Math 210, Spring 2001: Day By
Day Part 3
A day by day outline of some topics what we cover.
These are not intended to be balanced or comprehensive notes. They
are to supply motivation and fill gaps. Thus, standard material
easily available in traditional texts will be skipped, while brief
(quick and dirty) notes about computer or Internet issues may be here.
In principle, the Wednesday classes will involve more computer issues.
But real life is never that tidy.
Contents
- Tues. March 27
- Model for asymmetry in snails: dp/dt = ap( 1 - p)(1 - 1/2)
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.
- Wednesday. March 28
- Voting and Elections. There are two completely different
issues:
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]
- Thurs. March 29
- Review for dx/dt = f(x):
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
dR/dt = R - (R 2 + aRL) and
dL/dt = L - (L 2 + aLR),
where a > 0 measures the intensity of the interaction between the
number R(t) and L(t) of each species. [Note that if a = 0 these
equations "uncouple" and both are just the logistic equation.]
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
( R - (R 2 + aRL), L - (L 2 +
aLR) )
in the R L plane.
- Tues. April 3
- We continued discussing the previous example, particularly one
case of strong interraction (a=2) and weak interraction (a=1/2).
There are dramatic differences in the long-time behavior.
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.
- Wednesday April 4
- Some computer aspects of differential equations. One possibility
is to use Maple to plot the vector fields.
- Thursday April 5
- For a system
dx/dt = f1 (x,y)
dy/dt = f2 (x,y)
an equilibrium point (x, y) is f1 (x, y) = 0 and
f2 (x, y)= 0. We
seek a criterion for when an equilibrium point is stable.
Important Special Case
dx/dt = ax + by dy/dt = cx + dy,
where a, b, c, and d are all constants.
Step 1. Find the equilibrium point. This means solving
ax + by = 0 and cx + dy = 0.
Assuming that ad - bc is not zero, these only equilibrium boint is the
origin, (0, 0).
This problem is much clearer if one uses matrix notation:
dX/dt = A X(t)
where X(t) is the (column) vector X(t) = (x(t), y(t)) and A is the 2
× 2 matrix with entries a, b, c, and d. The assumption ad - bc
is not zero means that det (A) is not zero, so A is invertible.
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
p e pt V = A e pt V .
Therefore
AV = pV
so p must be an eigenvalue of A and V a corresponding eigenvector.
- Tues. April 10
- We showed that for a two-component linear system dX/dt = A X(t),
with A a 2 × 2 invertible matrix (so det(A) is not zero), the
origin is a stable equilibrium point if and only if trace(A) := a + d
< 0 and det(A) := ad - bc > 0. It is unstable if either
trace(A) < 0 or det(A) < 0. If trace(A) = 0 it is either unstable
( det(A) < 0 ) or "meta-stable" ( det(A) > 0 ).
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:
F(X) = F(Z) + F'(Z)(X - Z) + ... = F'(Z)(X - Z) + ...,
where ... means terms that vanish faster than linearly near X = Z.
Then near X = Z the differential equation is approximately
dX/dt = F'(Z)(X-Z).
Letting W = X-Z, so dW/dt = dX/dt, and writing A = F'(Z), we obtain:
dW/dt = A W,
which is a linear equation to which we can apply our earlier
result, assuming that det(F'(Z)) is not zero at the equilibrium points.
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.
- Wed. April 11
- Digital privacy: encryption. The current standard method uses
number theory.
A
very brief introduction as a starter (web page by Jeremy T. Teitelbaum).
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.
- Thurs. April 12
- Special Lecture: Searching the web with eigenvectors
by Professor Herbert Wilf of our Mathematics Department.
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
- Tues. April 17
- Encrypting messages on the web. Electronic signatures
Class notes:
pdf version
and PostScript version
- Wed. April 18
- (continued from Tuesday)
- Thurs. April 19
- The Theory of Voting
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.
- Tues. April 24
- Using Kendall-Wei Ranking to
rank football teams (Maple).
Some other voting methods: Borda count and the Hare system.
- Wed. April 25
- [no meeting]
- Thurs. April 26
- A review of the course.
Final Exam
The Final Exam will be held on Friday, May 4, 11:00
- 1:00. Anyone who cannot take the exam at this time please see me
before May 4 to arrange to take the exam. on Thursday, May 10,
11:00 - 1:00.