Pi Mu Epsilon Undergraduate Math Society at USC
Spring 2003 Semester
Organizers: Sergey Lototsky and Royce Peng.
January 17.
We discussed problems A1-A5 from the 2002 Putnam exam. One unsolved question is
about problem A4 (determinant tic-tac-toe): can you deduce that zero always wins
(with optimal play)
from the fact that the regular tic-tac-toe is a draw (again, with optimal play)?
January 24.
We discussed problems A6 and B1-B4 from the 2002 Putnam exam. We had an interesting
discussion even though there was only
one student attending. The hope is that more will show up
next time.
January 31.
- We finished with 2002 Putnam problems. Problem B-6 is related to
Vandermonde and Moore determinants. Something to think about.
- As a possible topic for the future discussions, the chaos and related
questions was suggested. Next time we will talk about a simple one-dimensional
chaotic system defined by the logistic equation.
February 7.
We started our discussion of chaos and fractals. Main topics discussed:
- Three one-dimensional system that can exhibit chaotic behavior (the
set of limit points is dense in some interval).
- The logistic equation and its bifurcation diagram.
- Fractal dimension vs topological dimension.
February 14.
We watched a movie about real-life chaotic systems. A double pendulum (initially
folded and displaced at 90 degrees) provides
an easy demonstration of chaotic behavior. Lorentz system can be used to compose
music variations and to provide secure communications.
February 21.
- We started with an easy high-school Olympiad-type problem.
- As a continuation of our discussion of chaos and fractals, we talked
about
- the Sierpinski triangle
- the Mandelbrot and Julia sets, and the difference between them.
- For the next time:
- Three types of connectedness for sets (connected, locally connected,
path-wise connected) and relation between them. The question arose in the
discussion of the properties of the Mandelbrot set.
- Henon, Lorenz, and other chaotic systems in several dimensions,
and their connection with
fractals.
February 28.
- A warm-up problem.
- A short biography of B. Mandelbrot (Born 1924 in Poland; now Emeritus Professor
at Yale and an IBM Fellow)
- The Mandelbrot set is connected and simply connected, but is it not
known whether it is locally connected or path connected.
- "Tumbling box" - how rotation of a box with all dimensions different is
stable about two of the axes, shortest and longest, and is unstable about the
remaining; how this is related to a flow on an ellipsoid and the Euler characteristic
of the ellipsoid.
- Lorenz system vs Lorenz map (the map can be constructed for various
chaotic systems and can predict the size of the next max).
March 7.
This time we just did some problems:
- An easy one: characterize the sequence {a_n} in which, for n bigger than two,
a_n is the arithmetical mean of the previous n-1 terms.
- More difficult: if x_i are integers from the set {-1,0,1,2} so that
the sum of x_i is 19 and the sum of the squares of x_i is 99, what are
the biggest and smallest possible values of the sum of the cubes of x_i?
Royce also talked about the Newton-Gerald relations between the sums of
powers of n numbers and the n symmetric polynomials. In the end, we did the problem
without that, though.
- Even more difficult: for an integer a, characterize the integer solutions of the
equation x^3+y^3-3axy=a^3.
March 14.
Spring break starts early for us.
March 21.
Spring break.
March 28.
I was away. Royce was in charge and talked about ordinal numbers.
April 4.
With me back, Royce was in charge again and talked about
- Ordinal numbers (Hardy's hierarchy and
how a seemingly fast growing base-bumping sequence eventually
becomes zero)
- How a turtle can finish a race
faster than a hare even if for an observer at a fixed point
the hare is running 10 times faster: they pass the point at different times and
both should run so that their speed is zero on a Cantor set; if the observer
happens to be at a point where the speeds are zero, that's fine too since
0=10*0.
April 11.
Again, Royce was the principal speaker. We discussed
- Easy examples of the base-bumping sequences (1 --> 0; 2 --> 3-1=2 --> 1 --> 0;
3=2^1+1 --> 3^1 +1 -1 = 3 --> 4-1=3 --> 2 --> 1 --> 0; but for 4, the number of
steps is really big)
- How to assign an ordinal number to a tree.
- Surreal numbers and their connection with games and ordinals.
We used the example of the Hackenbush game to
assign a number to a game and to define the sum and difference of several games.
To be continued.
April 18.
With no Royce present, I distributed
notes about surreal numbers. The notes are by Claus Tondering, and I found them
on the web. After a short discussion, we decided to read the
first half and talk about it next time.
April 25.
Here are some useful facts about surreal numbers we derived using the notes and
Royce's help:
- When we write x={X_l|X_r}, we expect X_l < x < X_r.
- In the expressions {a| } we treat the empty set on the right as +infinity;
in the expressions { |b}, we treat the empty set on the left as -infinity.
- If n>0, then { |n} =0 (as the oldest number in between -infinity and n).
Similarly, {-n| }=0.
- More generally, { |b} is less than or equal to 0={ | } and 0 is less than
or equal to {a| } for every a,b (it trivially follows
from the general definition).
- If s={S_l|S_r} and n is the smallest integer between S_l and S_r, then
S=n.
May 2.
We discussed various aspects of the surreal numbers.
- Royce demonstrated the general procedure for deriving formulas corresponding
to various operations on surreal numbers. The key is that X_l < x < X_R if (and,
essentially, only if) x={X_l | X_R}. That way we easily get the addition and
multiplication formulas.
- The picture of surreal numbers looks like as a very-very large binary tree,
with 0={ | } at the root.
As a result, every nonzero surreal number can be identified with a
sequence of "+" and
"-", depending on the route from the root to the number ("+" if you go right).
For example, 1/2={0|1}=+-. This is also mentioned on the last page of the notes.
- For surreal numbers, the birthday is the analog of the ordinal number.
- Complex surreal numbers, or surcomplex numbers, are obtained
by adjoining the square root of -1 to the field of surreal numbers.
Initially, there is not surreal number x so that x^2={ |0}.
- While limit,
continuity, derivative, etc. can be defined for any surreal-valued
function of a surreal variable, there is nothing useful we can do with
any function rather than rational.
Since the surreal numbers make a field, all algebraic formulas remain
valid, and one compute the derivatives of the rational functions and get the
familia results from calculus.
- For surreal numbers,
the collection of natural numbers is not "infinite enough". As a result,
Taylor-type expansions are not easily defined, and extensions to transcendental
functions is problematic.
- Different people are trying to define various analogs of integration and
differentiation with surreal numbers. We are not going that far and will
stop here for now.
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