate Parameters • Apply Mathematical Methods • Pure Mathematical Solution • Simulation and Validation (the inverse of the abstraction process) • Sensitivity Analysis • Relax Assumptions • Iterate • Assess Model Limitations

4. Homing Pigeons, Revisited • Recall from last time that we were attempting to write out a bird-speed function, ignoring wind and directional variation, in order to integrate for total distance covered. • We developed several ways to proceed: piecewise structures, damped harmonic oscillation, superposition of random processes with either one. • How do we decide between competing ideas for models?

5. Choice of Mathematical Structure • What governs our choice of stucture? Well, we consider several things, in order: • Does it do what we expect it should? • Does it behave correctly at known values and endpoints? • Ease of mathematics involved • Comparison with similar systems – other endurance models? • Ease of validation • Accuracy of system

6. Naïve and Advanced Models • Our first model for a given system shouldn’t try and address all, or even many, of the factors involved in the system. A crude representation is fine. • This is known as the “naïve” or “base” model, and will provide a baseline for comparison and for future model development. • For instance, in bird models, we can be satisfied even if we ignored random variation. • Later, we’ll attempt to make our model more realistic. Any model beyond our base model is termed an “advanced model”, and we may have several classes of models depending on which way we choose to expand the base model. • Often, we know the shortcomings of our model even without any additional analysis.

7. Modeling Beyond the Homing Pigeon • The mathematics we used here were intended to be very simple, but this isn’t what we’ll use very often in our models. • If you've thought at all about your project, you may have realized that your system probably doesn’t lend itself to a mathematical construction based on functional calculus. • Model construction isn’t restricted to the methods we’ve just used. • In modeling for an advanced system, we put together any system that works from all branches of mathematics. • As mathematics majors, we get only a few broad classes of methods – calculus and differential equations, linear algebra, and some group theory. • What about... optimization / graph theory / nonlinear systems / queueing theory / stochastics and random processes / cellular automata / decisionanalysis / game theory?

8. A few words on researching literature • Don’t begin by reading everything you can on a subject! • List specific questions you’d like to answer before you start your research. • Google Scholar, MathSciNet, and the mathematics and engineering libraries are good places to start.

9. A few words on time management... • Because this class will prepare us for the MCM, we will need to be able to derive models very rapidly (~20 minutes). • As such, every day we'll work on a little bit of modeling and see how fast we can move from open-ended question to mathematical solution.

10. Algorithms versus Models • Often in a given situation or MCM problem, we are asked to develop an algorithm instead of a model. • This difference is not hard to understand, but we'll make it explicit just so we don't trip up in the future. • An algorithm is a procedure which takes in a set of input variables, operates step-by-step on the inputs, and returns a set of output variables • A model is a mathematical structure used to predict the behavior of a real-world system. • In other words, our algorithm should operate by trying to capitalize on the predictions made by a model.

11. Evans Hall Elevators • Evans Hall has twelve floors and three main elevators centrally located on each floor. Everyone complains that the elevators often behave unintelligently, sending multiple elevators to a given floor where only one is required, and forcing individuals to wait when there are free elevators. • Develop a model to represent elevator operation and design an optimal algorithm for control of these elevators to improve performance. Use your model to assess the effectiveness of your algorithm. Keep in mind the limited amount of information available to the elevator.

12. Problems within the Purview of Modeling • Most disciplines are well-established, so the models used have been in existence for hundreds of years. As undergraduates we often consider only existing models in class. • This process is so established that we encounter models on a daily basis without always recognizing them as models. • However, glory lies in development of our own models. Let's try some samples.

13. How can we hold fair elections in a dangerous or uncertain environment? • Iraq's electoral system is based loosely on our own. Iraq is divided into a certain set of precincts, but turnout is expected to be fairly low, and perhaps unfairly biased towards certain groups. • As an added complication, nobody knows who's really running. • In the US election, random votes were lost on certain machines in certain areas. User and machine error, and rarely, fraud, also contributed to mis-votes. Rerunning an election is costly and highly undesirable. • How can we still determine a “fair” winner in these situations?

14. How Did Velociraptors Hunt? • Part 1. Assuming the velociraptor is a solitary hunter, design a mathematical model that describes a hunting strategy for a single velociraptor stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Assume that the thescelosaurus can always detect the velociraptor when it comes within 15 meters, but may detect the predator at even greater ranges (up to 50 meters) depending upon the habitat and weather conditions. Additionally, due to its physical structure and strength, the velociraptor has a limited turning radius when running at full speed. This radius is estimated to be three times the animal's hip height. On the other hand, the thescelosaurus is extremely agile and has a turning radius of 0.5 meters. • Part 2. Assuming more realistically that the velociraptor hunted in pairs, design a new model that describes a hunting strategy for two velociraptors stalking and chasing a single thescelosaurus as well as the evasive strategy of the prey. Use the other assumptions given in Part 1. • Here, some constraints and assumptions are given to us by the problem. • What sort of mathematical approaches might we use to solve this problem?

15. How many people should you date in your life? • Consider the sequence of individuals we choose to date over the course of a lifetime (100 or so). • Assume for a simple model that once we dump someone, we never see them again. • What is the optimal number of people to date before “settling down” to maximize the probability that we'll find Mr./Ms. Right? • In the advanced model, we can “play the field”...

16. How can we detect moving objects in an ambient noise field? • The world's oceans contain an ambient noise field. Seismic disturbances, surface shipping, and marine mammals are sources that, in different frequency ranges, contribute to this field. We wish to consider how this ambient noise might be used to detect large maving objects, e.g., submarines located below the ocean surface. Assuming that a submarine makes no intrinsic noise, develop a method for detecting the presence of a moving submarine, its speed, its size, and its direction of travel, using only information obtained by measuring changes to the ambient noise field. Begin with noise at one fixed frequency and amplitude.

17. Fin