Model for global warming: dT/dt = fincoming(T) - foutgoing(T), where fincoming(T) is the net incoming radiation to earth as a function of the temperature T.
Determinism (uniqueness): Given any starting point, x(t0) = c, there is exactly one solution x(t) of dx/dt = f(x) with x(t0) = c. To prove the uniqueness we assumed that the derivative of f(x) is bounded near x = c.
Consequence: If u(t) and v(t) are both solutions of dx/dt = f(x), then their graphs cannot cross.
Some References:
Modeling Differential Equations in Biology by Clifford H. Taubes,
Nonlinear Dynamics and Chaos by Steven H. Strogatz.
a). The mechanics of voting.
b). Interpreting the results of the election (see for instance Item 8 in Math 210 Topics).
Today's discussion focussed on the mechanical issues of enabling people to vote, say for the winner of an Academy Award or for the President of your senior class. In particular we discussed aspects of an electronic election. [See also On-Line Voting]
1. The qualitative behavior of the solutions of dx/dt = f(x) can be obtained from the graph of y = f(x).
2. The points where f(x) = 0 are constant solutions (equilibrium solutions).
3. Determinism for a solution of dx/dt = f(x) with x(t0) = c specified. Stability of solutions of dx/dt = f(x). What happens for large t? Examples.
Systems of Differential Equations
Example: R(t) and L(t) satisfy the system
What is the behavior for large t? As we will see, a critical issue is if the interraction is strong (a > 1) or weak (a < 1). We study the vector field
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A second example is the classical and important Lotka-Volterra Model for the population of a field with foxes and hares. Here in the long-run the population spirals to a stable equilibrium between the foxes and hares.
Important Special Case
This problem is much clearer if one uses matrix notation:
After looking at lots of specific examples, one is led to suspect that most of the time the solutions have the specific form X(t) = e pt V, where p is a constant and V is a constant vector. Then by differentiating we see that dX/dt = p e pt V. Thus, if this X(t) is to be a solution of our equation dX/dt = A X(t), then
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For a nonlinear two-component linearsystem dX/dt = F(X), where F(X) is the (column) vector F(X) = (f1 (x,y), (f2 (x,y) ), an equilibrium point is where F(X) = 0, that is, f1 (x,y) = 0 and f2 (x,y) = 0. After finding the equilibrium points, we check the stability of each one as follows. Say we are investigating the particular equilibrium point X = Z, so F(Z) = 0.
1. Linearize F(X) about X = Z by using only the beginning term of a Taylor series:
Note that if the linearized equations give stability or instability, then the same holds for the nonlinear equations. However if the linear equations gives "meta-stability," then the non-linear equation might be stable, unstable, or meta-stable; in this case there is not enough information to decide.
The derivative matrix F'(X) is computed using the first partial derivatives of f1(x,y) and f2(x,y):
In class we worked some specific numerical examples involving the hares-foxes dynamical system.
While our course has emphasized applications to the biological sciences, there are many others. For applications of some of these ideas to economics, see Chapter 9 of the game theory book Fun and Games by K. Binmore published by D.C. Heath.
Supplementary references (see also the course bibliography): What is the RSA cryptosystem?
Two online (non-mathematical) references are to the PGP web site and Netscape web site.
Special Room: DRL 3C2
A brief description of the Kendall-Wei ranking method. It's a good motivator for classes in matrix algebra since it shows how search engines like Google use eigenvectors to do their searches.
Some notes: as a PostScript file or as an Acrobat file
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Class notes: pdf version and PostScript version
Votes where there are only two candidates and those with more than two candidates. May's theorem: if there are opnly two candidates, under most circumstances a "majority vote" is best..
Examples of elections: political, corporate decisions, ranking tennis players, ranking football teams, giving grades in a course, Academy Awards, choosing an undergraduate college,...
Examples of difficulties with various election procedures. Some elementary references: A fourth grade class, Part IV of the elementary text For All Practical Purposes, and An Economist Article.
Using Kendall-Wei Ranking to rank football teams. This takes into account that a team should be ranked more highly if it beats strong teams than if it wins over weak teams. One ends up computing a positive eigenvector of a non-negative matrix. See the next lecture for an example.
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Some other voting methods: Borda count and the Hare system.
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