Math 570

Class Summary Page

for

Mathematical Modeling

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Thursday, October 2. Class was cancelled for Rosh Hashanah. Students were encouraged to make use of the computer laboratory to work on the homework set which involved extensive Maple computations.

Tuesday, October 7. We began to discuss dimensional analysis by first reviewing the flagpole problem. We contrasted the method with "inspectional" analysis. Then we did a dimensional analysis treatment of a simple pendulum. Greg Larson began his report on the University of Alaska, Fairbanks solution to the Coal Tipple Problem.

Thursday, October 9. Greg Larson finished his report on the University of Alaska, Fairbanks solution to the Coal Tipple Problem (and Dr. Barsky was very happy to see a dynamic rule). Keith Dunbar intoduced the paper scoring problem, and presented the St. Bonaventure solution. David Trigg began to present the Fudan University solution to this problem.

Tuesday, October 14. We completed the dimensional analysis of the flagpole problem, and began a dimensional analysis of the period of a pendulum. We came up with a list of factors that might affect the period (tau), and after some discussion decided to retain four of these: the length of the cord (l), the mass of the bob (m), the initial angle of displacement (alpha), and the gravitational acceleration (g). We found that there were (only) two independent dimensionless products that could be formed from tau, l, m, alpha and g: Pi_1 = (g/l)*tau^2 and Pi_2=alpha. From this we deduced that the equation of the period should have the form sqrt(l/g)*f(alpha), where f is some function of the initial angle of displacement. On Thursday, we will try to flesh out the method that we used here to get a systematic way of doing dimensional analyses.

Thursday, October 16. We looked at a systematic way of doing dimensional analyses. Given a modeling process involving n+1 variables/parameters X_1, X_2, ..., X_n, Y, we want to understand how Y depends on X_1, X_2, ..., X_n, i.e., we want to write Y=f(X_1, X_2, ..., X_n). We begin by forming a complete set of dimensionless products: a maximal set of independent dimensionless products. ............

Tuesday, October 21. We completed our dimensional analysis of the terminal velocity of a falling raindrop, using Buckingham's Theorem, and then compared the predictions of this analysis with the explicit models that we had studied for a falling raindrop at the beginning of the semester. In fact, an even more general air-drag model than the one discussed on September 11 was presented which showed how the density of the air influenced the terminal velocity. We then returned to the problem of the dimensional analysis of the period of a pendulum, but this time we added as to our list of parameters a frictional force, F. [See the summary for the next class to see why this was not quite right.]

Thursday, October 23. We began by correcting our work from last class. When we do a dimensional analysis, the quantities that we analyze must be ones that can be assigned a definite value. Last lecture we added a "force" to our analysis, without completely thinking through what that force was. It can't be a frictional force, since the frictional force is a varying function (it's zero when the bob is at it's greatest height because there the velocity is zero, and it's greatest when the cord is vertical because the bob's velocity is greates there). The mathematical work that we did on Tuesday was fine, the fault lies with how we interpreted it. The "F" that we introduced last class needs to be some characteristic force in the problem, say the maximum frictional force or the force with which we push the bob (if we don't release it from rest, as we had previously supposed that we would). We proposed two different ways to introduce a parameter to measure the frictional force. First, if we supposed that the drag on the pendulum was proportional to the velocity of the bob, i.e., F=k*v for some proportionality constant, then there are dimensions attached to k, and we can add this parameter to the list that we used in doing the dimensional analysis. Second, we might also try to see if the drag forces obeyed a velocity-squared law, F=(~k)*v^2, in which case we use the parameter (~k) instead of k; note that the two parameters have different dimensions, so we get different dimensional analyses. In the first case we found that tau = sqrt(l/g)*f(alpha,sqrt(l/g)*k/m) while in the second we had tau = sqrt(l/g)*(~f)(alpha,l*(~k)/m) The first case leads to the prediction tau(4*l,2*m) = 2*tau(l,m), while the second leads to tau(4*l,4*m) = 2*tau(l,m). We tested these by swinging a cup contining various amounts of Play-Doh from a fixed initial angle alpha, with a long piece of cord whose length we quartered. We obtained the following data (the times are how long it took for the bob to make 6 complete swings):

length of cord mass of bob	First trial	Second trial
1 unit of length 1 unit of mass	8.52 sec	8.53 sec
4 units of length 2 units of mass	16.84 sec	17.33 sec
1 unit of length 1/2 unit of mass	9.08 sec	9.06 sec

Afterwards we realized that the weight of the cup might not be negligible in comparison with the different masses of Play-Doh; this effect should be most pronounced for the last measurement.

Tuesday, October 28. We began to study deterministic population models. After beginning with a brief review of some standard continuous-time models (the exponential growth model, and the logistic growth model), we looked at the basic geometric growth model, x(t+1)=k*x(t), which led to a brief discussion on a solution method for linear, homogeneous, constant-coefficient difference equations (-- look for geometric solutions). Here k=1+rho, where rho is a net birth/death rate. After critiquing this simple model, we looked at a variant in which the population was stratified into age cohorts. Each cohort had its own reproductive and death rates. We were able to write this as a linear system: X(t+1)=K*X(t), and ended the lecture portion of the class by observing that the general solution to this system is X(t)=K^t*C, where C is the vector containing the initial population profile. We then examined some of the population pyramids found at the US Census Web site. Finally, at the very end of the period, we began a modeling exercise in which we examined the social structure of the Natchez Indians.

Thursday, October 30. We specialized our treatment population cohorts to the case of only two cohorts; here the matrix K (called the Leslie matrix) is [[alpha_1, alpha_2],[beta_1 0]], which we rewrote as [[alpha, gamma],[beta, 0]]. Calculating K^t for a few small powers of t was enough to convince us that we needed a better way to understand the dynamics of these populations. We decided to try an eigenvalue/eigenvector analysis. For this matrix we found that unless alpha=0 and either beta=0 or gamma=0 (both of which lead to K^2=0), there are two distinct eigenvalues (positive) lambda_+ and (negative) lambda_- with |lambda_-|<=|lambda_a|; recall that alpha>=0, gamma>=0, and 0<=beta<=1. Corresponding to these two eigenvalues are a pair of eigenvectors v_+ and v_-, which are necessarily independent, and which must thus span R^2. So we can write the initial population vector C as a linear combination of v_+ and v_-, say, C = c_+*v_+ + c_-*v_-. Then X(t) = K^t*C = c_+*(lambda_+)^t*v_+ + c_-*(lambda_-)^t*v_-. If c_+ > 0, then the long-term behavior of the population is determined the first eigenvalue and eigenvector; the population will diverge, remain bounded or tend to zero depending on whether lambda_1 exceeds, equals or is less than one. Furthermore, if c_+ > 0, then the limit proportion of the population in any one of the cohorts is equal to the ration of the corresponding component of v_+ to the sum of the components of v_+. We then demonstrated these general concepts in the special case where alpha=beta=1 and gamma=2, which gave lambda_+ = 2, lambda_- = -1, v_+ = [2, 1], and v_- = [1, -1]. We spent the last hour of class working on the Natchez Indian modeling exercise.

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This page last modified 10/30/97