Pamela B. Pierce

Bounded variation in the mean

It is shown that the concept of bounded variation in the mean is not a meaningful generalization of ordinary bounded variation. In fact, it is a characterization of functions which differ from functions of bounded variation on a zero set. Let f be a real-valued function in L1 on the circle group T. We define the corresponding interval function by f (I) = f (b) -f (a), where I denotes the interval [a, b]. Let 0 = to < tj < ... < t4 = 27r be a partition of [0, 27r], and Ikx = [x + tkl-, X+tk]. If ,n Vm(f) =sSUP{j E If(Ikx) I dx} < oo, Tk=1 where the supremum is taken over all partitions, then f is said to be of bounded variation in the mean (or of bounded variation in the L1 norm). We denote the class of all functions which are of bounded variation in the mean by BVM. This concept was introduced by Moricz and Siddiqi [MS], who investigated the convergence in the mean of the partial sums of S[f], the Fourier series of f. If f is of bounded variation (f E BV) with variation V(f, T), then

Conquer the World with Markov Chains

Pamela Pierce and Robert Wooster W e all get a little bit power hungry now and then, and for board game enthusiasts, the game Risk provides a nice setting in which to enact these fantasies of world domination. To help inform good strategy in Risk, we bring into play a mathematical tool called a Markov chain. The theory of Markov chains is both powerful and rich with applications. Some of these applications include modeling stock prices, genetic mutation in a population, and strategies in a baseball game; it could even include investigating the mysterious behavior of black holes. Lighter, but equally interesting, applications of Markov chains involve studying games. For instance, Markov chains have been used to determine the most valuable properties in Monopoly (see [1] and [3]), to find the average length of a game of Chutes and Ladders ([2]), and to prove that the card game War will end in finite time ([5]). An article by Baris Tan first looked at possible battle outcomes in Risk through the lens of Markov chains ([7]). Jason Osborne revised some of Tan’s probabilities ([6]). Here we use Markov chains to provide another perspective on probable outcomes of a battle, so that the readers of Math Horizons will be fully armed as they set out to conquer the world.

The Circle-Squaring Problem Decomposed

Mikl6s Laczkovich, however, shocked mathematicians around the world with an affirmative response to Tarskis question. In 1990, Laczkovich proved that any circle in the plane is equidecomposable with a square of equal area. He succeeded because he allowed pieces that are difficult to imagine-dustings of points that are selected using the controversial Axiom of Choice. Laczkovichs proof shows that such a decomposition is theoretically possible, but there is no picture to help us understand how this is accomplished. He gives an upper bound of 1 050 for the number of pieces that are required in this decomposition, and he shows that the rearrangement of the pieces can be accomplished using translations alone. None of the pieces require a rotation or a reflection.

On some high indices theorems III

We show that if the series, ∑ane−λns, with Hadamard gaps, represents a function of Φ-bounded variation on (0, ∞), then ∑Φ(|an|) converges.