# American Mathematical Monthly - January 2002

## JANUARY 2002

Financial Derivatives and Partial Differential Equations
by Robert Almgren
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The pricing and risk hedging of derivative securities is one of the most successful applications of mathematics in the modern world. Trillions of dollars move around the world every day, in transactions whose details are determined by stochastic models based on continuous functions that are nowhere differentiable, and on the solution of partial differential equations with free boundaries. In this note, I briefly review the assets that are traded in markets and the economic principles that go into the mathematical modeling; I describe the Black-Scholes-Merton hedging strategy and the resulting partial differential equation; then I describe a few extensions.

Lucy and Lily: a Game of Geometry and Number Theory
by Richard Evan Schwartz
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In this article I describe the mathematics behind "Lucy and Lily," a computer game I created. One plays the game by moving two pentagons in the plane via reflection-type moves. Watching only one pentagon move, the player stumbles blindly through what turns out to be the shadow of a 4-dimensional grid. Watching both pentagons at the same time, the player can expertly navigate the grid. Thus, the two pentagons together give a kind of stereographic picture of the 4th dimension. The article explains the number-theoretic secret behind this phenomenon. If you want to play "Lucy and Lily'" on the internet, you should check out my website: www.math.umd.edu/~res.

Numerical Integration of Periodic Functions: A Few Examples
by J.A.C. Weideman
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It comes as a surprise to many students that the very basic trapezoidal and midpoint rules for numerical integration can yield more accurate results than more sophisticated methods such as the Simpson and Gauss-Legendre rules. This happens in particular when the integrand is smooth and periodic. In a series of examples explicit error formulas for trapezoidal/midpoint approximations to a few integrals are derived. These examples were chosen to demonstrate a variety of convergence behaviors, most of them quicker than what the standard error estimates predict.

Harmonic Functions in the Unit Disk
by Victor L. Shapiro
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A "best possible" uniqueness theorem for harmonic functions in the unit disk is established. This theorem could have been conjectured but not proved by either Riemann or Gauss. The proof, which is provided in the article, depends upon topological notions not available in their lifetimes. It does, however, make use of some of Riemann’s, as well as Cantor’s, ideas.

The Algorithmic Theory of Randomness
by Sergio B. Volchan
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The notion of randomness is fundamental in many pure and applied scientific fields. However, it is also very difficult to give it a precise definition, even a mathematical one. In this paper we review three main notions of random sequences: chaoticness, incompressibility, and typicality. In particular we highlight the surprising connection of the concept of randomness to central ideas of modern mathematical logic such as algorithms, computability and Turing machines.

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