# 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 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.